User:Wvbailey/Propositional formula

2545:: The first thing one needs is values i.e. constants for the variables. If they are known, these constants (e.g. T or F, or 1 or 0) are substituted for their variables one after another until all variables are accounted for. If a variable has no value, the formula is indeterminate. But the formula can be repeated and T plugged into the first formula and F plugged into the second formula and thereby produce both evaluations. In the extreme, where all 2 combinations of n variables are presented in one table, this is the method of 7579: 7762: 1329:. The recommended procedure is to start at the top or right with a labelling of the input-variables. Assign variable names to each operator's output (e.g. u, v, w in the drawing). Start at the top and connect the inputs to their operators. Connect the outputs of the operators to operators further down the line. Make appropriate substitutions starting at the top to derive the final formula(s) at the bottom (e.g. q in the drawing). 6574:. For example, squares #3 and #7 abut. These two abutting squares can loose one literal (e.g. "p" from squares #3 and #7), four squares in a rectangle or square loose two literals, eight squares in a rectangle loose 3 literals, etc. (One seeks out the largest square or rectangles.) This process continues until all abutting squares are accounted for, at which point the propositional formula is said to be minimized. 5261: 1813:

leaves point upward to the first connectives, etc. (Enderton 2002:17 and 29ff).</ref> (i) The left end is enclosed in a (, the right end by a ) followed by =, (ii) The parentheses come in pairs, and (iii) there is an equal number of left ( and right ) parentheses. With respect to the other symbols, ~s, ~&, ~V, &&, &V, VV, V&, &s, Vs are not allowed ("s" is any variable).

7783: 943: 9320:

one after another for the presence or absence of the assertion -- then the law is considered intuitionistically appropriate. Thus an assertion such as: "This object must either BE or NOT BE (in the collection)", or "This object must either have this QUALITY or NOT have this QUALITY (relative to the objects in the collection)" is acceptable. See more at

9398:. It is here that, what we consider "modern" propositional logic, first appeared. In particular, PM introduces the notions of NOT and OR and the assertion symbol ⊦ as primitives. In terms of these notions they defined IMPLICATION → ( def. *1.01: ~p V q ), then AND (def. *3.01: ~(~p V ~q) ), then EQUIVALENCE p ←→ q (*4.01: (p → q) & ( q → p ) ). 9375:(i.e. x = x) for logic: it stated that the mental acts of choosing the property x and choosing x again and again is the same as choosing x once... As consequence of it he formed the equations x•(1-x)=0 and x+(1-x)=1 which for him expressed respectively the law of contradiction and the law of excluded middle." For Boole "1" was the 831:: At least one ternary relation ((a, b), c) is required if we are to combine two propositions, each with their own truth or falsity, into the single assertion with a single truth or falsity. As noted above, as algebraic operators these both behave according to the commutative law, i.e. (&(f, p), e) = (&(p, f), e). 693:) is unwilling to accept simultaneous assertions of BEING (Cow!) and NOT-BEING (Not-cow!), or assertions of QUALITY (Blue cow!) and NOT-QUALITY (Not-blue cow!). When applied to the same object simultaneously in place and time we say these paired assertions { BEING, NOT-BEING } and { QUALITY, NOT-QUALITY } are 788:(also called logical connectives, operators, logic gates). The connectives derive their intuitive meanings from their use in common language. The most common ones are NOT (IT'S NOT THE CASE THAT ...), OR (inclusive-or: "a" OR "b" or both simultaneously), AND (also "but"), and IF ... THEN ... ("a IMPLIES b"). 9319:

The use of the word "everything" in the law of excluded middle renders Russell's expression of this law open to debate. If restricted to an expression about BEING or QUALITY with reference to a finite collection of objects (a finite "universe of discourse") -- the members of which can be investigated

8473:

The formula known as "clocked flip-flop" memory ("c" is the "clock" and "d" is the "data") is given below. It works as follows: When c = 0 the data d (either 0 or 1) cannot "get through" to affect output q. When c = 1 the data d "gets through" and output q "follows" d's value. When c goes from 1 to 0

824:

conventional arithmetic, only one of two values are assigned to any variable and to the outcome of an operation[ref> Engineering sometimes uses the + and ∙ when no confusion can result.</ref>. Because only 0 or 1 can result from an operation, there is no "carry" for addition. So a choice has

674:

Example: "This sentence is either compound or simple, but indeed it is a run-on sentence." Rephrased before breaking into simple sentences this could be written: "This sentence is compound OR this sentence is simple AND this sentence is a run-on sentence." Broken into its simple sentences it is "This

332:

Example: Start with proposition "a". Equate (a V 0) = a. Substitute (b & ~(b))=0 for 0: (a V (b & ~(b)))=a. Distribute "a V" across (b & ~(b)): (a V b) & (a V ~(b) ) = a. Observe that (a V ~(b)) = (~(b) V a) ≡ (b → a). Likewise observe that (a V b) = (b V a) and b = ~(~(b)) so (b V a)

7481:

In the abstract (ideal) case the simplest oscillating formula is a NOT fed back to itself: ~(~(p=q)) = q. Analysis of an abstract (ideal) propositional formula in a truth-table reveals an inconsistency for both p=1 and p=0 cases: When p=1, q=0, this cannot be because p=q; ditto for when p=0 and q=1.

946:

The input variables p, c, d can represent any three propositions. The other, intermediate variables u, v, and w are for purposes of substitution. The symbols shown are typical to much engineering documentation, but other symbols are used as well. Usually a little circle represents "inversion" (NOT).

9567:

McCluskey comments that "it could be argued that the analysis is still incomplete because the word statement "The outputs are equal to the previous values of the inputs" has not been obtained"; he goes on to dismiss such worries because "English is not a formal language in a mathematical sense, it

7998:

the output q=1 so when and if (s=0 & r=1) the flip-flop will be reset. Or, if (s=1 & r=0) the flip-flop will be set. In the abstract (ideal) instance in which s=1 => s=0 & r=1 => r=0 simultaneously, the formula q will be indeterminate (undecidable). Due to delays in "real" OR, AND

6585:

Observe that by the Idempotency law (A V A) = A, we can create more terms. Then by association and distributive laws the variables to disappear can be paired, and then "disappeared" with the Law of contradiction (x & ~x)=0. The following uses brackets only to keep track of the terms; they have

2070:

In a binary (i.e. two-symbol) evaluation, any two unique symbols can work. In general, v variables create tables with 2 rows. If a value is not known in a binary evaluation, it may be written as "u" or if not important, as "d" (don't care); but a truth table is still required with a full assignment

7816:

About the simplest memory results when the output of an OR feeds back to one of its inputs, in this case output "q" feeds back into "p". Given that the formula is first evaluated (initialized) with p=0 & q=0, it will "flip" once when "set" by s=1. Thereafter, output "q" will sustain "q" in the

7803:

at the input and q at the output. After "breaking" the feed-back, the truth table construction proceeds in the conventional manner. But afterwards, in every row the output q is compared to the now-independent input p and any inconsistencies between p and q are noted (i.e. p=0 together with q=1, or

1436:

are, alternately called "the grammar", or "the syntax". The rules must be absolutely mechanical and must leave no chance for ambiguity. They proceed by an induction process that begins with a basis (i) and is followed by formation rule (ii). A third rule -- substitution -- is also required in this

1353:

the evaluation of one of the switches based upon behavior in the circuit: { { switch_#1_active = +5 volts at node B17, switch_#1_inactive = 0 volts node B17, }, { switch_#2_active = 0 volts at node B18, switch_#2_inactive = +5 volts node B18 } }. Also, { T, F } should be reserved for abbreviations

7786:

A "clocked flip-flop" memory ("c" is the "clock" and "d" is the "data"). The data can change at any time when clock c=0; when clock c=1 the output q "tracks" the value of data d. When c goes from 1 to 0 it "traps" d = q's value and this continues to appear at q no matter what d does (as long as c

7595:

Analysis requires a delay to be inserted and then the loop cut between the delay and the input "p". The delay must be viewed as a kind of proposition that has "qd" (q-delayed) as output for "q" as input. This new proposition adds another column to the truth table. The inconsistency is now between

5334:

In the same way that a 2-row truth table displays the valuation of a propositiional formula for all 2, n variables produces a 2-square Karnaugh map (even though we cannot draw it). For example, 3 variables produces 2 = 8 squares ; each of corresponding truth-table row and Karnaugh-map square

894:

formula ( p → f ). In the following the first formula (f → p) is called the NECESSARY condition, and the converse ( p → f ) is called the SUFFICIENT condition. When conjoined by AND they are said to be NECESSARY and SUFFICIENT and thus EQUIVALENT. If one jokes that "Having a basket full of bread,

649:

must be about specific objects or specific states of mind. Each must have at least a noun (an immediate object of thought or observation), a verb (in the active voice, present tense preferred), and perhaps an adjective or adverb. "Dog!" probably implies "I see a dog" but should be rejected as too

7430:

The notion of a propositional formula appearing as one of its own variables requires a formation rule that allows the assignment of the formula to a variable. In general there is no stipulation (either axiomatic or truth-table systems of objects and relations) that forbids this from happening.

2552:

Evaluation procedes by finding the "inner-most" and evaluating it (or them) from e.g. left to right if there are more than one. Given the evaluation, the new inner-most (s) must be again found by a process similar to the parenthesis checker and evaluated. This process is repeated until the last

1812:

Some observations are possible about a well-formed formula assembled per a grammar such as one described above.[ref> Enderton sketches an algorithm that constructs an upside-down tree with the formula at the root; its leaves will be the variables in the formula, e.g. (c, d, q, c, d) and these

682:: In all cases a propositional system must, either directly or indirectly, include a means to express NEGATION. As a connective NEGATION occurs as NOR, i.e. in the phrase "NEITHER cold NOR heat stops me." The NOT-AND or NAND has no usage in English but appears indirectly as NEITHER... NOR.... . 1701:

The following example works for either system. Given the collection of formulas derived from either speech (example: c = "It's noon", d = "It's a nice day", p = "I have food", q = "We're going picnicking" ) or from engineering (example: c = "Clock-signal is active", d = "Motor-start switch is

7765:

About the simplest memory results when the output of an OR feeds back to one of its inputs, in this case output "q" feeding back into "p". The next simplest is the "flip-flop" shown below the once-flip. Analysis of these sorts of formulas can be done by either cutting the feedback path(s) or

1527:: Substitution is the method by which one plugs a constant or a formula inside a formula in place of a variable. The rule is: Throughout the formula, wherever the variable-to-be-substituted is found, the substitution of the formula or constant for the variable must take place. 7993:

shown below the once-flip. Given that r=0 & s=0 and q=0 at the outset, it is "set" (s=1) in a manner similar to the once-flip. It however has a provision to "reset" q=0 when "r"=1. And additional complication occurs if both set=1 and reset=1. In this formula, the set=1

794:

Electrical engineering creates specific operator-symbols for NOT, AND, and OR, NOR (OR followed by NOT), NAND (AND followed by NOT; also called "the stroke" | ), XOR (eXclusive-OR: a or b but not both, "not-equivalent", "addition without carry"), and XNOR (EQUIVALENCE).

670:, the common ones being OR, AND, and IF ..., THEN. BUT is treated the same as AND, and OR and AND sometimes are confused so care is required. A compound sentence can usually be reworded into a series of simple sentences, although the result will probably sound stilted. 7457:. Propositional formulas with feedback lead, in their simplest form, to state machines; they also lead to memories in the form of Turing tapes and counter-machine counters. From combinations of these elements one can build any sort of bounded computational model (e.g. 933:

There are at least two possible approaches to the development of a propositional calculus: (1) The notion of substitution together with the definition of the connectives (operators) by use of truth tables, or (ii) A formal axiomatic system (cf Tarski 1941:146-147).

840:

Example: "I have food" OR "I am picnicking." This is ambiguous as to exactly what is happening. I may be walking down the street with a armful of groceries, OR I may be picnicking with a friend who's brought the food, OR I have my food spread out on my picnic

5335:

represents one minterm. Thus 4 variables produces 16 rows and 16 squares, and thus 16 minterms. Any propositional formula can be reduced to a "logical sum" (OR) of the active (i.e. "1"- or "T"-valued) minterms. When in this form the formula is said to be in

6393:

Use the values of the formula (e.g. "p") found by the truth-table method and place them in their into their respective (associated) Karnaugh squares. If values of "d" for "don't care" appear in the table, this adds flexibility during the reduction phase.

5264:

A truth table will contain 2 rows, where n is the number of variables (e.g. three variables "p", "d", "c" produce 2 rows. Each row represents a minterm. Each minterm can be found on the Hass diagram, on the Veitch diagram, and on the Karnaugh map.

9351:

had critically analyzed the syllogistic logic with a sympathy torward Locke's semiotics. George Bentham's work (1827) resulted in the notion of "quantification of the predicate" (1827) (nowdays symbolized as ∀ ≡ "for all"). A "row" instigated by

7434:

The simplest case occurs when an OR formula becomes one its own inputs e.g. p = q. Begin with (p V s) = q, then let p = q. Observe that q's "definition" depends on itself "q" as well as on "s" and the OR connective; this definition of q is thus

7592:. If either of the delay and NOT are not abstract (i.e. not ideal), the type of analysis to be used will be dependent upon the exact nature of the objects that make up the oscillator; such things fall outside mathematics and into engineering. 889:

identity. In the following example, "picnicking" is not identical to "having food"; "having food" is identical to "having food". But the two propositions are rendered "equivalent" by the conjunction (the AND) of formula ( f → p ) with its

1821:

else add 1 to counter and continue. Proceed to the right, ignoring all symbols but ( and ); add 1 each time a ( occurs, and subtract 1 if ) occurs. If the count goes to 0, check to see that an = sign is the next symbol to the right else

819:

The propositional algebra closely resembles an arithmetic that starts with three rules symbolized by OR, AND, and NEGATE , variables (e.g. a, b, c ... x, y, z,...), two constants 0, 1, the equate-symbol =, and parentheses ( , ) . But

9598:

The notion of delay and the principle of local causation as caused ultimately by the speed of light appears in Robin Gandy (1980), "Church's thesis and Principles for Mechanisms", in J. Barwise, H. J. Keisler and K. Kunen, eds.,

5575:

Technically, the propositional function has been reduced to its (unminimized) conjunctive normal form: each row has its minterm expression and these can be OR'd to produce the formula in its (unminimized) conjunctive normal

9603:, North-Holland Publishing Company (1980) 123-148. Gandy considered this to be the most important of his principles: "Contemporary physics rejects the possibility of instantaneous action at a distance" (p. 135). Gandy was 717:

Example: If this morning I look out my window and observe something in my field and utter: "That cow is blue!", then I am not allowed (within the boundaries of sanity) to go on in the same breath and assert: "That cow is

2074:

The primitive notion of NOT results in the two axioms ~※ ≡ ℥, and ~℥ ≡ ※ that indicate that only two symbols will be allowed in the system. 16 operators (connectives) with 2 symbols as input and one output are possible:

1348:

Caution is advised. It is quite common to reverse the "sense" of the values even in the same system. For example, switches come in two varieties: normally-open and normally-closed. So a system of two switches might

7791:

Without delay inconsistencies must be eliminated from a truth table analysis. With the notion of “delay”, this condition presents itself as a momentary inconsistency between the fed-back output variable q and p =

915:

Example: IF "I have food" THEN "I am going on a picnic" ELSE (otherwise) "I am taking a nap." = ( "I have food" AND "I am going on a picnic" ) OR ( "Its not the case that 'I have food'" AND "I am taking a nap." )

1325:: A typical treatment used by engineers employs diagrams made from one-input NOT (inverter) symbols, and 2 input AND, OR, NOT, NOR, XOR and AND-OR-SELECT operator-symbols. These are linked by lines to indicate 1816:

Example: A simple parenthesis checker (does not test for errors such as " ~V " and is easily fooled) starts at the left end of the symbol string and sets a counter to 0. If the left-most symbol is not ( then

773:

Example: "If the motor's over-temp switch "OTM" is closed and start button "PB1" is pushed, OR if the motor's over-temp switch "OTM" is open and the emergency-start botton "ES" is held in, THEN the motor "M"

1343:, }, { open, closed }, { up, down }, { +5 volts, 0 volts } etc. The usual process is to "map" (put into 1-1 correspondence) definitions such as { door_open = +5 volts, door_closed = 0 volts } to { 1, 0 }. 860:

Common usage expects that, at least intuitively, "my having food" is a causitive factor in my "going-on-a-picnic" behavior. But in the propositional logic this is not the case: "IF "f" THEN "p" " is just

5556:

Complex formulas such as those for binary adders typically begin with a stage that decode the input binary code into its minterms. Any propositional formula can be reduced to its conjuctive normal form.

9616:

McKlusky p. 194-5 discusses "breaking the loop" and inserts "amplifiers" to do this; Wickes (p. 118-121) discuss inserting delays. McCluskey p. 195ff discusses the problem of "races" caused by delays.

1828:

This method locates the "deepest-in" pair(s) of parentheses. It also locates the principal (outermost) connective, V in this example, which has a count of 1 (the same count as the outer parentheses).

9394:

he studied the Frege's work and made a (famous, notorious) ememdation with respect to it (1904). This work led to a collatoration with Whitehead that, in the year 1912, produced the first volume of

654:

Example: "This sentence is simple"; "This squirrel is dead"; "This dog runs", "That cow is blue", "Switch M31 is closed", "This cap is off", "Tomorrow is Friday", "This statement is false", etc.

816:

to represent constants (numbers), (ii) presents rules for creating formulas from formulas, (iii) presents rules for simplifying formulas, (iv) presents rules for the evaluation of a formula.

615:). The synthesis (creation) and analysis of propositional formulas appear in philosophy, mathematics, and electrical engineering. Contributions to the art have come from all three disciplines. 1364:

The following formal axiomatization follows a treatment in Suppes 1957. It provides all the necessary ingredients to generate the important theorems and to evaluate (a V b), (a & b) and

911:: This operator appears in computer science as the simplest CASE operator and in electrical engineering as the AND-OR-SELECT operator. Although it sounds like two implications it is not. 1533:: If "nothing" (emptiness) is symbolized by the sign "0" (the empty set) and "all" (fullness, completeness) is symbolized by the sign "1" then the law of contradiction can be used to 5342:

In the following table, observe the peculiar sequence of rows: (0, 1, 3, 2, 6, 7, 5, 4, 0). the first column is the decimal equivalent of the binary equivalent of the digits cba,

6577:

Example: The map method usually is done by inspection. The following example expands the algebraic method to show the "trick" behind the combining of terms on a Karnaugh map:

730:

If for some reason, such as the consequence of an analysis or investigation, both assertions about an object exist in a place and time, then the observations are said to be

4465:: A truth table is just a generalization of the above example. It creates 2 rows, one for each unique combination of variable-values, where n is the number of variables: 7804:

p=1 and q=0); when the "line" is "remade" both are rendered impossible by the Law of contradiction ~(p & ~p)). Rows revealing inconsistences are either considered

1998:

in a deductive system assigns "values" to the variables (the propositions). In n-ary (e.g. ternary or 3-symbol) systems v variables will create tables with n rows.

835:

Example: "I have food" AND "I am picnicking." f = "I have food", p = "I am picnicking": (f & p) = e. Given this conjunction of events I am probably eating "e".

9386:'s massive undertaking resulted in a formal calculus of propositions, but his symbolism is so daunting that it had little influence excepting on one person: 7766:

inserting (ideal) delay in the path. A cut path and an assumption that no delay occurs anywhere in the "circuit" results in inconsistencies for some of the

5570:

Produce the formula's truth table. Number its rows using the binary-equivalents of the variables (usually just sequentially 0 through n-1) for n variables.

865:

as NOT-"f" OR "p" (~(f) V p ). As this is not common English speech the use of this operator can be confusing. It is not used in electrical engineering.

9558:

Both NOT and AND can be combined into a single connective such as NAND ( NOT "a AND b" )) but the resulting formulas are unwieldy. Ditto for NOT and OR.

765:

Example: "If I have some food and it's noon and it's not raining, then we are going on a picnic; but if it is raining then we will eat at Fishy's."

559:

Quite often, if a particular place in the table is unimportant the table's designer uses a third symbol "d" ("don't care") for added flexibility.

7450:(condition) of the "hidden" variable p (i.e. the value of q fed back and assigned to p). When this is known the apparent inconsistency goes away. 5320:

Example: a, b, c, d are variables. ((( a & ~(b) ) & ~(c)) & d) is a term. This can be abbreviated as (a & ~b & ~c & d).

9639:

and developed some notable theorems with Quine and on his own. For those interested in the history, the book contains a wealth of references .

7770:(combination of inputs and outputs, e.g. (p=0, s=1, r=1) results in an inconsistency). When delay is present these inconsistencies are merely 7588:: If an delay (ideal or non-ideal) is inserted in the abstract formula between p and q then p will oscillate between 1 and 0: 101010...101... 1702:

closed", p = "Motor-TEMP okay", q = "Power is ON to motor" ) the following set of simple formulas will be used to derive a final formula q:

6581:

Minterms #3 and #7 abut, #7 and #6 abut, and #4 and #6 abut (because the table's edges wrap around). So each of these pairs can be reduced.

2619:

As found by a 5th application of the parenthesis algorithm, there are two innermost formulas. Start with the one on the left: (0 & 1)=0

1674:

Into the above for 0 and 1, substitute (a & ~(a))=0 and (a V (~a))=1 : ((a & ~(a)) & (a V (~a)) = ( 0 & (a V ~(a))) =?

954:: The following are to be regarded as definitions. As only 16 tables (2) are possible, the choice of symbolization is necessarily limited. 947:

For example, and AND gate with a little circle drawn at its output becomes a NAND; an OR with a little circle drawn at its output is a NOR.

852:

behave according to the commutative law. Thus (→(f, p), e) ≠ (→(p, f), e). In spoken language it also contributes a sense of causality.

5323:

Example: p, q, r, s are variables. (((p & ~(q) ) & r) & ~(s) ) is an alterm. This can be abbreviated as (p V ~q V r V ~s).

7442:

Without knowledge of what is going on “inside” the formula, it would appear (from the "outside") as if the the output is no longer a

9331:

applied to propositions had to wait until the early 1800's. In an (adverse) reaction to the 2000 year tradition of Aristotle's

7446:

of the inputs alone. That is, sometimes when looks at q one sees 0 and other times 1. To avoid this problem one has to know the

5597:

However, this formula be reduced both in the number of terms (from 4 to 3) and in the total count of its literals (12 to 6).

2557:

Example: Given that (p, d, c) = (1, 1, 0) and the formula ((c & d) V (p & ~((c & ~(d)))))=q, find the value of q:

2006:

Either a small set of axioms can be used to define the behavior of the symbols, or truth tables for each symbol can be used.

769:"I have some food" = f; "It's noon" = n; "It is raining" = r, "We are going on a picnic" = p, "We will eat at Fishy's." = h. 8474:

the last value of the data remains "trapped" at output "q". As long as c=0, d can change value without causing q to change.

895:

wine and cheese is same as goin' on a picnic" they are asserting the (effective) equivalence of the two simple sentences.

804:: Once a "proposition" (sentence) has been converted into a propositional formula, the formula can be manipulated in an 798:

Computer science adds the CASE operator also used by engineering (IF-THEN-ELSE: ( ("c" AND "a") OR ("NOT c" AND "b") ).

9540:

PM p. 91 eschews "the" because they require a clear-cut "object of sensation"; they stipulate the use of "this" (p. 91)

9687:, Dover Publications, Inc., Mineola, New York. ISBN 0-486-28462-X (pbk.). This book is in print and readily available. 9663:, Dover Publications, Inc., Mineola, New York. ISBN 0-486-40687-3 (pbk.). This book is in print and readily available. 9635:, McGraw-Hill Book Company, NY. No ISBN. Library of Congress Catalog Card Number 65-17394. McCluskey was a student of 9589:

will occur (cf McCluskey p. 191-2). In abstract (idealized) mathematical systems adequate loop gain is not a problem.

293:

Logical implication does not behave according to the commutative or distributive, associative or rules listed above.

791:

Mathematical logic adds EQUIVALENCE ("a" IF AND ONLY IF "b", "a" IFF "b", "a" IS NECESSARY AND SUFFICIENT FOR "b").

685:"NOT" is unlike the other connectives. In a sense it is a degenerative binary operator. It derives from Aristotle's 9364:

to write up his ideas on logic, and to publish them as MAL in 1847" (Grattin-Guinness and Bornet 1997:xxviii).

761:. If a simple sentence appears more than once in a compound sentence it is always assigned the same variable. 5304:

In electrical engineering a variable v or its negation ~(v) is lumped together into a single notion called a

307:

that is laid down according to a specification. Another approach uses following "theorems" and the notion of

8500:

The state diagram is similar in shape to the flip-flop's state diagram, but with different labelling on the

5336: 585: 581: 5370:

is derived from this notion. It can be extended to renderings of three and four-dimensional objects called

9391: 9297: 6570:

Minterms of adjacent (abutting) 1-squares (T-squares) can be reduced with respect to the number of their

2577:

As found by a 2nd application of the parenthesis algorithm, the innermost formula now is r=(0 & 0)= 0

9376: 785: 7761: 7578: 592: 333:= (~(~b)) V a) ≡ (~(b) → a). Put both of these back into the equivalence: (~(b) → a) & (b → a) = a. 9669: 1335:: Any two pairings of symbols is possible, for example: { ℥, ※ } , { T, F }, { 1, 0 ), { ON, OFF }, { 722:

blue!". Similar difficulties arrise if I assert "That cow is blue", but my wife asserts: "That cow is

9293: 1454:-- is an example and by no means the only possible system. Another system can be based upon objects v 686: 899:

Example: (IF "I have food" THEN "I am picnicking") & (IF "I am picnicking" THEN "I have food").

690: 645:

object of sensation e.g. "This cow is blue", "There's a coyote!" ". . Thus the simple "primitive"

611:

The topic presupposes a philosophy of the nature of sensations of objectsand and their being (i.e.

565:

is the opposite process whereby, when given a propositional formula, the truth table and perhaps a

280:: The other connectives are usually defined in terms of the three simple connectives NOT, OR, AND: 7453:

To understand the behavior of formulas with feedback requires the more sophisticated analysis of

6600:

q = ( (~p & d & c ) V (p & d & c) V (p & d & ~c) V (p & ~d & ~c) )

5589:( (~p & d & c ) V (p & d & c) V (p & d & ~c) V (p & ~d & ~c) ) = q 7470: 7454: 9549:

NEITHER... NOR... is a form of DeMorgan's law that asserts (A NAND B): (~A V ~B) = ~(A & B).

9427: 5260: 9521: 2605:

As found by a 4rd application of the parenthesis algorithm, the innermost formula is ~(0) = 1:

2591:

As found by a 3rd application of the parenthesis algorithm, the innermost formula is (0) = 0:

9644: 9387: 9353: 9285: 7466: 2562:

Plug in all the constants for their variables: ((0 & 1) V (1 & ~((0 & ~(1)))))=q

675:

sentence is compound" OR "This sentence is simple" AND "This sentence is a run-on sentence."

641:

are declarative in nature, that is, they make assertions about the condition or nature of a

142: 2647:

6th application of the parenthesis algorithm locates (0 V 1) = 1; this ends the evaluation.

9628: 9517: 9348: 7462: 150: 146: 7782: 942: 9656: 9418: 9408: 7458: 1706:{ ~(d) = u, (c & d) = s, (c & u ) = r, ~(r) = v, (p & v) = q, (s V w) = q } 9680: 9636: 9503: 7818: 7775: 5581:

Example: ((c & d) V (p & ~(c & (~d)))) = q in conjuctive normal form is:

5371: 5293: 5274: 694: 17: 7817:"flipped" condition (state q=1). This behavior, now time-dependent, is shown by the 5366:

This sequence comes about because in each row, only one variable changes at a time.

2567:

As found by the parenthesis algorithm, the innermost formula is u = ~(d) = ~(1) = 0

289:

BICONDITIONAL or LOGICAL EQUIVALENCE: (a ←→ b) ≡ ( (a & b) V ( ~(a) & ~(b) )

9507: 9412: 9361: 9321: 5375: 5289: 5285: 5278: 869:

Example: "It's not the case that 'I have food'" OR "I am picnicking" is considered

731: 698: 566: 9604: 9494: 5270: 4456: 2546: 702: 304: 9674:

transacctions of the American Mathematical Socienty, Vol. 5 91904) pp. 288-309.

2055:

Axiom: ~(~(a) & a) = 1 ; thus ~(~(0) & 0) = 1, ~(~(1) & 1) = 1

738:

Example: My assertion: "That cow is blue" and my wife's assertion "That cow is

9336: 9327:

Although a propositional calculus originated with Aristotle, the notion of an

710: 6382:= (~p&d&c) V (~p&d&c) V (p&d&~c ) V (p&d&c ) 9357: 9344: 9332: 9289: 7999:

and NOT the result will be unknown at the outset but thereafter predicable.

7990: 5367: 1668:

Prove the theorems 9, 10, 11, 12, 18, and 19 and use these in the following:

778:"OTM is closed"; "PB1 is pushed"; "OTM is NOT-closed "; "ES is held in "; 646: 6658:

Distributive law ( x & (y V z) ) = ( (x & y) V (x & z) ) :

1400:: = (equates, "is the same as", used for assignment of a name to a formula) 7774:

and expire when the delay(s) expire. The drawings on the right are called

705:, this knowing about contradiction to be "self-evident" i.e. derived from 689:: Our intuition (that is, our knowing based upon repeated observation and 9511: 7443: 6694:

Law of identity ( x V 0 ) = x leading to the reduced form of the formula:

6676:

Commutative law and law of contradiction (x & ~x) = (~x & x) = 0:

6590:

Put the formula in conjuctive normal form with the formula to be reduced:

919:"I have food" = f, "I am going on a picnic" = p, "I am taking a nap" = n: 813: 784:: The variables that stand in place of the sentences are linked by their 612: 7799:

A truth table reveals the rows where inconsistencies occur between p = q

713:

such as Locke and Hume disagreed saying it is derived from experience.

580:

or simplification of the formula, or its conversion to what is known as

6571: 5328: 5305: 805: 922:("f" AND "p") OR ( NOT-"f" AND "n"): ( (f & p) V (~(f) & a) ) 9304:

Example: Here O is an expression about an objects BEING or QUALITY:

7425: 1537:

the notion of nothing and the law of excluded middle can be used to

825:

to be made: either 1 + 1 = 1 (OR) or 1 + 1 = 0 (exclusive-or, XOR).

588:. A number of tools (methods, algorithms) are available to do this. 156:

with one MAJOR exception: a distributive law exists for + over ∙ (!)

2052:

Axiom: (a & a) = a ; thus (T & T) = T, (F & F) = F

9685:

Introduction to Logic and to the Methodology of Deductive Sciences

9514:(1953) develop map-methods for simplifying propositional functions 9383: 7781: 7760: 7577: 5259: 1578:

5 Distributive law: (a & (b V c) = ((a & b) V (a & c))

941: 9668:

On his page 204 in a footnote he references his set of axioms to

1616:

19 Associative law: ((a & b) & c) = (a & (b & c))

311:

to create as complex a formula as desired (cf McClusky 1965:85):

2027:

Axiom: (~(a) V a) = 1 ; thus (~(0) V 0) = 1, (~(1) V 1) = 1

1421:

Example: ))&~(ab))V) is a string of symbols (This string is

7439:. Either of two conditions can result: oscillation or memory. 1581:

6 Distributive law: (a V (b & c) = ((a V b) & (a V c))

1520:(iii) The only formulas are those given by rules (i) and (ii). 9672:, "Sets of Independent Postulates for the Algebra of Logic," 856:

Example: IF "I have food" THEN "I am picnicking": (f → p) = e

9367:

About his contribution Grattin-Guinness and Bornet comment:

7596:"qd" and "p" as shown in red; two stable states resulting: 9500:

Textbooks about "switching circuits" appear in early 1950's

7808:

or just eliminated as inconsistent and hence "impossible".

1368:

Primitive notions: "object" (existence), "is the same as"

9415:(1919) describe a "trigger relay" made from a vaccuum tube 5374:. Hasse diagrams flattened into two dimensions these are 1677:

By 6 (distributive law): ((0 & a) V (0 & ~(a)) =?

591:

Besides the theorems above, The single most important of

6243: 6164: 6086: 6007: 5928: 5850: 5772: 5694: 9292:: (1) The law of identity: "Whatever is, is.", (2) The 9288:(1912:74) lists three laws of thought that derive from 6399: 5288:: for small number of variables (6 or less) the use of 2024:

Axiom: (a V a) = a ; thus (0 V 0) = 0, (1 V 1) = 1

1911: 1559:(Huntington 1908, Suppes 1957:204, McCluskey 1965:85): 1547:(2) Incompatibility (contradiction): (O & ~(O)) ≡ 0 1474:

is a binary relation e.g. the ordered-pair (a,c), and R

753:

simple sentence (proposition, assertion) is assigned a

8509: 1575:

4 Commutative law for &: (a & b) = (b & a)

9274:

state 1 with ( d=1 & c=1 ): q=1 is following d=1

8814:

state 0 with ( d=0 & c=1 ): q=0 is following d=0

7426:

Director's cut: Propositional formula with "feedback"

5561:

Reduction by use of the Map method (Veitch, Karnaugh)

5292:. A Karnaugh map is a flattened (i.e. 2-dimensional) 4455:

The above example is evaluated in the 6th row of the

1680:

By 4 (commutation): ((a & 0) V (~(a) & 0)) =?

1640:

By 5 (distributive law): ((a V a) & (a V ~a)) = a

602:(a V b V ... V z) = ~(~a & ~b & ... & ~z) 599:(a & b & ... & z) = ~(~a V ~b V ... V ~z) 595:

is the following (cf McClusky p. 87, PM p. 119-120):

569:

are created from the table. The above theorems also

9649:

A Mathematical Introduction to Logic: Second Edition

5312:. A string of literals connected by OR is called an 5308:. A string of literals connected by AND is called a 627:(sentences, assertions) are considered to be either 9371:"Boole's principal single innovation was law (6.3) 9296:: "Nothing cannot both be and not be", and (3) The 8506: 8001: 7823: 7598: 7484: 6717: 6396: 5599: 5380: 4467: 2659: 2077: 2071:of all combinations of variable-value assignments. 1833: 1442:

The following system -- based on the objects ~(a)=v

1425:

well-formed per the process to be described below.)

956: 336: 161: 106:(0 ⋀ 0) = 0, (0 ⋀ 1) = 0, (1 ⋀ 0) = 1, (1 ⋀ 1) = 1 80:(0 V 0) = 0, (0 V 1) = 1, (1 V 0) = 1, (1 V 1) = 1 30: 9572:procedure for obtaining word statements" (p. 185). 2633:Evaluate the formula on the right: (1 & 1) = 1 1497:is a variable and s and t are formulas then ~(s)=v 576:: Both synthesis and analysis quite often lead to 9581:More precisely, given enough "loop gain", either 6704:q = ( (d & c) V (p & d) V (p & ~c) ) 1613:18 Associative law: ((a V b) V c) = (a V (b V c)) 662:sentence is a series of simple sentences joined ( 9633:Introduction to the Theory of Switching Circuits 5316:. Typically the literal ~(v) is abbreviated ~v. 1591:Major theorems derived from this set of axioms: 701:. Russell (1912) believed, in the manner of the 623:For the purposes of the propositional calculus, 6629:Associative law (x V (y V z)) = ( (x V y) V z ) 1796:((c & d) V (p & ~((c & ~(d))))) = q 1697:Synthesis of a formula by substitution: Example 1683:By substitution: ((a & 0) V (u & 0)) =? 1490:(i) If s is a variable then (s)=s is a formula; 238:((a & b) & c) = (a & (b & c) ) 9044:state 1 with (d =0 & c=0 ), 1 is trapped 1643:By 7 (LoEM: (a V ~a)=1): ((a V a) & 1) = a 1622:24 DeMorgan's law: ~(a & b) = ~(a) V ~(b)) 1432:: The rules that dictate what is and is not a 873:to "IF 'I have food' THEN 'I am picnicking.'" 9197:state 1 with (d =1 & c=0 ), 1 is trapped 8891:state 0 with ( d=1 & r=0 ), 0 is trapped 8737:state 0 with ( s=0 & r=0 ), 0 is trapped 802:Algebra of propositions and their connectives 8: 9497:builds a multiplier using relays (1937-1938) 9421:(1937) invents the binary adder using relays 7370: 6452: 6449: 6436: 6433: 6324: 6313: 6311: 6309: 6307: 6305: 6302: 6300: 6297: 6294: 6291: 6289: 6287: 6284: 6281: 6278: 6276: 6273: 6271: 6268: 6265: 6262: 6260: 6258: 6255: 6252: 6249: 6246: 6234: 6232: 6230: 6228: 6226: 6223: 6221: 6218: 6215: 6212: 6210: 6208: 6205: 6202: 6199: 6197: 6194: 6192: 6189: 6186: 6183: 6181: 6179: 6176: 6173: 6170: 6167: 6156: 6154: 6152: 6150: 6148: 6145: 6143: 6140: 6137: 6134: 6132: 6130: 6127: 6124: 6121: 6119: 6116: 6114: 6111: 6108: 6105: 6103: 6101: 6098: 6095: 6092: 6089: 6077: 6075: 6073: 6071: 6069: 6066: 6064: 6061: 6058: 6055: 6053: 6051: 6048: 6045: 6042: 6040: 6037: 6035: 6032: 6029: 6026: 6024: 6022: 6019: 6016: 6013: 6010: 5998: 5996: 5994: 5992: 5990: 5987: 5985: 5982: 5979: 5976: 5974: 5972: 5969: 5966: 5963: 5961: 5958: 5956: 5953: 5950: 5947: 5945: 5943: 5940: 5937: 5934: 5931: 5920: 5918: 5916: 5914: 5912: 5909: 5907: 5904: 5901: 5898: 5896: 5894: 5891: 5888: 5885: 5883: 5880: 5878: 5875: 5872: 5869: 5867: 5865: 5862: 5859: 5856: 5853: 5842: 5840: 5838: 5836: 5834: 5831: 5829: 5826: 5823: 5820: 5818: 5816: 5813: 5810: 5807: 5805: 5802: 5800: 5797: 5794: 5791: 5789: 5787: 5784: 5781: 5778: 5775: 5764: 5762: 5760: 5758: 5756: 5753: 5751: 5748: 5745: 5742: 5740: 5738: 5735: 5732: 5729: 5727: 5724: 5722: 5719: 5716: 5713: 5711: 5709: 5706: 5703: 5700: 5697: 5535: 5518: 5501: 5484: 5467: 5450: 5433: 5416: 5399: 5382: 5185: 5118: 5051: 4984: 4917: 4850: 4783: 4716: 4639: 4581: 4469: 4387: 4318: 4249: 4186: 4113: 4039: 3976: 3899: 3821: 3758: 3678: 3597: 3534: 3454: 3373: 3310: 3226: 3141: 3078: 2991: 2903: 2816: 2729: 2661: 2499: 2443: 2387: 2331: 2275: 2224: 1983: 1980: 1977: 1974: 1971: 1968: 1965: 1962: 1959: 1956: 1953: 1950: 1947: 1944: 1941: 1938: 1935: 1932: 1929: 1926: 1923: 1920: 1917: 1914: 1835: 1258: 1198: 1138: 1078: 1018: 958: 929:Formal development of propositional formulas 747:Assignment of variables for simple sentences 517: 479: 441: 403: 365: 338: 269:a & (b V c) = (a & b) V (a & c) 258: 243: 227: 211: 195: 179: 163: 109: 83: 58: 40:Propositional (sentential-logic) connective 27:Informal development: propositional formulas 9341:Essay concerning human understanding (1690) 6497: 6463: 6415: 6413: 6322: 2996:Find deepest formula e.g. u = ~(d) = ~(1): 1637:Substitute 7 into 1: (a V (a & ~a)) = a 1598:10 Absorption, idempotency: (a & a) = a 103:(0∙0) = 0, (0∙1) = 0, (1∙0) = 0, (1∙1) = 1 77:(0+0) = 0, (0+1) = 1, (1+0) = 0, (1+1) = 1 8569: 7989:The next simplest case is the "set-reset" 4470: 2570:((0 & 1) V (1 & ~((0 & 0))))=q 2058:Definition: ( a V b ) ≡ ~(~(a) & ~(b)) 2030:Definition: ( a & b ) ≡ ~(~(a) V ~(b)) 1686:By 12 (null element): ((a & 0) V 0) =? 1619:23 DeMorgan's law: ~(a V b) = ~(a) V ~(b)) 1572:3 Commutative law for V: (a V b) = (b V a) 1550:(3) Law of excluded middle: (O V ~(O)) ≡ 1 1354:having to do with "Truth" and "Falsehood". 252:a V (b & c) = (a V b) & (a V c) 9524:develop a method for simplifying circuits 9310:(2) Law of contradiction: ~(O & ~(O)) 8567: 8565: 8563: 8561: 8559: 8557: 8555: 8552: 8549: 8547: 8545: 8543: 8540: 8537: 8535: 8533: 8530: 8528: 8526: 8523: 8521: 8519: 8517: 8515: 8513: 8511: 7368: 6531: 6431: 1646:Substitute u for (a V a): (u & 1) = a 1587:8 law of contradiction:(a & ~(a)) = 0 902:( ( f → p ) & ( p → f ) ) ≡ ( f = p ) 327:~(~(x)) = x (Involution, double negation) 9347:(theory of the use of symbols). By 1826 8462:state 1 with s & r simultaneously 1 3083:Evaluate "deepest formula" e.g. u = 0: 2080: 1774:): (s V (p & ~((c & ~(d))))) = q 1584:7 Law of excluded middle: (a V ~(a)) = 1 9533: 7487: 6714:(4) Verify reduction with a truth table 6426: 6423: 6420: 6417: 6410: 6407: 6404: 6401: 2662: 2066:Axioms based on truth table definitions 1762:Substitute (p & ~((c & ~(d))))= 1478:is a ternary relation, i.e. ((a,b), d). 339: 303:creates a propositional formula from a 164: 33: 9313:(3) Law of excluded middle: (O V ~(O)) 8004: 7826: 7601: 6720: 6611:Idempotency (absorption) [ A V A) = A: 5602: 1836: 1595:9 Absorption, idempotency: (a V a) = a 1569:2 Right-hand identity: (a & 1) = a 808:not too different from arithmetic. An 782:Sentential (propositional) connectives 324:(x V ~(x)) = 1 (Complements, (x → x) ) 6389:(2) Create the formula's Karnaugh map 5566:(1) Produce the formula's truth table 1375:: ≡ (is identical to by definition). 709:(intrinsic, built-in) awareness; the 7: 5383: 1793:V (p & ~((c & ~(d))))) = q: 1649:By 2 (identity): (u & 1) = u = a 959: 139:The "laws" of propositional calculus 5690:Formula in conjunctive normal form 1757:): (p & ~((c & ~(d)))) = w. 1417:of symbols into a "symbol string". 812:(i) generalizes formulas by use of 742:blue!" are inconsistent assertions. 55:Example: Propositional evaluations 9300:: "Everything must be or not be." 8493:& ( ~( c & ~( d ) ) ) ) = 8481:& ( ~( c & ~( d ) ) ) ) = 2584:((0 & 1) V (1 & ~((0))))=q 2061:Definition: ( a → b ) ≡ (~(a) V b) 2033:Definition: ( a → b ) ≡ (~(a) V b) 1566:1 Right-hand identity: (a V 0) = a 1323:Engineering-symbol formation rules 284:IMPLY: ((~(a) V (a)) ≡ ((a) → (b)) 24: 9568:is not really possible to have a 9379:and "0" was nothing. (p. xxviiff) 1484:Definition of well-formed formula 5384:decimal equivalent of (c, b, a) 4392:Evaluate new "deepest formula": 4191:Evaluate new "deepest formula": 3981:Evaluate new "deepest formula": 3763:Evaluate new "deepest formula": 3539:Evaluate new "deepest formula": 3315:Evaluate new "deepest formula": 2598:((0 & 1) V (1 & ~(0)))=q 1689:By substitution: ( w & 0) =? 1604:12 Null element: (a & 0) = 0 1394:: V (OR), & (AND), ~ (NOT). 321:(x & ~(x)) = 0 (Complements) 9402:Computation and switching logic 8415:state 1 with ( s=1 & r=0 ) 8322:state 1 with ( s=0 & r=0 ) 8183:state 0 with ( s=0 & r=1 ) 8136:state 0 with ( s=0 & r=0 ) 7821:to the right of the once-flip. 2002:Binary (two-symbol) evaluations 1610:14 Double negation: ~(~(a)) = a 222:((a V b) V c) = (a V (b V c) ) 1664:(a & b) for a = 0, b = 1: 1652:By substitution of (a V a)=u: 206:( a & b ) = ( b & a ) 1: 9356:over a priority dispute with 8489:Example: ( ( c & d ) V ( 8477:Example: ( ( c & d ) V ( 7419: 7417: 7415: 7413: 7411: 7409: 7407: 7405: 7400: 7398: 7396: 7394: 7392: 7390: 7388: 7386: 7384: 7382: 7380: 7378: 7376: 7374: 7372: 6547: 6545: 6512: 6478: 6447: 6445: 6379: 6377: 6375: 6373: 6371: 6369: 6367: 6365: 6363: 6361: 6359: 6357: 6355: 6353: 6351: 6349: 6347: 6342: 6340: 6338: 6336: 6334: 6332: 6330: 6328: 6326: 6318: 6239: 6160: 6082: 6003: 5924: 5922: 5846: 5844: 5768: 5766: 2821:Plug in values for variables 2612:((0 & 1) V (1 & 1))=q 1411:Concatenation-formation rules 1388:: letters {a, b, ... z, ...}, 255:Distribution of OR over AND 9607:'s student and close friend. 9200: 9123: 9047: 8970: 8894: 8817: 8740: 8663: 8575: 8574: 8418: 8371: 8325: 8278: 8232: 8186: 8139: 8092: 8040: 8003: 7954: 7927: 7901: 7874: 7846: 7825: 7727: 7701: 7675: 7649: 7621: 7600: 7552: 7529: 7505: 7486: 7402: 7296: 7225: 7154: 7083: 7012: 6941: 6870: 6799: 6719: 6558: 6555: 6552: 6549: 6542: 6539: 6536: 6533: 6526: 6523: 6520: 6517: 6514: 6509: 6506: 6503: 6500: 6492: 6489: 6486: 6483: 6480: 6475: 6472: 6469: 6466: 6458: 6455: 6442: 6439: 6388: 6381: 6344: 6315: 6236: 6158: 6079: 6000: 5601: 4526: 4525: 4323:Find new "deepest formula": 4118:Find new "deepest formula": 3904:Find new "deepest formula": 3683:Find new "deepest formula": 3459:Find new "deepest formula": 3231:Find new "deepest formula": 2175: 2133: 2079: 1745:Substitute ~((c & ~(d))= 1601:11 Null element: (a V 1) = 1 1501:is a formula, then (s V t)=v 938:System based on truth tables 176: 173: 54: 52:Example: Boolean arithmetic 51: 32: 9683:1941 (1995 Dover edition), 9659:1957 (1999 Dover edition), 8035: 8033: 8031: 8029: 8027: 8025: 8023: 8019: 8017: 8014: 8012: 8010: 7841: 7839: 7837: 7834: 7832: 7830: 7828: 7616: 7614: 7612: 7610: 7607: 7603: 7500: 7498: 7496: 7494: 7491: 6723: 5689: 5686: 5605: 5565: 5549:(~a & ~b & ~c) 5413:(~c & ~b & ~a) 5395: 5392: 5389: 5386: 5353:cba = (c=1, b=0, a=0) = 101 4493: 4491: 4483: 4481: 2726: 2685: 2670: 2665: 2225: 2134: 2123: 2120: 2114: 2111: 2108: 2105: 2099: 2096: 2093: 2087: 2038:Axioms based on NOT and AND 1906: 1563:0 Existence: ∃(a)∃(b) a ≠ b 994: 991: 988: 985: 362: 360: 351: 343: 341: 170: 48: 39: 36: 9702: 9390:. First as the student of 9307:(1) Law of Identity: O = O 8021: 8008: 8006: 7605: 7489: 6795: 6792: 6789: 6786: 6783: 6780: 6777: 6774: 6771: 6768: 6765: 6762: 6759: 6756: 6753: 6750: 6747: 6744: 6741: 6738: 6735: 5683: 5680: 5677: 5674: 5671: 5668: 5665: 5662: 5659: 5656: 5653: 5650: 5647: 5644: 5641: 5638: 5635: 5632: 5629: 5626: 5623: 5620: 5617: 5532:(c & ~b & ~a) 5430:(~c & ~b & a) 5269:> Requires some tools: 4522: 4520: 4518: 4516: 4514: 4512: 4510: 4508: 4506: 4504: 4502: 4500: 4498: 4496: 4489: 4487: 4485: 4478: 2724: 2722: 2720: 2718: 2716: 2714: 2712: 2710: 2707: 2704: 2702: 2700: 2698: 2695: 2692: 2690: 2683: 2681: 2678: 2676: 2674: 2672: 2668: 2090: 2082: 2010:Axioms based on NOT and OR 1903: 1900: 1897: 1894: 1891: 1888: 1885: 1882: 1879: 1876: 1873: 1870: 1867: 1864: 1861: 1858: 1855: 1852: 1849: 1846: 1843: 1840: 1838: 1728:Substitute (c & ~(d))= 1692:By 12 (null element): 0 =? 997: 977: 974: 971: 968: 965: 358: 356: 354: 349: 347: 345: 318:(x & 1) = x (Identity) 9201: 9124: 9048: 8971: 8895: 8818: 8741: 8664: 8571: 8419: 8372: 8326: 8279: 8233: 8187: 8140: 8093: 8041: 8037: 7955: 7928: 7902: 7875: 7847: 7843: 7728: 7702: 7676: 7650: 7622: 7618: 7553: 7530: 7506: 7502: 7297: 7226: 7155: 7084: 7013: 6942: 6871: 6804:( ~p & ~d & ~c ) 6800: 6586:no special significance: 5698:( ~p & ~d & ~c ) 5536: 5519: 5515:(c & ~b & a) 5502: 5485: 5481:(c & b & ~a) 5468: 5464:(~c & b & ~a) 5451: 5447:(~c & b & a) 5434: 5417: 5400: 5186: 5119: 5052: 4985: 4918: 4851: 4784: 4717: 4640: 4582: 4388: 4319: 4250: 4187: 4114: 4040: 3977: 3900: 3822: 3759: 3679: 3598: 3535: 3455: 3374: 3311: 3227: 3142: 3079: 2992: 2904: 2817: 2730: 2500: 2444: 2388: 2332: 2276: 2176: 1808:Well-formed formula (wff) 1634:Start with 1: (a V 0) = a 1531:Development of the axioms 1259: 1199: 1139: 1079: 1019: 885:: Logical equivalence is 518: 480: 442: 404: 366: 259: 244: 228: 212: 196: 180: 167: 110: 84: 59: 8469:Clocked flip-flop memory 7724:qd & p inconsistent 7698:qd & p inconsistent 7088:( p & ~d & ~c ) 6946:( ~p & d & ~c ) 6732: 6729: 6726: 6011:( p & ~d & ~c ) 5854:( ~p & d & ~c ) 5614: 5611: 5608: 5300:Literal, term and alterm 5256:Reduction to normal form 4475: 4472: 2688: 2553:evaluation is complete. 2129: 2126: 2117: 2102: 2084: 1740:)=v: ~((c & ~(d)))=v 1014: 1011: 1008: 1005: 1002: 999: 982: 979: 962: 278:Other common connectives 266:a∙(b+c) = (a∙b) + (a∙c) 263:a∙(b+c) = (a∙b) + (a∙c) 249:a+(b∙c) = (a+b) ∙ (a+c) 235:( (a∙b)∙c) = (a∙(b∙c) ) 232:( (a∙b)∙c) = (a∙(b∙c) ) 129:(1 -(1))=0, (1- (0)) =1 46:Alternate logic symbols 45: 42: 9120:q & p inconsistent 8967:q & p inconsistent 8368:q & p inconsistent 8275:q & p inconsistent 8229:q & p inconsistent 7924:q & p inconsistent 7572:q & p inconsistent 7549:q & p inconsistent 7230:( p & d & ~c ) 7159:( p & ~d & c ) 7017:( ~p & d & c ) 6875:( ~p & ~d & c) 6602:= ( #3 V #7 V #6 V #4 ) 6168:( p & d & ~c ) 6090:( p & ~d & c ) 5932:( ~p & d & c ) 5776:( ~p & ~d & c) 5337:disjunctive normal form 1781:Substitute (c & d)= 1430:Well-formed formula wff 1413:: An object is created 952:Truth table definitions 586:conjunctive normal form 582:disjunctive normal form 219:((a+b)+c) =( (a+(b+c)) 216:((a+b)+c) =( (a+(b+c)) 174:Propositional formulas 9392:Alfred North Whitehead 9298:law of excluded middle 9281:Historical development 7788: 7779: 7586:Oscillation with delay 7582: 7301:( p & d & c ) 6316:( p & d & c ) 6247:( p & d & c ) 5498:(c & b & a) 5266: 1671:((a=0) & (b=1)) =? 1333:Evaluation of formulas 948: 786:sentential connectives 668:sentential connectives 315:(x V 0) = x (Identity) 297:Synthesis and Analysis 203:( a ∙ b ) = ( b ∙ a ) 200:( a ∙ b ) = ( b ∙ a ) 190:( a V b ) = ( b V a ) 187:( a + b ) = ( b + a ) 184:( a + b ) = ( b + a ) 9661:Introduction to Logic 9396:Principia Mathematica 9377:universe of discourse 8485:, but now let p = q: 7785: 7764: 7581: 6237:(p & d & ~c) 6080:(~p & d & c) 6001:(~p & d & c) 5357:= 1*2 + 0*2 + 1*2 = 5 5263: 2734:Start with known wff 945: 848:: This operator does 666:) by what are called 49:Venn diagram concept 9601:The Kleene Symposium 9294:law of contradiction 4463:Truth table-analysis 1723:): (c & ~(d))=r. 1371:Non-axiomatic sign: 909:IF ... THEN ... ELSE 876:(~(f) V p) ≡ (f → p) 871:logically equivalent 687:law of contradiction 584:( or less commonly, 7978:state 1 with s = 1 7951:state 1 with s = 0 7471:Macintosh computers 7455:sequential circuits 6566:(3) Reduce minterms 5350:= c*2 + b*2 + a*2: 4254:Count parentheses: 4044:Count parentheses: 3826:Count parentheses: 3602:Count parentheses: 3378:Count parentheses: 3146:Count parentheses: 2626:(0 V (1 & 1))=q 1541:the notion of all: 1434:well-formed formula 883:Logical EQUIVALENCE 691:inductive reasoning 34:Arithmetic concept 7789: 7780: 7583: 5267: 2908:count parentheses 1392:connective-symbols 949: 607:Formal development 593:DeMorgan's theorem 299:: One process for 37:Arithmetic symbol 9442:their summation Σ 9278: 9277: 8466: 8465: 7982: 7981: 7754: 7753: 7576: 7575: 7467:register machines 7423: 7422: 6620:( #3 V V V #4 ) 6563: 6562: 6386: 6385: 5553: 5552: 5253: 5252: 4453: 4452: 2540: 2539: 1987: 1986: 1525:Substitution rule 1319: 1318: 556: 555: 275: 274: 171:Boolean formulas 168:Algebra formulas 136: 135: 9693: 9670:E. V. Huntington 9645:Herbert Enderton 9617: 9614: 9608: 9596: 9590: 9579: 9573: 9565: 9559: 9556: 9550: 9547: 9541: 9538: 9388:Bertrand Russell 9354:William Hamilton 9286:Bertrand Russell 8507: 8002: 7985:Flip-flop memory 7824: 7812:Once-flip memory 7806:transient states 7599: 7485: 7463:counter machines 6718: 6397: 5687:Active minterms 5600: 5381: 4468: 2660: 2139:incompatibility 2078: 1990:Evaluating a wff 1834: 1711:Substitute ~(d)= 1398:Predicate symbol 1360:Axiomatic system 957: 863:formally defined 757:symbol called a 639:Simple sentences 337: 162: 151:commutative laws 143:distributive law 132:￢(1)=0, ￢(0)=1 31: 9701: 9700: 9696: 9695: 9694: 9692: 9691: 9690: 9629:E. J. McCluskey 9625: 9620: 9615: 9611: 9597: 9593: 9580: 9576: 9566: 9562: 9557: 9553: 9548: 9544: 9539: 9535: 9531: 9518:E. J. McCluskey 9486: 9482: 9478: 9474: 9468: 9464: 9460: 9456: 9449: 9445: 9441: 9437: 9433: 9426:Example: Given 9374: 9349:Richard Whately 9283: 8471: 7987: 7814: 7802: 7795: 7759: 7479: 7459:Turing machines 7428: 6716: 6568: 6391: 5568: 5563: 5555: 5360: 5356: 5349: 5332: 5302: 5273:and indirectly 5258: 2068: 2049:Axiom: ~(F) = T 2046:Axiom: ~(T) = F 2040: 2021:Axiom: ~(0) = 1 2018:Axiom: ~(1) = 0 2012: 2004: 1992: 1810: 1804: 1699: 1516: 1512: 1508: 1505:, (s & t)=v 1504: 1500: 1496: 1477: 1473: 1469: 1465: 1461: 1457: 1453: 1449: 1445: 1367: 1362: 1321: 940: 931: 846:IF ... THEN ... 621: 609: 572: 558: 165:Algebraic laws 147:associative law 29: 22: 21: 20: 12: 11: 5: 9699: 9697: 9689: 9688: 9677: 9676: 9665: 9664: 9657:Patrick Suppes 9653: 9652: 9641: 9640: 9624: 9621: 9619: 9618: 9609: 9591: 9574: 9560: 9551: 9542: 9532: 9530: 9527: 9526: 9525: 9515: 9501: 9498: 9491: 9490: 9489: 9488: 9484: 9480: 9476: 9472: 9469: 9466: 9462: 9461:) XOR carry_in 9458: 9454: 9447: 9443: 9439: 9435: 9431: 9423: 9422: 9419:George Stibitz 9416: 9409:William Eccles 9381: 9380: 9372: 9343:used the word 9317: 9316: 9315: 9314: 9311: 9308: 9282: 9279: 9276: 9275: 9272: 9269: 9267: 9265: 9263: 9261: 9259: 9256: 9254: 9251: 9248: 9245: 9243: 9241: 9238: 9235: 9232: 9230: 9227: 9225: 9222: 9219: 9216: 9214: 9212: 9209: 9206: 9203: 9199: 9198: 9195: 9192: 9190: 9188: 9186: 9184: 9182: 9179: 9177: 9174: 9171: 9168: 9166: 9164: 9161: 9158: 9155: 9153: 9150: 9148: 9145: 9142: 9139: 9137: 9135: 9132: 9129: 9126: 9122: 9121: 9118: 9116: 9114: 9112: 9110: 9108: 9106: 9103: 9101: 9098: 9095: 9092: 9090: 9088: 9085: 9082: 9079: 9077: 9074: 9072: 9069: 9066: 9063: 9061: 9059: 9056: 9053: 9050: 9046: 9045: 9042: 9039: 9037: 9035: 9033: 9031: 9029: 9026: 9024: 9021: 9018: 9015: 9013: 9011: 9008: 9005: 9002: 9000: 8997: 8995: 8992: 8989: 8986: 8984: 8982: 8979: 8976: 8973: 8969: 8968: 8965: 8963: 8961: 8959: 8957: 8955: 8953: 8950: 8948: 8945: 8942: 8939: 8937: 8935: 8932: 8929: 8926: 8924: 8921: 8919: 8916: 8913: 8910: 8908: 8906: 8903: 8900: 8897: 8893: 8892: 8889: 8886: 8884: 8882: 8880: 8878: 8876: 8873: 8871: 8868: 8865: 8862: 8860: 8858: 8855: 8852: 8849: 8847: 8844: 8842: 8839: 8836: 8833: 8831: 8829: 8826: 8823: 8820: 8816: 8815: 8812: 8809: 8807: 8805: 8803: 8801: 8799: 8796: 8794: 8791: 8788: 8785: 8783: 8781: 8778: 8775: 8772: 8770: 8767: 8765: 8762: 8759: 8756: 8754: 8752: 8749: 8746: 8743: 8739: 8738: 8735: 8732: 8730: 8728: 8726: 8724: 8722: 8719: 8717: 8714: 8711: 8708: 8706: 8704: 8701: 8698: 8695: 8693: 8690: 8688: 8685: 8682: 8679: 8677: 8675: 8672: 8669: 8666: 8662: 8661: 8658: 8655: 8652: 8649: 8646: 8643: 8640: 8637: 8634: 8631: 8628: 8625: 8622: 8619: 8616: 8613: 8610: 8607: 8604: 8601: 8598: 8595: 8592: 8589: 8586: 8583: 8580: 8577: 8573: 8572: 8570: 8568: 8566: 8564: 8562: 8560: 8558: 8556: 8554: 8551: 8548: 8546: 8544: 8542: 8539: 8536: 8534: 8532: 8529: 8527: 8525: 8522: 8520: 8518: 8516: 8514: 8512: 8510: 8498: 8497: 8470: 8467: 8464: 8463: 8460: 8457: 8455: 8453: 8451: 8448: 8446: 8443: 8440: 8437: 8435: 8432: 8429: 8427: 8424: 8421: 8417: 8416: 8413: 8410: 8408: 8406: 8404: 8401: 8399: 8396: 8393: 8390: 8388: 8385: 8382: 8380: 8377: 8374: 8370: 8369: 8366: 8364: 8362: 8360: 8358: 8355: 8353: 8350: 8347: 8344: 8342: 8339: 8336: 8334: 8331: 8328: 8324: 8323: 8320: 8317: 8315: 8313: 8311: 8308: 8306: 8303: 8300: 8297: 8295: 8292: 8289: 8287: 8284: 8281: 8277: 8276: 8273: 8271: 8269: 8267: 8265: 8262: 8260: 8257: 8254: 8251: 8249: 8246: 8243: 8241: 8238: 8235: 8231: 8230: 8227: 8225: 8223: 8221: 8219: 8216: 8214: 8211: 8208: 8205: 8203: 8200: 8197: 8195: 8192: 8189: 8185: 8184: 8181: 8178: 8176: 8174: 8172: 8169: 8167: 8164: 8161: 8158: 8156: 8153: 8150: 8148: 8145: 8142: 8138: 8137: 8134: 8131: 8129: 8127: 8125: 8122: 8120: 8117: 8114: 8111: 8109: 8106: 8103: 8101: 8098: 8095: 8091: 8090: 8088: 8085: 8082: 8079: 8076: 8073: 8070: 8067: 8064: 8061: 8058: 8055: 8052: 8049: 8046: 8043: 8039: 8038: 8036: 8034: 8032: 8030: 8028: 8026: 8024: 8022: 8020: 8018: 8016: 8013: 8011: 8009: 8007: 8005: 7986: 7983: 7980: 7979: 7976: 7973: 7971: 7968: 7965: 7962: 7960: 7957: 7953: 7952: 7949: 7946: 7944: 7941: 7938: 7935: 7933: 7930: 7926: 7925: 7922: 7920: 7918: 7915: 7912: 7909: 7907: 7904: 7900: 7899: 7896: 7893: 7891: 7888: 7885: 7882: 7880: 7877: 7873: 7872: 7870: 7867: 7864: 7861: 7858: 7855: 7852: 7849: 7845: 7844: 7842: 7840: 7838: 7836: 7833: 7831: 7829: 7827: 7813: 7810: 7800: 7793: 7776:state diagrams 7758: 7755: 7752: 7751: 7748: 7745: 7743: 7740: 7738: 7735: 7733: 7730: 7726: 7725: 7722: 7719: 7717: 7714: 7712: 7709: 7707: 7704: 7700: 7699: 7696: 7693: 7691: 7688: 7686: 7683: 7681: 7678: 7674: 7673: 7670: 7667: 7665: 7662: 7660: 7657: 7655: 7652: 7648: 7647: 7645: 7642: 7639: 7636: 7633: 7630: 7627: 7624: 7620: 7619: 7617: 7615: 7613: 7611: 7609: 7606: 7604: 7602: 7574: 7573: 7570: 7567: 7565: 7562: 7560: 7557: 7555: 7551: 7550: 7547: 7544: 7542: 7539: 7537: 7534: 7532: 7528: 7527: 7525: 7522: 7519: 7516: 7513: 7510: 7508: 7504: 7503: 7501: 7499: 7497: 7495: 7493: 7490: 7488: 7478: 7475: 7427: 7424: 7421: 7420: 7418: 7416: 7414: 7412: 7410: 7408: 7406: 7404: 7401: 7399: 7397: 7395: 7393: 7391: 7389: 7387: 7385: 7383: 7381: 7379: 7377: 7375: 7373: 7371: 7369: 7366: 7365: 7363: 7361: 7358: 7356: 7353: 7350: 7347: 7345: 7342: 7340: 7337: 7334: 7331: 7329: 7326: 7324: 7321: 7318: 7315: 7313: 7311: 7308: 7305: 7302: 7299: 7295: 7294: 7292: 7290: 7287: 7285: 7282: 7279: 7276: 7274: 7271: 7269: 7266: 7263: 7260: 7258: 7255: 7253: 7250: 7247: 7244: 7242: 7240: 7237: 7234: 7231: 7228: 7224: 7223: 7221: 7219: 7216: 7214: 7211: 7208: 7205: 7203: 7200: 7198: 7195: 7192: 7189: 7187: 7184: 7182: 7179: 7176: 7173: 7171: 7169: 7166: 7163: 7160: 7157: 7153: 7152: 7150: 7148: 7145: 7143: 7140: 7137: 7134: 7132: 7129: 7127: 7124: 7121: 7118: 7116: 7113: 7111: 7108: 7105: 7102: 7100: 7098: 7095: 7092: 7089: 7086: 7082: 7081: 7079: 7077: 7074: 7072: 7069: 7066: 7063: 7061: 7058: 7056: 7053: 7050: 7047: 7045: 7042: 7040: 7037: 7034: 7031: 7029: 7027: 7024: 7021: 7018: 7015: 7011: 7010: 7008: 7006: 7003: 7001: 6998: 6995: 6992: 6990: 6987: 6985: 6982: 6979: 6976: 6974: 6971: 6969: 6966: 6963: 6960: 6958: 6956: 6953: 6950: 6947: 6944: 6940: 6939: 6937: 6935: 6932: 6930: 6927: 6924: 6921: 6919: 6916: 6914: 6911: 6908: 6905: 6903: 6900: 6898: 6895: 6892: 6889: 6887: 6885: 6882: 6879: 6876: 6873: 6869: 6868: 6866: 6864: 6861: 6859: 6856: 6853: 6850: 6848: 6845: 6843: 6840: 6837: 6834: 6832: 6829: 6827: 6824: 6821: 6818: 6816: 6814: 6811: 6808: 6805: 6802: 6798: 6797: 6794: 6791: 6788: 6785: 6782: 6779: 6776: 6773: 6770: 6767: 6764: 6761: 6758: 6755: 6752: 6749: 6746: 6743: 6740: 6737: 6734: 6731: 6728: 6725: 6722: 6715: 6712: 6711: 6710: 6709: 6708: 6707: 6706: 6696: 6695: 6691: 6690: 6689: 6688: 6687: 6686: 6678: 6677: 6673: 6672: 6671: 6670: 6669: 6668: 6660: 6659: 6655: 6654: 6653: 6652: 6651: 6650: 6639: 6631: 6630: 6626: 6625: 6624: 6623: 6622: 6621: 6613: 6612: 6608: 6607: 6606: 6605: 6604: 6603: 6592: 6591: 6583: 6582: 6567: 6564: 6561: 6560: 6557: 6554: 6551: 6548: 6546: 6544: 6541: 6538: 6535: 6532: 6529: 6528: 6525: 6522: 6519: 6516: 6513: 6511: 6508: 6505: 6502: 6499: 6495: 6494: 6491: 6488: 6485: 6482: 6479: 6477: 6474: 6471: 6468: 6465: 6461: 6460: 6457: 6454: 6451: 6448: 6446: 6444: 6441: 6438: 6435: 6432: 6429: 6428: 6425: 6422: 6419: 6418:(~d & ~c) 6416: 6414: 6412: 6409: 6406: 6403: 6402:(~d & ~c) 6400: 6390: 6387: 6384: 6383: 6380: 6378: 6376: 6374: 6372: 6370: 6368: 6366: 6364: 6362: 6360: 6358: 6356: 6354: 6352: 6350: 6348: 6346: 6343: 6341: 6339: 6337: 6335: 6333: 6331: 6329: 6327: 6325: 6323: 6320: 6319: 6317: 6314: 6312: 6310: 6308: 6306: 6304: 6301: 6299: 6296: 6293: 6290: 6288: 6286: 6283: 6280: 6277: 6275: 6272: 6270: 6267: 6264: 6261: 6259: 6257: 6254: 6251: 6248: 6245: 6241: 6240: 6238: 6235: 6233: 6231: 6229: 6227: 6225: 6222: 6220: 6217: 6214: 6211: 6209: 6207: 6204: 6201: 6198: 6196: 6193: 6191: 6188: 6185: 6182: 6180: 6178: 6175: 6172: 6169: 6166: 6162: 6161: 6159: 6157: 6155: 6153: 6151: 6149: 6147: 6144: 6142: 6139: 6136: 6133: 6131: 6129: 6126: 6123: 6120: 6118: 6115: 6113: 6110: 6107: 6104: 6102: 6100: 6097: 6094: 6091: 6088: 6084: 6083: 6081: 6078: 6076: 6074: 6072: 6070: 6068: 6065: 6063: 6060: 6057: 6054: 6052: 6050: 6047: 6044: 6041: 6039: 6036: 6034: 6031: 6028: 6025: 6023: 6021: 6018: 6015: 6012: 6009: 6005: 6004: 6002: 5999: 5997: 5995: 5993: 5991: 5989: 5986: 5984: 5981: 5978: 5975: 5973: 5971: 5968: 5965: 5962: 5960: 5957: 5955: 5952: 5949: 5946: 5944: 5942: 5939: 5936: 5933: 5930: 5926: 5925: 5923: 5921: 5919: 5917: 5915: 5913: 5911: 5908: 5906: 5903: 5900: 5897: 5895: 5893: 5890: 5887: 5884: 5882: 5879: 5877: 5874: 5871: 5868: 5866: 5864: 5861: 5858: 5855: 5852: 5848: 5847: 5845: 5843: 5841: 5839: 5837: 5835: 5833: 5830: 5828: 5825: 5822: 5819: 5817: 5815: 5812: 5809: 5806: 5804: 5801: 5799: 5796: 5793: 5790: 5788: 5786: 5783: 5780: 5777: 5774: 5770: 5769: 5767: 5765: 5763: 5761: 5759: 5757: 5755: 5752: 5750: 5747: 5744: 5741: 5739: 5737: 5734: 5731: 5728: 5726: 5723: 5721: 5718: 5715: 5712: 5710: 5708: 5705: 5702: 5699: 5696: 5692: 5691: 5688: 5685: 5682: 5679: 5676: 5673: 5670: 5667: 5664: 5661: 5658: 5655: 5652: 5649: 5646: 5643: 5640: 5637: 5634: 5631: 5628: 5625: 5622: 5619: 5616: 5613: 5610: 5607: 5604: 5595: 5594: 5593: 5592: 5591: 5590: 5579: 5578: 5567: 5564: 5562: 5559: 5551: 5550: 5547: 5544: 5541: 5538: 5534: 5533: 5530: 5527: 5524: 5521: 5517: 5516: 5513: 5510: 5507: 5504: 5500: 5499: 5496: 5493: 5490: 5487: 5483: 5482: 5479: 5476: 5473: 5470: 5466: 5465: 5462: 5459: 5456: 5453: 5449: 5448: 5445: 5442: 5439: 5436: 5432: 5431: 5428: 5425: 5422: 5419: 5415: 5414: 5411: 5408: 5405: 5402: 5398: 5397: 5394: 5391: 5388: 5385: 5372:Hasse diagrams 5364: 5363: 5362: 5361: 5358: 5354: 5347: 5331: 5326: 5325: 5324: 5321: 5301: 5298: 5257: 5254: 5251: 5250: 5248: 5246: 5244: 5242: 5239: 5237: 5234: 5231: 5228: 5226: 5223: 5220: 5217: 5215: 5212: 5210: 5207: 5204: 5201: 5199: 5197: 5194: 5191: 5188: 5184: 5183: 5181: 5179: 5177: 5175: 5172: 5170: 5167: 5164: 5161: 5159: 5156: 5153: 5150: 5148: 5145: 5143: 5140: 5137: 5134: 5132: 5130: 5127: 5124: 5121: 5117: 5116: 5114: 5112: 5110: 5108: 5105: 5103: 5100: 5097: 5094: 5092: 5089: 5086: 5083: 5081: 5078: 5076: 5073: 5070: 5067: 5065: 5063: 5060: 5057: 5054: 5050: 5049: 5047: 5045: 5043: 5041: 5038: 5036: 5033: 5030: 5027: 5025: 5022: 5019: 5016: 5014: 5011: 5009: 5006: 5003: 5000: 4998: 4996: 4993: 4990: 4987: 4983: 4982: 4980: 4978: 4976: 4974: 4971: 4969: 4966: 4963: 4960: 4958: 4955: 4952: 4949: 4947: 4944: 4942: 4939: 4936: 4933: 4931: 4929: 4926: 4923: 4920: 4916: 4915: 4913: 4911: 4909: 4907: 4904: 4902: 4899: 4896: 4893: 4891: 4888: 4885: 4882: 4880: 4877: 4875: 4872: 4869: 4866: 4864: 4862: 4859: 4856: 4853: 4849: 4848: 4846: 4844: 4842: 4840: 4837: 4835: 4832: 4829: 4826: 4824: 4821: 4818: 4815: 4813: 4810: 4808: 4805: 4802: 4799: 4797: 4795: 4792: 4789: 4786: 4782: 4781: 4779: 4777: 4775: 4773: 4770: 4768: 4765: 4762: 4759: 4757: 4754: 4751: 4748: 4746: 4743: 4741: 4738: 4735: 4732: 4730: 4728: 4725: 4722: 4719: 4715: 4714: 4711: 4708: 4705: 4702: 4699: 4696: 4693: 4690: 4687: 4684: 4681: 4678: 4675: 4672: 4669: 4666: 4663: 4660: 4657: 4654: 4651: 4648: 4645: 4642: 4638: 4637: 4635: 4633: 4631: 4629: 4627: 4625: 4622: 4619: 4617: 4615: 4612: 4609: 4607: 4605: 4602: 4600: 4598: 4595: 4593: 4591: 4589: 4587: 4585: 4583: 4580: 4579: 4577: 4575: 4573: 4571: 4569: 4567: 4565: 4563: 4561: 4559: 4557: 4555: 4553: 4551: 4548: 4546: 4544: 4542: 4540: 4538: 4536: 4533: 4530: 4527: 4524: 4523: 4521: 4519: 4517: 4515: 4513: 4511: 4509: 4507: 4505: 4503: 4501: 4499: 4497: 4495: 4492: 4490: 4488: 4486: 4484: 4482: 4480: 4477: 4476:switch closed 4474: 4471: 4451: 4450: 4447: 4444: 4442: 4440: 4438: 4436: 4434: 4432: 4430: 4428: 4426: 4424: 4422: 4420: 4418: 4416: 4414: 4412: 4409: 4407: 4405: 4403: 4401: 4399: 4397: 4395: 4393: 4390: 4386: 4385: 4382: 4379: 4376: 4374: 4372: 4370: 4368: 4366: 4364: 4362: 4360: 4358: 4356: 4354: 4352: 4349: 4347: 4345: 4342: 4340: 4338: 4335: 4333: 4331: 4328: 4326: 4324: 4321: 4317: 4316: 4314: 4312: 4309: 4307: 4305: 4303: 4301: 4299: 4297: 4295: 4293: 4291: 4289: 4287: 4285: 4282: 4280: 4278: 4275: 4273: 4271: 4268: 4266: 4264: 4261: 4258: 4255: 4252: 4248: 4247: 4245: 4243: 4241: 4239: 4237: 4235: 4233: 4231: 4229: 4227: 4225: 4223: 4221: 4219: 4217: 4214: 4212: 4210: 4208: 4206: 4204: 4202: 4200: 4198: 4196: 4194: 4192: 4189: 4185: 4184: 4181: 4178: 4175: 4172: 4170: 4168: 4166: 4164: 4162: 4160: 4158: 4156: 4154: 4152: 4149: 4146: 4143: 4140: 4137: 4135: 4133: 4130: 4128: 4126: 4123: 4121: 4119: 4116: 4112: 4111: 4109: 4106: 4103: 4100: 4098: 4096: 4094: 4092: 4090: 4088: 4086: 4084: 4082: 4080: 4077: 4074: 4071: 4068: 4065: 4063: 4061: 4058: 4056: 4054: 4051: 4048: 4045: 4042: 4038: 4037: 4035: 4033: 4031: 4029: 4027: 4025: 4023: 4021: 4019: 4017: 4015: 4013: 4011: 4009: 4007: 4005: 4003: 4001: 3999: 3997: 3995: 3992: 3990: 3988: 3986: 3984: 3982: 3979: 3975: 3974: 3971: 3968: 3965: 3962: 3960: 3958: 3956: 3954: 3952: 3950: 3948: 3946: 3944: 3942: 3939: 3936: 3933: 3930: 3927: 3924: 3921: 3918: 3915: 3912: 3909: 3907: 3905: 3902: 3898: 3897: 3895: 3892: 3889: 3886: 3884: 3882: 3880: 3878: 3876: 3874: 3872: 3870: 3868: 3866: 3863: 3860: 3857: 3854: 3851: 3848: 3845: 3842: 3839: 3836: 3833: 3830: 3827: 3824: 3820: 3819: 3817: 3815: 3813: 3811: 3809: 3807: 3805: 3803: 3801: 3799: 3797: 3795: 3793: 3791: 3788: 3786: 3784: 3782: 3780: 3778: 3776: 3774: 3772: 3770: 3768: 3766: 3764: 3761: 3757: 3756: 3753: 3750: 3747: 3744: 3741: 3739: 3737: 3735: 3733: 3731: 3728: 3726: 3724: 3721: 3718: 3715: 3712: 3709: 3706: 3703: 3700: 3697: 3694: 3691: 3688: 3686: 3684: 3681: 3677: 3676: 3674: 3671: 3668: 3665: 3662: 3660: 3658: 3656: 3654: 3652: 3649: 3647: 3645: 3642: 3639: 3636: 3633: 3630: 3627: 3624: 3621: 3618: 3615: 3612: 3609: 3606: 3603: 3600: 3596: 3595: 3593: 3591: 3589: 3587: 3585: 3583: 3581: 3579: 3577: 3575: 3572: 3570: 3568: 3566: 3564: 3562: 3560: 3558: 3556: 3554: 3552: 3550: 3548: 3546: 3544: 3542: 3540: 3537: 3533: 3532: 3529: 3526: 3523: 3520: 3517: 3515: 3513: 3511: 3509: 3507: 3504: 3502: 3500: 3497: 3494: 3491: 3488: 3485: 3482: 3479: 3476: 3473: 3470: 3467: 3464: 3462: 3460: 3457: 3453: 3452: 3450: 3447: 3444: 3441: 3438: 3436: 3434: 3432: 3430: 3428: 3425: 3423: 3421: 3418: 3415: 3412: 3409: 3406: 3403: 3400: 3397: 3394: 3391: 3388: 3385: 3382: 3379: 3376: 3372: 3371: 3369: 3367: 3365: 3363: 3361: 3359: 3357: 3355: 3353: 3351: 3348: 3346: 3344: 3342: 3340: 3338: 3336: 3334: 3332: 3330: 3328: 3326: 3324: 3322: 3320: 3318: 3316: 3313: 3309: 3308: 3305: 3302: 3299: 3296: 3293: 3290: 3288: 3286: 3284: 3281: 3278: 3275: 3272: 3269: 3266: 3263: 3260: 3257: 3254: 3251: 3248: 3245: 3242: 3239: 3236: 3234: 3232: 3229: 3225: 3224: 3222: 3219: 3216: 3213: 3210: 3207: 3205: 3203: 3201: 3198: 3195: 3192: 3189: 3186: 3183: 3180: 3177: 3174: 3171: 3168: 3165: 3162: 3159: 3156: 3153: 3150: 3147: 3144: 3140: 3139: 3137: 3135: 3133: 3131: 3129: 3127: 3125: 3123: 3121: 3118: 3116: 3114: 3112: 3110: 3108: 3106: 3104: 3102: 3100: 3098: 3096: 3094: 3092: 3090: 3088: 3086: 3084: 3081: 3077: 3076: 3073: 3070: 3067: 3064: 3061: 3058: 3055: 3052: 3049: 3046: 3043: 3040: 3037: 3034: 3031: 3028: 3025: 3022: 3019: 3016: 3013: 3010: 3007: 3004: 3001: 2999: 2997: 2994: 2990: 2989: 2987: 2984: 2981: 2978: 2975: 2972: 2969: 2966: 2963: 2960: 2957: 2954: 2951: 2948: 2945: 2942: 2939: 2936: 2933: 2930: 2927: 2924: 2921: 2918: 2915: 2912: 2909: 2906: 2902: 2901: 2898: 2895: 2892: 2889: 2886: 2883: 2880: 2877: 2874: 2871: 2868: 2865: 2862: 2859: 2856: 2853: 2850: 2847: 2844: 2841: 2838: 2835: 2832: 2829: 2826: 2824: 2822: 2819: 2815: 2814: 2811: 2808: 2805: 2802: 2799: 2796: 2793: 2790: 2787: 2784: 2781: 2778: 2775: 2772: 2769: 2766: 2763: 2760: 2757: 2754: 2751: 2748: 2745: 2742: 2739: 2737: 2735: 2732: 2728: 2727: 2725: 2723: 2721: 2719: 2717: 2715: 2713: 2711: 2709: 2706: 2703: 2701: 2699: 2697: 2694: 2691: 2689: 2687: 2684: 2682: 2680: 2677: 2675: 2673: 2671: 2669: 2667: 2664: 2658: 2657: 2656: 2655: 2649: 2648: 2644: 2643: 2642: 2641: 2635: 2634: 2630: 2629: 2628: 2627: 2621: 2620: 2616: 2615: 2614: 2613: 2607: 2606: 2602: 2601: 2600: 2599: 2593: 2592: 2588: 2587: 2586: 2585: 2579: 2578: 2574: 2573: 2572: 2571: 2564: 2563: 2559: 2558: 2538: 2537: 2536:~(~a & a) 2534: 2532: 2530: 2528: 2526: 2524: 2522: 2520: 2518: 2516: 2514: 2512: 2510: 2508: 2506: 2503: 2501: 2498: 2497: 2494: 2491: 2488: 2485: 2482: 2479: 2476: 2473: 2470: 2467: 2464: 2461: 2458: 2455: 2452: 2449: 2446: 2442: 2441: 2438: 2435: 2432: 2429: 2426: 2423: 2420: 2417: 2414: 2411: 2408: 2405: 2402: 2399: 2396: 2393: 2390: 2386: 2385: 2382: 2379: 2376: 2373: 2370: 2367: 2364: 2361: 2358: 2355: 2352: 2349: 2346: 2343: 2340: 2337: 2334: 2330: 2329: 2326: 2323: 2320: 2317: 2314: 2311: 2308: 2305: 2302: 2299: 2296: 2293: 2290: 2287: 2284: 2281: 2278: 2274: 2273: 2271: 2268: 2266: 2263: 2260: 2257: 2254: 2251: 2248: 2245: 2242: 2240: 2237: 2235: 2232: 2230: 2227: 2223: 2222: 2220: 2217: 2216:(b IMPLIES a) 2214: 2211: 2210:(a IMPLIES b) 2208: 2205: 2202: 2199: 2196: 2193: 2190: 2188: 2186: 2184: 2181: 2179: 2177: 2174: 2173: 2170: 2168: 2166: 2164: 2162: 2160: 2157: 2155: 2153: 2150: 2148: 2146: 2144: 2142: 2140: 2137: 2135: 2132: 2131: 2128: 2125: 2122: 2119: 2116: 2113: 2110: 2107: 2104: 2101: 2098: 2095: 2092: 2089: 2086: 2083: 2081: 2067: 2064: 2063: 2062: 2059: 2056: 2053: 2050: 2047: 2044: 2039: 2036: 2035: 2034: 2031: 2028: 2025: 2022: 2019: 2016: 2011: 2008: 2003: 2000: 1991: 1988: 1985: 1984: 1982: 1979: 1976: 1973: 1970: 1967: 1964: 1961: 1958: 1955: 1952: 1949: 1946: 1943: 1940: 1937: 1934: 1931: 1928: 1925: 1922: 1919: 1916: 1913: 1909: 1908: 1905: 1902: 1899: 1896: 1893: 1890: 1887: 1884: 1881: 1878: 1875: 1872: 1869: 1866: 1863: 1860: 1857: 1854: 1851: 1848: 1845: 1842: 1839: 1837: 1809: 1806: 1802: 1801: 1800: 1799: 1798: 1797: 1776: 1775: 1759: 1758: 1742: 1741: 1725: 1724: 1708: 1707: 1698: 1695: 1694: 1693: 1690: 1687: 1684: 1681: 1678: 1675: 1672: 1669: 1658: 1657: 1650: 1647: 1644: 1641: 1638: 1635: 1624: 1623: 1620: 1617: 1614: 1611: 1608: 1605: 1602: 1599: 1596: 1589: 1588: 1585: 1582: 1579: 1576: 1573: 1570: 1567: 1564: 1554: 1553: 1552: 1551: 1548: 1522: 1521: 1518: 1514: 1510: 1506: 1502: 1498: 1494: 1491: 1481: 1480: 1475: 1471: 1470:(a, b) where R 1467: 1463: 1459: 1455: 1451: 1447: 1443: 1427: 1426: 1408: 1407: 1401: 1395: 1389: 1383: 1361: 1358: 1357: 1356: 1317: 1316: 1313: 1310: 1307: 1304: 1301: 1298: 1296: 1293: 1290: 1287: 1284: 1281: 1278: 1276: 1273: 1270: 1267: 1264: 1261: 1257: 1256: 1253: 1250: 1247: 1244: 1241: 1238: 1236: 1233: 1230: 1227: 1224: 1221: 1218: 1216: 1213: 1210: 1207: 1204: 1201: 1197: 1196: 1193: 1190: 1187: 1184: 1181: 1178: 1176: 1173: 1170: 1167: 1164: 1161: 1158: 1156: 1153: 1150: 1147: 1144: 1141: 1137: 1136: 1133: 1130: 1127: 1124: 1121: 1118: 1116: 1113: 1110: 1107: 1104: 1101: 1098: 1096: 1093: 1090: 1087: 1084: 1081: 1077: 1076: 1073: 1070: 1067: 1064: 1061: 1058: 1056: 1053: 1050: 1047: 1044: 1041: 1038: 1036: 1033: 1030: 1027: 1024: 1021: 1017: 1016: 1013: 1010: 1007: 1004: 1001: 998: 996: 993: 990: 987: 984: 981: 978: 976: 973: 970: 967: 964: 961: 939: 936: 930: 927: 926: 925: 924: 923: 920: 906: 905: 904: 903: 880: 879: 878: 877: 858: 857: 843: 842: 837: 836: 776: 775: 767: 766: 744: 743: 728: 727: 677: 676: 656: 655: 620: 617: 608: 605: 604: 603: 600: 554: 553: 550: 547: 544: 541: 538: 535: 532: 529: 526: 523: 520: 516: 515: 512: 509: 506: 503: 500: 497: 494: 491: 488: 485: 482: 478: 477: 474: 471: 468: 465: 462: 459: 456: 453: 450: 447: 444: 440: 439: 436: 433: 430: 427: 424: 421: 418: 415: 412: 409: 406: 402: 401: 398: 395: 392: 389: 386: 383: 380: 377: 374: 371: 368: 364: 363: 361: 359: 357: 355: 353: 350: 348: 346: 344: 342: 340: 335: 334: 329: 328: 325: 322: 319: 316: 291: 290: 286: 285: 273: 272: 270: 267: 264: 261: 257: 256: 253: 250: 247: 245: 242: 241: 239: 236: 233: 230: 226: 225: 223: 220: 217: 214: 210: 209: 207: 204: 201: 198: 194: 193: 191: 188: 185: 182: 178: 177: 175: 172: 169: 166: 134: 133: 130: 127: 124: 121: 118: 115: 112: 108: 107: 104: 101: 98: 95: 92: 89: 86: 82: 81: 78: 75: 72: 70: 67: 64: 61: 57: 56: 53: 50: 47: 44: 43:Logic symbols 41: 38: 35: 28: 25: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 9698: 9686: 9682: 9681:Alfred Tarski 9679: 9678: 9675: 9671: 9667: 9666: 9662: 9658: 9655: 9654: 9650: 9646: 9643: 9642: 9638: 9637:Willard Quine 9634: 9630: 9627: 9626: 9622: 9613: 9610: 9606: 9602: 9595: 9592: 9588: 9584: 9578: 9575: 9571: 9564: 9561: 9555: 9552: 9546: 9543: 9537: 9534: 9528: 9523: 9519: 9516: 9513: 9509: 9505: 9504:Willard Quine 9502: 9499: 9496: 9493: 9492: 9483:) = carry_out 9470: 9452: 9451: 9446:and carry_out 9429: 9425: 9424: 9420: 9417: 9414: 9410: 9407: 9406: 9405: 9403: 9399: 9397: 9393: 9389: 9385: 9378: 9370: 9369: 9368: 9365: 9363: 9359: 9355: 9350: 9346: 9342: 9338: 9334: 9330: 9325: 9323: 9312: 9309: 9306: 9305: 9303: 9302: 9301: 9299: 9295: 9291: 9287: 9280: 9273: 9270: 9268: 9266: 9264: 9262: 9260: 9257: 9255: 9252: 9249: 9246: 9244: 9242: 9239: 9236: 9233: 9231: 9228: 9226: 9223: 9220: 9217: 9215: 9213: 9210: 9207: 9204: 9196: 9193: 9191: 9189: 9187: 9185: 9183: 9180: 9178: 9175: 9172: 9169: 9167: 9165: 9162: 9159: 9156: 9154: 9151: 9149: 9146: 9143: 9140: 9138: 9136: 9133: 9130: 9127: 9119: 9117: 9115: 9113: 9111: 9109: 9107: 9104: 9102: 9099: 9096: 9093: 9091: 9089: 9086: 9083: 9080: 9078: 9075: 9073: 9070: 9067: 9064: 9062: 9060: 9057: 9054: 9051: 9043: 9040: 9038: 9036: 9034: 9032: 9030: 9027: 9025: 9022: 9019: 9016: 9014: 9012: 9009: 9006: 9003: 9001: 8998: 8996: 8993: 8990: 8987: 8985: 8983: 8980: 8977: 8974: 8966: 8964: 8962: 8960: 8958: 8956: 8954: 8951: 8949: 8946: 8943: 8940: 8938: 8936: 8933: 8930: 8927: 8925: 8922: 8920: 8917: 8914: 8911: 8909: 8907: 8904: 8901: 8898: 8890: 8887: 8885: 8883: 8881: 8879: 8877: 8874: 8872: 8869: 8866: 8863: 8861: 8859: 8856: 8853: 8850: 8848: 8845: 8843: 8840: 8837: 8834: 8832: 8830: 8827: 8824: 8821: 8813: 8810: 8808: 8806: 8804: 8802: 8800: 8797: 8795: 8792: 8789: 8786: 8784: 8782: 8779: 8776: 8773: 8771: 8768: 8766: 8763: 8760: 8757: 8755: 8753: 8750: 8747: 8744: 8736: 8733: 8731: 8729: 8727: 8725: 8723: 8720: 8718: 8715: 8712: 8709: 8707: 8705: 8702: 8699: 8696: 8694: 8691: 8689: 8686: 8683: 8680: 8678: 8676: 8673: 8670: 8667: 8659: 8656: 8653: 8650: 8647: 8644: 8641: 8638: 8635: 8632: 8629: 8626: 8623: 8620: 8617: 8614: 8611: 8608: 8605: 8602: 8599: 8596: 8593: 8590: 8587: 8584: 8581: 8578: 8508: 8505: 8503: 8496: 8492: 8488: 8487: 8486: 8484: 8480: 8475: 8468: 8461: 8458: 8456: 8454: 8452: 8449: 8447: 8444: 8441: 8438: 8436: 8433: 8430: 8428: 8425: 8422: 8414: 8411: 8409: 8407: 8405: 8402: 8400: 8397: 8394: 8391: 8389: 8386: 8383: 8381: 8378: 8375: 8367: 8365: 8363: 8361: 8359: 8356: 8354: 8351: 8348: 8345: 8343: 8340: 8337: 8335: 8332: 8329: 8321: 8318: 8316: 8314: 8312: 8309: 8307: 8304: 8301: 8298: 8296: 8293: 8290: 8288: 8285: 8282: 8274: 8272: 8270: 8268: 8266: 8263: 8261: 8258: 8255: 8252: 8250: 8247: 8244: 8242: 8239: 8236: 8228: 8226: 8224: 8222: 8220: 8217: 8215: 8212: 8209: 8206: 8204: 8201: 8198: 8196: 8193: 8190: 8182: 8179: 8177: 8175: 8173: 8170: 8168: 8165: 8162: 8159: 8157: 8154: 8151: 8149: 8146: 8143: 8135: 8132: 8130: 8128: 8126: 8123: 8121: 8118: 8115: 8112: 8110: 8107: 8104: 8102: 8099: 8096: 8089: 8086: 8083: 8080: 8077: 8074: 8071: 8068: 8065: 8062: 8059: 8056: 8053: 8050: 8047: 8044: 8000: 7997: 7992: 7984: 7977: 7974: 7972: 7969: 7966: 7963: 7961: 7958: 7950: 7947: 7945: 7942: 7939: 7936: 7934: 7931: 7923: 7921: 7919: 7916: 7913: 7910: 7908: 7905: 7898:state 0, s=0 7897: 7894: 7892: 7889: 7886: 7883: 7881: 7878: 7871: 7868: 7865: 7862: 7859: 7856: 7853: 7850: 7822: 7820: 7819:state diagram 7811: 7809: 7807: 7797: 7784: 7777: 7773: 7769: 7763: 7756: 7749: 7746: 7744: 7741: 7739: 7736: 7734: 7731: 7723: 7720: 7718: 7715: 7713: 7710: 7708: 7705: 7697: 7694: 7692: 7689: 7687: 7684: 7682: 7679: 7671: 7668: 7666: 7663: 7661: 7658: 7656: 7653: 7646: 7643: 7640: 7637: 7634: 7631: 7628: 7625: 7597: 7593: 7591: 7587: 7580: 7571: 7568: 7566: 7563: 7561: 7558: 7556: 7548: 7545: 7543: 7540: 7538: 7535: 7533: 7526: 7523: 7520: 7517: 7514: 7511: 7509: 7483: 7476: 7474: 7472: 7468: 7464: 7460: 7456: 7451: 7449: 7445: 7440: 7438: 7437:impredicative 7432: 7367: 7364: 7362: 7359: 7357: 7354: 7351: 7348: 7346: 7343: 7341: 7338: 7335: 7332: 7330: 7327: 7325: 7322: 7319: 7316: 7314: 7312: 7309: 7306: 7303: 7300: 7293: 7291: 7288: 7286: 7283: 7280: 7277: 7275: 7272: 7270: 7267: 7264: 7261: 7259: 7256: 7254: 7251: 7248: 7245: 7243: 7241: 7238: 7235: 7232: 7229: 7222: 7220: 7217: 7215: 7212: 7209: 7206: 7204: 7201: 7199: 7196: 7193: 7190: 7188: 7185: 7183: 7180: 7177: 7174: 7172: 7170: 7167: 7164: 7161: 7158: 7151: 7149: 7146: 7144: 7141: 7138: 7135: 7133: 7130: 7128: 7125: 7122: 7119: 7117: 7114: 7112: 7109: 7106: 7103: 7101: 7099: 7096: 7093: 7090: 7087: 7080: 7078: 7075: 7073: 7070: 7067: 7064: 7062: 7059: 7057: 7054: 7051: 7048: 7046: 7043: 7041: 7038: 7035: 7032: 7030: 7028: 7025: 7022: 7019: 7016: 7009: 7007: 7004: 7002: 6999: 6996: 6993: 6991: 6988: 6986: 6983: 6980: 6977: 6975: 6972: 6970: 6967: 6964: 6961: 6959: 6957: 6954: 6951: 6948: 6945: 6938: 6936: 6933: 6931: 6928: 6925: 6922: 6920: 6917: 6915: 6912: 6909: 6906: 6904: 6901: 6899: 6896: 6893: 6890: 6888: 6886: 6883: 6880: 6877: 6874: 6867: 6865: 6862: 6860: 6857: 6854: 6851: 6849: 6846: 6844: 6841: 6838: 6835: 6833: 6830: 6828: 6825: 6822: 6819: 6817: 6815: 6812: 6809: 6806: 6803: 6713: 6705: 6702: 6701: 6700: 6699: 6698: 6697: 6693: 6692: 6684: 6683: 6682: 6681: 6680: 6679: 6675: 6674: 6666: 6665: 6664: 6663: 6662: 6661: 6657: 6656: 6648: 6645: 6642: 6640: 6637: 6636: 6635: 6634: 6633: 6632: 6628: 6627: 6619: 6618: 6617: 6616: 6615: 6614: 6610: 6609: 6601: 6598: 6597: 6596: 6595: 6594: 6593: 6589: 6588: 6587: 6580: 6579: 6578: 6575: 6573: 6565: 6530: 6496: 6462: 6430: 6427:(d & ~c) 6421:(~d & c) 6411:(d & ~c) 6405:(~d & c) 6398: 6395: 6321: 6242: 6163: 6085: 6006: 5927: 5849: 5771: 5693: 5598: 5588: 5587: 5586: 5585: 5584: 5583: 5582: 5577: 5573: 5572: 5571: 5560: 5558: 5548: 5545: 5542: 5539: 5531: 5528: 5525: 5522: 5514: 5511: 5508: 5505: 5497: 5494: 5491: 5488: 5480: 5477: 5474: 5471: 5463: 5460: 5457: 5454: 5446: 5443: 5440: 5437: 5429: 5426: 5423: 5420: 5412: 5409: 5406: 5403: 5379: 5377: 5376:Karnaugh maps 5373: 5369: 5352: 5351: 5345: 5344: 5343: 5340: 5338: 5330: 5327: 5322: 5319: 5318: 5317: 5315: 5311: 5307: 5299: 5297: 5295: 5294:Hasse diagram 5291: 5290:Karnaugh maps 5287: 5282: 5280: 5276: 5275:Hasse diagram 5272: 5262: 5255: 5249: 5247: 5245: 5243: 5240: 5238: 5235: 5232: 5229: 5227: 5224: 5221: 5218: 5216: 5213: 5211: 5208: 5205: 5202: 5200: 5198: 5195: 5192: 5189: 5182: 5180: 5178: 5176: 5173: 5171: 5168: 5165: 5162: 5160: 5157: 5154: 5151: 5149: 5146: 5144: 5141: 5138: 5135: 5133: 5131: 5128: 5125: 5122: 5115: 5113: 5111: 5109: 5106: 5104: 5101: 5098: 5095: 5093: 5090: 5087: 5084: 5082: 5079: 5077: 5074: 5071: 5068: 5066: 5064: 5061: 5058: 5055: 5048: 5046: 5044: 5042: 5039: 5037: 5034: 5031: 5028: 5026: 5023: 5020: 5017: 5015: 5012: 5010: 5007: 5004: 5001: 4999: 4997: 4994: 4991: 4988: 4981: 4979: 4977: 4975: 4972: 4970: 4967: 4964: 4961: 4959: 4956: 4953: 4950: 4948: 4945: 4943: 4940: 4937: 4934: 4932: 4930: 4927: 4924: 4921: 4914: 4912: 4910: 4908: 4905: 4903: 4900: 4897: 4894: 4892: 4889: 4886: 4883: 4881: 4878: 4876: 4873: 4870: 4867: 4865: 4863: 4860: 4857: 4854: 4847: 4845: 4843: 4841: 4838: 4836: 4833: 4830: 4827: 4825: 4822: 4819: 4816: 4814: 4811: 4809: 4806: 4803: 4800: 4798: 4796: 4793: 4790: 4787: 4780: 4778: 4776: 4774: 4771: 4769: 4766: 4763: 4760: 4758: 4755: 4752: 4749: 4747: 4744: 4742: 4739: 4736: 4733: 4731: 4729: 4726: 4723: 4720: 4712: 4709: 4706: 4703: 4700: 4697: 4694: 4691: 4688: 4685: 4682: 4679: 4676: 4673: 4670: 4667: 4664: 4661: 4658: 4655: 4652: 4649: 4646: 4643: 4636: 4634: 4632: 4630: 4628: 4626: 4623: 4620: 4618: 4616: 4613: 4610: 4608: 4606: 4603: 4601: 4599: 4596: 4594: 4592: 4590: 4588: 4586: 4584: 4578: 4576: 4574: 4572: 4570: 4568: 4566: 4564: 4562: 4560: 4558: 4556: 4554: 4552: 4549: 4547: 4545: 4543: 4541: 4539: 4537: 4534: 4531: 4528: 4479:clock active 4466: 4464: 4460: 4459:shown below. 4458: 4448: 4445: 4443: 4441: 4439: 4437: 4435: 4433: 4431: 4429: 4427: 4425: 4423: 4421: 4419: 4417: 4415: 4413: 4410: 4408: 4406: 4404: 4402: 4400: 4398: 4396: 4394: 4391: 4383: 4380: 4377: 4375: 4373: 4371: 4369: 4367: 4365: 4363: 4361: 4359: 4357: 4355: 4353: 4350: 4348: 4346: 4343: 4341: 4339: 4336: 4334: 4332: 4329: 4327: 4325: 4322: 4315: 4313: 4310: 4308: 4306: 4304: 4302: 4300: 4298: 4296: 4294: 4292: 4290: 4288: 4286: 4283: 4281: 4279: 4276: 4274: 4272: 4269: 4267: 4265: 4262: 4259: 4256: 4253: 4246: 4244: 4242: 4240: 4238: 4236: 4234: 4232: 4230: 4228: 4226: 4224: 4222: 4220: 4218: 4215: 4213: 4211: 4209: 4207: 4205: 4203: 4201: 4199: 4197: 4195: 4193: 4190: 4182: 4179: 4176: 4173: 4171: 4169: 4167: 4165: 4163: 4161: 4159: 4157: 4155: 4153: 4150: 4147: 4144: 4141: 4138: 4136: 4134: 4131: 4129: 4127: 4124: 4122: 4120: 4117: 4110: 4107: 4104: 4101: 4099: 4097: 4095: 4093: 4091: 4089: 4087: 4085: 4083: 4081: 4078: 4075: 4072: 4069: 4066: 4064: 4062: 4059: 4057: 4055: 4052: 4049: 4046: 4043: 4036: 4034: 4032: 4030: 4028: 4026: 4024: 4022: 4020: 4018: 4016: 4014: 4012: 4010: 4008: 4006: 4004: 4002: 4000: 3998: 3996: 3993: 3991: 3989: 3987: 3985: 3983: 3980: 3972: 3969: 3966: 3963: 3961: 3959: 3957: 3955: 3953: 3951: 3949: 3947: 3945: 3943: 3940: 3937: 3934: 3931: 3928: 3925: 3922: 3919: 3916: 3913: 3910: 3908: 3906: 3903: 3896: 3893: 3890: 3887: 3885: 3883: 3881: 3879: 3877: 3875: 3873: 3871: 3869: 3867: 3864: 3861: 3858: 3855: 3852: 3849: 3846: 3843: 3840: 3837: 3834: 3831: 3828: 3825: 3818: 3816: 3814: 3812: 3810: 3808: 3806: 3804: 3802: 3800: 3798: 3796: 3794: 3792: 3789: 3787: 3785: 3783: 3781: 3779: 3777: 3775: 3773: 3771: 3769: 3767: 3765: 3762: 3754: 3751: 3748: 3745: 3742: 3740: 3738: 3736: 3734: 3732: 3729: 3727: 3725: 3722: 3719: 3716: 3713: 3710: 3707: 3704: 3701: 3698: 3695: 3692: 3689: 3687: 3685: 3682: 3675: 3672: 3669: 3666: 3663: 3661: 3659: 3657: 3655: 3653: 3650: 3648: 3646: 3643: 3640: 3637: 3634: 3631: 3628: 3625: 3622: 3619: 3616: 3613: 3610: 3607: 3604: 3601: 3594: 3592: 3590: 3588: 3586: 3584: 3582: 3580: 3578: 3576: 3573: 3571: 3569: 3567: 3565: 3563: 3561: 3559: 3557: 3555: 3553: 3551: 3549: 3547: 3545: 3543: 3541: 3538: 3530: 3527: 3524: 3521: 3518: 3516: 3514: 3512: 3510: 3508: 3505: 3503: 3501: 3498: 3495: 3492: 3489: 3486: 3483: 3480: 3477: 3474: 3471: 3468: 3465: 3463: 3461: 3458: 3451: 3448: 3445: 3442: 3439: 3437: 3435: 3433: 3431: 3429: 3426: 3424: 3422: 3419: 3416: 3413: 3410: 3407: 3404: 3401: 3398: 3395: 3392: 3389: 3386: 3383: 3380: 3377: 3370: 3368: 3366: 3364: 3362: 3360: 3358: 3356: 3354: 3352: 3349: 3347: 3345: 3343: 3341: 3339: 3337: 3335: 3333: 3331: 3329: 3327: 3325: 3323: 3321: 3319: 3317: 3314: 3306: 3303: 3300: 3297: 3294: 3291: 3289: 3287: 3285: 3282: 3279: 3276: 3273: 3270: 3267: 3264: 3261: 3258: 3255: 3252: 3249: 3246: 3243: 3240: 3237: 3235: 3233: 3230: 3223: 3220: 3217: 3214: 3211: 3208: 3206: 3204: 3202: 3199: 3196: 3193: 3190: 3187: 3184: 3181: 3178: 3175: 3172: 3169: 3166: 3163: 3160: 3157: 3154: 3151: 3148: 3145: 3138: 3136: 3134: 3132: 3130: 3128: 3126: 3124: 3122: 3119: 3117: 3115: 3113: 3111: 3109: 3107: 3105: 3103: 3101: 3099: 3097: 3095: 3093: 3091: 3089: 3087: 3085: 3082: 3074: 3071: 3068: 3065: 3062: 3059: 3056: 3053: 3050: 3047: 3044: 3041: 3038: 3035: 3032: 3029: 3026: 3023: 3020: 3017: 3014: 3011: 3008: 3005: 3002: 3000: 2998: 2995: 2988: 2985: 2982: 2979: 2976: 2973: 2970: 2967: 2964: 2961: 2958: 2955: 2952: 2949: 2946: 2943: 2940: 2937: 2934: 2931: 2928: 2925: 2922: 2919: 2916: 2913: 2910: 2907: 2899: 2896: 2893: 2890: 2887: 2884: 2881: 2878: 2875: 2872: 2869: 2866: 2863: 2860: 2857: 2854: 2851: 2848: 2845: 2842: 2839: 2836: 2833: 2830: 2827: 2825: 2823: 2820: 2812: 2809: 2806: 2803: 2800: 2797: 2794: 2791: 2788: 2785: 2782: 2779: 2776: 2773: 2770: 2767: 2764: 2761: 2758: 2755: 2752: 2749: 2746: 2743: 2740: 2738: 2736: 2733: 2666:Description: 2653: 2652: 2651: 2650: 2646: 2645: 2639: 2638: 2637: 2636: 2632: 2631: 2625: 2624: 2623: 2622: 2618: 2617: 2611: 2610: 2609: 2608: 2604: 2603: 2597: 2596: 2595: 2594: 2590: 2589: 2583: 2582: 2581: 2580: 2576: 2575: 2569: 2568: 2566: 2565: 2561: 2560: 2556: 2555: 2554: 2550: 2548: 2544: 2535: 2533: 2531: 2529: 2527: 2525: 2523: 2521: 2519: 2517: 2515: 2513: 2511: 2509: 2507: 2505:(~a & a) 2504: 2502: 2495: 2492: 2489: 2486: 2483: 2480: 2477: 2474: 2471: 2468: 2465: 2462: 2459: 2456: 2453: 2450: 2447: 2439: 2436: 2433: 2430: 2427: 2424: 2421: 2418: 2415: 2412: 2409: 2406: 2403: 2400: 2397: 2394: 2391: 2383: 2380: 2377: 2374: 2371: 2368: 2365: 2362: 2359: 2356: 2353: 2350: 2347: 2344: 2341: 2338: 2335: 2327: 2324: 2321: 2318: 2315: 2312: 2309: 2306: 2303: 2300: 2297: 2294: 2291: 2288: 2285: 2282: 2279: 2272: 2269: 2267: 2264: 2261: 2258: 2255: 2252: 2249: 2246: 2243: 2241: 2238: 2236: 2233: 2231: 2228: 2221: 2218: 2215: 2212: 2209: 2206: 2203: 2200: 2197: 2194: 2191: 2189: 2187: 2185: 2182: 2180: 2178: 2171: 2169: 2167: 2165: 2163: 2161: 2158: 2156: 2154: 2151: 2149: 2147: 2145: 2143: 2141: 2138: 2136: 2076: 2072: 2065: 2060: 2057: 2054: 2051: 2048: 2045: 2042: 2041: 2037: 2032: 2029: 2026: 2023: 2020: 2017: 2014: 2013: 2009: 2007: 2001: 1999: 1997: 1989: 1910: 1832: 1829: 1826: 1825: 1820: 1814: 1807: 1805: 1795: 1794: 1792: 1788: 1784: 1780: 1779: 1778: 1777: 1773: 1769: 1765: 1761: 1760: 1756: 1752: 1748: 1744: 1743: 1739: 1735: 1731: 1727: 1726: 1722: 1718: 1714: 1710: 1709: 1705: 1704: 1703: 1696: 1691: 1688: 1685: 1682: 1679: 1676: 1673: 1670: 1667: 1666: 1665: 1663: 1655: 1651: 1648: 1645: 1642: 1639: 1636: 1633: 1632: 1631: 1629: 1621: 1618: 1615: 1612: 1609: 1606: 1603: 1600: 1597: 1594: 1593: 1592: 1586: 1583: 1580: 1577: 1574: 1571: 1568: 1565: 1562: 1561: 1560: 1558: 1549: 1546: 1545: 1544: 1543: 1542: 1540: 1536: 1532: 1528: 1526: 1519: 1517:are formulas. 1492: 1489: 1488: 1487: 1485: 1479: 1440: 1439: 1438: 1435: 1431: 1424: 1420: 1419: 1418: 1416: 1415:concatenation 1412: 1405: 1402: 1399: 1396: 1393: 1390: 1387: 1384: 1381: 1378: 1377: 1376: 1374: 1369: 1365: 1359: 1355: 1352: 1346: 1345: 1344: 1342: 1338: 1334: 1330: 1328: 1324: 1314: 1311: 1308: 1305: 1302: 1299: 1297: 1294: 1291: 1288: 1285: 1282: 1279: 1277: 1274: 1271: 1268: 1265: 1262: 1254: 1251: 1248: 1245: 1242: 1239: 1237: 1234: 1231: 1228: 1225: 1222: 1219: 1217: 1214: 1211: 1208: 1205: 1202: 1194: 1191: 1188: 1185: 1182: 1179: 1177: 1174: 1171: 1168: 1165: 1162: 1159: 1157: 1154: 1151: 1148: 1145: 1142: 1134: 1131: 1128: 1125: 1122: 1119: 1117: 1114: 1111: 1108: 1105: 1102: 1099: 1097: 1094: 1091: 1088: 1085: 1082: 1074: 1071: 1068: 1065: 1062: 1059: 1057: 1054: 1051: 1048: 1045: 1042: 1039: 1037: 1034: 1031: 1028: 1025: 1022: 955: 953: 944: 937: 935: 928: 921: 918: 917: 914: 913: 912: 910: 901: 900: 898: 897: 896: 893: 888: 884: 875: 874: 872: 868: 867: 866: 864: 855: 854: 853: 851: 847: 839: 838: 834: 833: 832: 830: 826: 823: 817: 815: 811: 807: 803: 799: 796: 792: 789: 787: 783: 779: 772: 771: 770: 764: 763: 762: 760: 756: 752: 748: 741: 737: 736: 735: 733: 725: 721: 716: 715: 714: 712: 708: 704: 700: 696: 695:contradictory 692: 688: 683: 681: 673: 672: 671: 669: 665: 661: 653: 652: 651: 650:ambiguous. 648: 644: 640: 636: 634: 630: 626: 618: 616: 614: 606: 601: 598: 597: 596: 594: 589: 587: 583: 579: 575: 570: 568: 564: 560: 551: 548: 545: 542: 539: 536: 533: 530: 527: 524: 521: 513: 510: 507: 504: 501: 498: 495: 492: 489: 486: 483: 475: 472: 469: 466: 463: 460: 457: 454: 451: 448: 445: 437: 434: 431: 428: 425: 422: 419: 416: 413: 410: 407: 399: 396: 393: 390: 387: 384: 381: 378: 375: 372: 369: 331: 330: 326: 323: 320: 317: 314: 313: 312: 310: 306: 302: 298: 294: 288: 287: 283: 282: 281: 279: 271: 268: 265: 262: 260:Distribution 254: 251: 248: 246: 240: 237: 234: 231: 224: 221: 218: 215: 208: 205: 202: 199: 192: 189: 186: 183: 160: 158: 157: 152: 148: 144: 140: 131: 128: 125: 122: 119: 116: 113: 105: 102: 99: 96: 93: 90: 87: 79: 76: 73: 71: 68: 65: 62: 26: 19: 18:User:Wvbailey 9684: 9673: 9660: 9648: 9632: 9612: 9600: 9594: 9586: 9582: 9577: 9569: 9563: 9554: 9545: 9536: 9510:1952 and M. 9508:E. W. Veitch 9479:) V carry_in 9438:and carry_in 9413:F. W. Jordan 9401: 9400: 9395: 9382: 9366: 9362:George Boole 9340: 9328: 9326: 9322:Venn diagram 9318: 9284: 8660:Description 8501: 8499: 8494: 8490: 8482: 8478: 8476: 8472: 7995: 7988: 7815: 7805: 7798: 7790: 7771: 7768:total states 7767: 7594: 7590:ad infinitum 7589: 7585: 7584: 7480: 7452: 7447: 7441: 7436: 7433: 7429: 6703: 6647: 6644: 6641: 6599: 6584: 6576: 6569: 6424:(d & c) 6408:(d & c) 6392: 5596: 5580: 5574: 5569: 5554: 5365: 5346:Example: cba 5341: 5333: 5313: 5309: 5303: 5286:Karnaugh map 5283: 5279:Karnaugh map 5268: 4462: 4461: 4454: 2551: 2547:truth tables 2542: 2541: 2253:(a & b) 2159:equivalence 2073: 2069: 2043:Axiom: a = a 2015:Axiom: a = a 2005: 1995: 1993: 1830: 1827: 1823: 1818: 1815: 1811: 1803: 1790: 1786: 1782: 1771: 1767: 1763: 1754: 1753:in (p & 1750: 1746: 1737: 1733: 1729: 1720: 1719:in (c & 1716: 1712: 1700: 1661: 1659: 1653: 1630:formula #9: 1627: 1625: 1590: 1556: 1555: 1538: 1534: 1530: 1529: 1524: 1523: 1513:, (s ←→ t)=v 1483: 1482: 1441: 1433: 1429: 1428: 1422: 1414: 1410: 1409: 1403: 1397: 1391: 1385: 1379: 1372: 1370: 1366: 1363: 1350: 1347: 1340: 1336: 1332: 1331: 1327:substitution 1326: 1322: 1320: 1072:(b & a) 1052:(b & a) 1032:(b & a) 951: 950: 932: 908: 907: 891: 886: 882: 881: 870: 862: 859: 849: 845: 844: 828: 827: 821: 818: 809: 801: 800: 797: 793: 790: 781: 780: 777: 768: 758: 754: 750: 746: 745: 739: 732:inconsistent 729: 723: 719: 706: 703:rationalists 699:inconsistent 684: 679: 678: 667: 664:concatenated 663: 659: 657: 642: 638: 637: 632: 628: 625:propositions 624: 622: 619:Propositions 610: 590: 577: 573: 571: 567:Karnaugh map 562: 561: 557: 309:substitution 308: 300: 296: 295: 292: 277: 276: 229:Association 213:Association 197:Commutation 181:Commutation 155: 154: 138: 137: 100:conjunction 74:disjunction 9605:Alan Turing 9583:oscillation 9506:1952, 1955 9495:Alan Turing 9428:binary bits 8502:transitions 7787:remains 0). 7477:Oscillation 6667:( V V ) 6638:( V V ) 5271:truth table 4457:truth-table 2204:(a XNOR b) 2198:(a NAND b) 1654:(a V a) = a 1509:, (s → t)=v 1404:parentheses 711:empiricists 305:truth table 111:difference 9623:References 9360:"inspired 9337:John Locke 9333:syllogisms 6685:( V V ) 4473:TEMP okay 2543:Evaluation 2201:(a AND b) 2195:(a XOR b) 2183:(a NOR b) 2172:tautology 2152:(a "+" b) 1996:evaluation 1607:13 (0 ≠ 1) 1373:Defined as 647:assertions 643:particular 9529:Footnotes 9358:De Morgan 9345:semiotics 9290:Aristotle 7991:flip-flop 7772:transient 6724:Minterms 5606:Minterms 5368:Gray code 4532:nice day 4494:motor ON 2640:(0 V 1)=q 2256:~(a ⊕ b) 2234:~(a V b) 2219:(a OR b) 1831:Example: 1660:Example: 1626:Example: 1493:(ii) if v 1462:(a) and v 1386:variables 1380:existence 1003:variable 1000:variable 983:variable 980:variable 963:variable 960:variable 814:variables 578:reduction 574:Reduction 301:synthesis 126:negation 9522:H. Shorr 9512:Karnaugh 7750:state 0 7672:state 1 7473:, etc). 7444:function 6572:literals 5396:minterm 5329:minterms 2270:(a V b) 2262:(a → b) 2247:(a ⊕ b) 1770:in (s V 1662:Evaluate 1656:. Q.E.D. 1446:and (a R 1437:system. 1075:(b V a) 1055:(b V a) 1035:(b V a) 892:converse 841:blanket. 774:starts." 759:variable 707:a priori 680:NEGATION 660:compound 633:compound 613:ontology 563:Analysis 373:( ( ( ~ 85:product 9475:& b 9329:algebra 7801:delayed 7794:delayed 5306:literal 4550:picnic 2192:NOT(a) 1785:in for 829:AND, OR 810:algebra 806:algebra 749:: Each 726:blue!". 9631:1965, 9587:memory 9570:formal 8630:& 8615:& 8597:& 8066:& 7996:forces 7757:Memory 6781:& 6763:& 6745:& 6510:row 6 6507:row 7 6504:row 5 6501:row 4 6476:row 2 6473:row 3 6470:row 1 6467:row 0 5660:& 5645:& 5627:& 5314:alterm 5277:, and 4692:& 4680:& 4662:& 4257:count 4148:& 4047:count 3938:& 3920:& 3829:count 3717:& 3699:& 3605:count 3493:& 3475:& 3381:count 3280:& 3265:& 3247:& 3149:count 3045:& 3030:& 3012:& 2911:count 2870:& 2855:& 2837:& 2783:& 2768:& 2750:& 2663:Step: 1912:count 1880:& 1868:& 1850:& 1824:error. 1557:Axioms 1539:define 1535:define 1406:: ), ( 1351:invert 822:unlike 755:unique 751:unique 629:simple 385:& 153:apply 149:, and 141:: The 97:& 9457:XOR b 9453:( ( a 9450:are: 9434:and b 9384:Frege 7448:state 5576:form. 5284:> 5281:. 4535:noon 4529:food 2244:~(a) 2239:~(b) 1819:error 1736:in ~( 1628:Prove 1069:~(a) 1066:~(b) 1049:~(a) 1046:~(b) 1029:~(a) 1026:~(b) 635:. 394:a) ) 16:< 9520:and 9465:)= Σ 9411:and 8576:row 8087:= q 7869:= q 7644:= q 7524:= q 6721:row 5603:row 5310:term 4641:row 2265:(b) 2259:(a) 2213:(b) 2207:(a) 1789:in ( 1766:for 1749:for 1732:for 1715:for 1450:b)=v 1012:AND 1009:NOT 1006:NOT 992:AND 989:NOT 986:NOT 972:AND 969:NOT 966:NOT 376:(b) 352:= a 117:NOT 91:AND 60:sum 9585:or 9471:( a 9404:: 9339:'s 9324:. 8657:=q 7623:qd 6559:~c 6550:~c 6543:~c 6534:~c 6481:~p 6464:~p 6453:~d 6450:~d 6437:~d 6434:~d 5378:. 4389:9c 4320:9b 4251:9a 4188:8c 4115:8b 4041:8a 3978:7c 3901:7b 3823:7a 3760:6c 3680:6b 3599:6a 3536:5c 3456:5b 3375:5a 3312:4c 3228:4b 3143:4a 3080:3c 2993:3b 2905:3a 2654:1=q 2250:b) 2130:~0 2127:~1 2124:~2 2121:~3 2118:~4 2115:~5 2112:~6 2109:~7 1994:An 1907:q2 1486:: 1423:not 1382:: ∃ 1015:OR 995:OR 975:OR 887:not 850:not 740:not 724:not 720:not 697:or 631:or 388:(b 382:a) 114:1- 66:OR 9647:, 9335:, 9271:1 9258:1 9253:0 9250:0 9247:1 9240:1 9237:1 9234:1 9229:1 9224:1 9221:1 9218:1 9211:1 9208:1 9205:1 9202:7 9194:1 9181:1 9176:0 9173:0 9170:0 9163:1 9160:1 9157:1 9152:1 9147:1 9144:0 9141:0 9134:0 9131:1 9128:1 9125:6 9105:0 9100:1 9097:1 9094:1 9087:0 9084:0 9081:1 9076:0 9071:0 9068:0 9065:1 9058:1 9055:0 9052:1 9049:5 9041:1 9028:0 9023:1 9020:0 9017:0 9010:1 9007:1 9004:1 8999:1 8994:0 8991:0 8988:0 8981:0 8978:0 8975:1 8972:4 8952:1 8947:0 8944:0 8941:1 8934:1 8931:0 8928:0 8923:1 8918:1 8915:1 8912:1 8905:1 8902:1 8899:0 8896:3 8888:0 8875:1 8870:0 8867:0 8864:0 8857:1 8854:0 8851:0 8846:0 8841:1 8838:0 8835:0 8828:0 8825:1 8822:0 8819:2 8811:0 8798:0 8793:1 8790:1 8787:1 8780:0 8777:0 8774:0 8769:0 8764:0 8761:0 8758:1 8751:1 8748:0 8745:0 8742:1 8734:0 8721:0 8716:1 8713:0 8710:0 8703:1 8700:0 8697:0 8692:0 8687:0 8684:0 8681:0 8674:0 8671:0 8668:0 8665:0 8654:) 8651:) 8648:) 8645:) 8642:) 8639:d 8636:( 8633:~ 8627:c 8624:( 8621:( 8618:~ 8612:q 8609:( 8606:V 8603:) 8600:d 8594:c 8591:( 8588:( 8585:c 8582:d 8579:q 8553:u 8550:r 8541:v 8538:w 8531:q 8524:s 8504:. 8459:1 8450:1 8445:0 8442:0 8439:1 8434:1 8431:1 8426:1 8423:1 8420:1 8412:1 8403:0 8398:1 8395:1 8392:1 8387:1 8384:1 8379:0 8376:1 8373:1 8357:1 8352:0 8349:0 8346:1 8341:0 8338:0 8333:1 8330:0 8327:1 8319:1 8310:0 8305:1 8302:1 8299:1 8294:1 8291:0 8286:0 8283:0 8280:1 8264:1 8259:0 8256:0 8253:0 8248:1 8245:1 8240:1 8237:1 8234:0 8218:0 8213:1 8210:0 8207:0 8202:1 8199:1 8194:0 8191:1 8188:0 8180:0 8171:1 8166:0 8163:0 8160:0 8155:0 8152:0 8147:1 8144:0 8141:0 8133:0 8124:0 8119:1 8116:0 8113:0 8108:0 8105:0 8100:0 8097:0 8094:0 8084:) 8081:) 8078:) 8075:r 8072:( 8069:~ 8063:p 8060:( 8057:V 8054:s 8051:( 8048:r 8045:s 8042:p 8015:q 7975:1 7970:1 7967:1 7964:1 7959:1 7956:1 7948:1 7943:1 7940:1 7937:0 7932:0 7929:1 7917:0 7914:1 7911:1 7906:1 7903:0 7895:0 7890:0 7887:0 7884:0 7879:0 7876:0 7866:) 7863:p 7860:V 7857:s 7854:( 7851:s 7848:p 7835:q 7796:. 7747:0 7742:1 7737:0 7732:1 7729:1 7721:1 7716:0 7711:1 7706:0 7703:1 7695:0 7690:1 7685:0 7680:1 7677:0 7669:1 7664:0 7659:1 7654:0 7651:0 7641:) 7638:p 7635:( 7632:~ 7629:( 7626:p 7608:q 7569:0 7564:1 7559:0 7554:1 7546:1 7541:0 7536:1 7531:0 7521:) 7518:p 7515:( 7512:~ 7507:p 7492:q 7469:, 7465:, 7461:, 7403:q 7360:1 7355:0 7352:0 7349:1 7344:1 7339:1 7336:1 7333:1 7328:1 7323:1 7320:1 7317:1 7310:1 7307:1 7304:1 7298:7 7289:0 7284:1 7281:1 7278:1 7273:1 7268:1 7265:1 7262:1 7257:1 7252:0 7249:0 7246:1 7239:0 7236:1 7233:1 7227:6 7218:1 7213:0 7210:0 7207:1 7202:0 7197:0 7194:0 7191:1 7186:0 7181:1 7178:0 7175:0 7168:1 7165:0 7162:1 7156:5 7147:0 7142:1 7139:1 7136:1 7131:1 7126:0 7123:0 7120:1 7115:0 7110:0 7107:0 7104:0 7097:0 7094:0 7091:1 7085:4 7076:1 7071:0 7068:0 7065:0 7060:1 7055:1 7052:0 7049:0 7044:1 7039:1 7036:1 7033:1 7026:1 7023:1 7020:0 7014:3 7005:0 7000:1 6997:0 6994:0 6989:0 6984:1 6981:0 6978:0 6973:0 6968:0 6965:0 6962:1 6955:0 6952:1 6949:0 6943:2 6934:1 6929:0 6926:0 6923:0 6918:0 6913:0 6910:0 6907:0 6902:0 6897:1 6894:0 6891:0 6884:1 6881:0 6878:0 6872:1 6863:0 6858:1 6855:0 6852:0 6847:0 6842:0 6839:0 6836:0 6831:0 6826:0 6823:0 6820:0 6813:0 6810:0 6807:0 6801:0 6796:) 6793:) 6790:c 6787:( 6784:~ 6778:p 6775:( 6772:V 6769:) 6766:d 6760:p 6757:( 6754:V 6751:) 6748:c 6742:d 6739:( 6736:( 6733:c 6730:d 6727:p 6646:V 6643:V 6556:c 6553:c 6540:c 6537:c 6527:1 6524:1 6521:0 6518:1 6515:p 6498:p 6493:0 6490:1 6487:0 6484:0 6459:d 6456:d 6443:d 6440:d 6345:q 6303:1 6298:0 6295:0 6292:1 6285:1 6282:1 6279:1 6274:1 6269:1 6266:1 6263:0 6256:1 6253:1 6250:1 6244:7 6224:1 6219:0 6216:0 6213:0 6206:1 6203:1 6200:1 6195:1 6190:1 6187:0 6184:0 6177:0 6174:1 6171:1 6165:6 6146:0 6141:1 6138:1 6135:1 6128:0 6125:0 6122:1 6117:0 6112:0 6109:0 6106:1 6099:1 6096:0 6093:1 6087:5 6067:0 6062:1 6059:0 6056:0 6049:1 6046:1 6043:1 6038:1 6033:0 6030:0 6027:0 6020:0 6017:0 6014:1 6008:4 5988:1 5983:0 5980:0 5977:1 5970:1 5967:0 5964:0 5959:1 5954:1 5951:1 5948:1 5941:1 5938:1 5935:0 5929:3 5910:1 5905:0 5902:0 5899:0 5892:1 5889:0 5886:0 5881:0 5876:1 5873:0 5870:0 5863:0 5860:1 5857:0 5851:2 5832:0 5827:1 5824:1 5821:1 5814:0 5811:0 5808:0 5803:0 5798:0 5795:0 5792:1 5785:1 5782:0 5779:0 5773:1 5754:0 5749:1 5746:0 5743:0 5736:1 5733:0 5730:0 5725:0 5720:0 5717:0 5714:0 5707:0 5704:0 5701:0 5695:0 5684:) 5681:) 5678:) 5675:) 5672:) 5669:d 5666:( 5663:~ 5657:c 5654:( 5651:( 5648:~ 5642:p 5639:( 5636:V 5633:) 5630:d 5624:c 5621:( 5618:( 5615:c 5612:d 5609:p 5546:0 5543:0 5540:0 5537:0 5529:0 5526:0 5523:1 5520:4 5512:1 5509:0 5506:1 5503:5 5495:1 5492:1 5489:1 5486:7 5478:0 5475:1 5472:1 5469:6 5461:0 5458:1 5455:0 5452:2 5444:1 5441:1 5438:0 5435:3 5427:1 5424:0 5421:0 5418:1 5410:0 5407:0 5404:0 5401:0 5393:a 5390:b 5387:c 5359:10 5339:. 5296:. 5241:1 5236:0 5233:0 5230:1 5225:1 5222:1 5219:1 5214:1 5209:1 5206:1 5203:0 5196:1 5193:1 5190:1 5187:7 5174:1 5169:0 5166:0 5163:0 5158:1 5155:1 5152:1 5147:1 5142:1 5139:0 5136:0 5129:0 5126:1 5123:1 5120:6 5107:0 5102:1 5099:1 5096:1 5091:0 5088:0 5085:1 5080:0 5075:0 5072:0 5069:1 5062:1 5059:0 5056:1 5053:5 5040:0 5035:1 5032:0 5029:0 5024:1 5021:1 5018:1 5013:1 5008:0 5005:0 5002:0 4995:0 4992:0 4989:1 4986:4 4973:1 4968:0 4965:0 4962:1 4957:1 4954:0 4951:0 4946:1 4941:1 4938:1 4935:1 4928:1 4925:1 4922:0 4919:3 4906:1 4901:0 4898:0 4895:0 4890:1 4887:0 4884:0 4879:0 4874:1 4871:0 4868:0 4861:0 4858:1 4855:0 4852:2 4839:0 4834:1 4831:1 4828:1 4823:0 4820:0 4817:0 4812:0 4807:0 4804:0 4801:1 4794:1 4791:0 4788:0 4785:1 4772:0 4767:1 4764:0 4761:0 4756:1 4753:0 4750:0 4745:0 4740:0 4737:0 4734:0 4727:0 4724:0 4721:0 4718:0 4713:) 4710:) 4707:) 4704:) 4701:d 4698:( 4695:~ 4689:c 4686:( 4683:~ 4677:p 4674:( 4671:V 4668:) 4665:d 4659:c 4656:( 4653:( 4650:c 4647:d 4644:p 4624:u 4621:r 4614:v 4611:w 4604:q 4597:s 4449:q 4446:= 4411:1 4384:q 4381:= 4378:) 4351:1 4344:V 4337:0 4330:( 4311:0 4284:1 4277:1 4270:1 4263:1 4260:0 4216:1 4183:q 4180:= 4177:) 4174:) 4151:1 4145:1 4142:( 4139:V 4132:0 4125:( 4108:0 4105:0 4102:1 4079:2 4076:2 4073:2 4070:2 4067:1 4060:1 4053:1 4050:0 3994:0 3973:q 3970:= 3967:) 3964:) 3941:1 3935:1 3932:( 3929:V 3926:) 3923:1 3917:0 3914:( 3911:( 3894:0 3891:0 3888:1 3865:2 3862:2 3859:2 3856:2 3853:1 3850:1 3847:2 3844:2 3841:2 3838:2 3835:1 3832:0 3790:1 3755:q 3752:= 3749:) 3746:) 3743:) 3730:0 3723:( 3720:~ 3714:1 3711:( 3708:V 3705:) 3702:1 3696:0 3693:( 3690:( 3673:0 3670:0 3667:1 3664:2 3651:3 3644:3 3641:2 3638:2 3635:2 3632:2 3629:1 3626:1 3623:2 3620:2 3617:2 3614:2 3611:1 3608:0 3574:0 3531:q 3528:= 3525:) 3522:) 3519:) 3506:0 3499:( 3496:~ 3490:1 3487:( 3484:V 3481:) 3478:1 3472:0 3469:( 3466:( 3449:0 3446:0 3443:1 3440:2 3427:3 3420:3 3417:2 3414:2 3411:2 3408:2 3405:1 3402:1 3399:2 3396:2 3393:2 3390:2 3387:1 3384:0 3350:0 3307:q 3304:= 3301:) 3298:) 3295:) 3292:) 3283:0 3277:0 3274:( 3271:( 3268:~ 3262:1 3259:( 3256:V 3253:) 3250:1 3244:0 3241:( 3238:( 3221:0 3218:0 3215:1 3212:2 3209:3 3200:4 3197:4 3194:4 3191:4 3188:3 3185:2 3182:2 3179:2 3176:2 3173:1 3170:1 3167:2 3164:2 3161:2 3158:2 3155:1 3152:0 3120:0 3075:q 3072:= 3069:) 3066:) 3063:) 3060:) 3057:) 3054:1 3051:( 3048:~ 3042:0 3039:( 3036:( 3033:~ 3027:1 3024:( 3021:V 3018:) 3015:1 3009:0 3006:( 3003:( 2986:0 2983:0 2980:1 2977:2 2974:3 2971:4 2968:5 2965:5 2962:4 2959:4 2956:4 2953:4 2950:3 2947:2 2944:2 2941:2 2938:2 2935:1 2932:1 2929:2 2926:2 2923:2 2920:2 2917:1 2914:0 2900:q 2897:= 2894:) 2891:) 2888:) 2885:) 2882:) 2879:1 2876:( 2873:~ 2867:0 2864:( 2861:( 2858:~ 2852:1 2849:( 2846:V 2843:) 2840:1 2834:0 2831:( 2828:( 2818:2 2813:q 2810:= 2807:) 2804:) 2801:) 2798:) 2795:) 2792:d 2789:( 2786:~ 2780:c 2777:( 2774:( 2771:~ 2765:p 2762:( 2759:V 2756:) 2753:d 2747:c 2744:( 2741:( 2731:1 2708:u 2705:r 2696:v 2693:w 2686:q 2679:s 2549:. 2496:1 2493:1 2490:1 2487:1 2484:1 2481:1 2478:1 2475:1 2472:0 2469:0 2466:0 2463:0 2460:0 2457:0 2454:0 2451:0 2448:1 2445:1 2440:1 2437:1 2434:1 2431:1 2428:0 2425:0 2422:0 2419:0 2416:1 2413:1 2410:1 2407:1 2404:0 2401:0 2398:0 2395:0 2392:1 2389:0 2384:1 2381:1 2378:0 2375:0 2372:1 2369:1 2366:0 2363:0 2360:1 2357:1 2354:0 2351:0 2348:1 2345:1 2342:0 2339:0 2336:0 2333:1 2328:1 2325:0 2322:1 2319:0 2316:1 2313:0 2310:1 2307:0 2304:1 2301:0 2298:1 2295:0 2292:1 2289:0 2286:1 2283:0 2280:0 2277:0 2229:b 2226:a 2106:7 2103:6 2100:5 2097:4 2094:3 2091:2 2088:1 2085:0 1981:0 1978:0 1975:1 1972:2 1969:3 1966:4 1963:4 1960:3 1957:3 1954:3 1951:3 1948:2 1945:2 1942:2 1939:2 1936:1 1933:1 1930:2 1927:2 1924:2 1921:2 1918:1 1915:0 1904:= 1901:) 1898:) 1895:) 1892:) 1889:d 1886:( 1883:~ 1877:c 1874:( 1871:~ 1865:q 1862:( 1859:V 1856:) 1853:d 1847:c 1844:( 1841:( 1466:=R 1458:=R 1339:, 1315:T 1312:T 1309:F 1306:F 1303:T 1300:T 1295:1 1292:1 1289:0 1286:0 1283:1 1280:1 1275:℥ 1272:℥ 1269:※ 1266:※ 1263:℥ 1260:℥ 1255:T 1252:F 1249:T 1246:F 1243:F 1240:T 1235:1 1232:0 1229:1 1226:0 1223:0 1220:1 1215:℥ 1212:※ 1209:℥ 1206:※ 1203:※ 1200:℥ 1195:T 1192:F 1189:F 1186:T 1183:T 1180:F 1175:1 1172:0 1169:0 1166:1 1163:1 1160:0 1155:℥ 1152:※ 1149:※ 1146:℥ 1143:℥ 1140:※ 1135:F 1132:F 1129:T 1126:T 1123:F 1120:F 1115:0 1112:0 1109:1 1106:1 1103:0 1100:0 1095:※ 1092:※ 1089:℥ 1086:℥ 1083:※ 1080:※ 1063:a 1060:b 1043:a 1040:b 1023:a 1020:b 734:. 658:A 552:1 549:1 546:1 543:1 540:1 537:1 534:1 531:1 528:1 525:0 522:1 519:1 514:0 511:1 508:0 505:0 502:1 499:0 496:0 493:1 490:1 487:0 484:0 481:1 476:1 473:1 470:1 467:1 464:0 461:1 458:1 455:1 452:0 449:1 446:1 443:0 438:0 435:1 432:0 429:1 426:0 423:0 420:0 417:0 414:0 411:1 408:0 405:0 400:a 397:= 391:→ 379:→ 370:a 367:b 159:. 145:, 123:~ 120:￢ 94:⋀ 88:∙ 69:V 63:+ 9651:. 9487:; 9485:i 9481:i 9477:i 9473:i 9467:i 9463:i 9459:i 9455:i 9448:i 9444:i 9440:i 9436:i 9432:i 9430:a 9373:2 8495:q 8491:q 8483:q 8479:p 7792:q 7778:. 6649:. 5355:2 5348:2 1791:s 1787:s 1783:s 1772:w 1768:w 1764:w 1755:v 1751:v 1747:v 1738:r 1734:r 1730:r 1721:u 1717:u 1713:u 1515:i 1511:i 1507:i 1503:i 1499:i 1495:i 1476:2 1472:1 1468:2 1464:i 1460:1 1456:i 1452:i 1448:2 1444:i 1341:O 1337:|

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Knowledge

User:Wvbailey/Propositional formula

Index