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arithmetic algebra, we don't have expressions such as (2 * 3 + 4) having one value (6 + 4) in one ranking (or according to one author) and another value (2 * 7) in a different ranking (or according to some other author). So one ranking must surely be "wrong", or at least highly unconventional. (Or is logic just one of those areas in which different practitioners define things differently??)
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treatment. I find Kleene's 1952 far better. It seems we should be able to construct a very simple induction-like method of how to construct the formulas. Even Kleene's is too complex because he's constructing an arithmetic+logic system. Kleene has some three little lemmas about parenthesis pairing and nesting etc.
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This section appears to contradict (significantly) the book, "Foundations Of
Computation" by Critchlow and Eck (2nd ed, 2011). Pg 3 of that book, which deals with the relative precedences (are precedences the same things as "ranks" / "seniorities" ?), states (but with my additional clarifications in
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One issue that the content above brings to mind is that this article is just about propositional formulas. It isn't meant to be about all of propositional logic. We can have a few sentences that link to other important concepts, like valuations. Here is a rough list of the things that I think should
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Jochen, thanks, they do seem more similar as reverses of each other. But they still remain different, and when did "almost the same" become good enough in maths? Use of these different rankings will lead to a given propositional formula evaluating to true for one ranking and false for the other. In
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Propositions from a propositional formula (simple or propositional variables or propositional constants) are denoted with small letters p,q in contrast with predicate letters P, Q, R, S, which asociated to individual constants form propositional constants which can also be present in propositional
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RE {T, F}. Problem is, folks other than mathematicians use propositional formulas. I advise anyone doing so (i.e. creating propositional formulas and then evaluating them) to stay away from T and F. If they are doing rhetoric, then the T & F need to come into play. Kleene also agrees, at least
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We had a conflict edit, so there is a lot more here now. I basically agree with what you've written. I will cut the "brainstorm" to reduce the clutter. I am evolving in the direction you are proposing. I am suffering from some confusions and frustrations: in particular I don't like
Enderton's 2002
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RE the "algebra" versus the "valuation": my understanding here is the "algebra" is the substitution rules that I have been trying to "axiomatize" with the inductive definitions, plus the reduction-rules such as De Morgan etc. Plus the notions of "distributive law" and "associative law" etc.
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with regards to "metamathematical consistency proofs for the calculus" (1952:125). My practical/engineering experience with {T, F} has been very, very poor. In practice, many folks use "reverse logic" and make a mess of T & F. Also, you cannot "compute" with it using what I would call
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I'm not sure what this means: "The use of symbols { T, F } is discouraged in mathematics because of the possibility of confusion with "Truth" and "Falsity",..." The symbols T and F are usually used in mathematics, since truth and falsity is exactly what they are supposed to
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RE: "term". I wouldn't have a problem with just using "formula" excepting engineers and computer scientists use the notion "term" for a conjunction of variables, i.e. (p & ~q & r) is a "term". In particular
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here, but we should be able to give a brief overview of the properties of formulas, mention truth assignments, etc. It will take a few days, and the rough draft I have needs a lot of wikilinks to be added. — Carl
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etc. It's an alternate universe to mathematics. So perhaps I can define "formula" inductively and then define "term" as an alternate usage for (a & b) or (a & b & ... & z). There's also an
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RE: NOT, OR, AND: I recommend we use NOT, OR, AND and define IMPLY, XOR in terms of these others. My thought was to put in the big table as a summary of the definitions of the operators, and be done with
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Some of them them group AND before OR, others the other way round. And then one example goes from "a & a → b" to "a & (a → b)" despite the ordering giving IMPLICATION a higher rank than AND.
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There is lot of nonsense in the sections mentioning "engineering"; you probably spotted some of them. I guess "engineering connectives" is supposed to mean terminology for logical connectives used in
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The mentioned example of a simple proposition (Five is greater than three.) includes implicitly a predicate letter G (greater than) which also has another (arithmetical) notation ": -->
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The section "Connective seniority (symbol rank)" appears to contradict (significantly) pgs 3 and 5 of the book, "Foundations Of
Computation" by Critchlow and Eck (2nd ed, 2011)
384:. I think sticking with that convention is fine. In any case the article will discuss how other connectives, that take arbitrary finite numbers of formulas, can be added.
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There is also a sentence with dubious status like "Engineers try to avoid the notions of truth and falsity that... the philosophers.". This sentence should be removed.--
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Simple propositions like "It rains tomorrow/in the next hours." can have a variabile truth value, not known at the present moment. Thus simple propositions can also be
391:. Nowadays that word is usually reserved for certain expression in first order logic; there are no more terms in propositional logic. Everything is a formula. — Carl
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To me, both rankings seem to be almost the same, except that they are presented in orders reverse to each other, (and that and/or are ranked differently). -
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A propositional formula, like any expression, can be used to define a function. However, this requires to fix the parameters and their order. For example,
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In summary, Critchlow and Eck (2nd ed, 2011) give the relative precedences of the logical operators, from highest to lowest, as follows:
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RE parsing, normal forms etc. I agree. A lot needs to be said about such things as ((A & B) & C) = (A & B & C)
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Very brief summary style discussion of valuations, just enough to give background for the rest of the article. Link to
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Propositional logic doesn't consider relation symbols; your analysis of "Five is greater than three" would presuppose
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is used in certain fields. But in any case it isn't necessary to define terms in order to merely define
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Boolean logic, i.e. the logic of Boole: ~a = (1-a), a & B = a * b, a V b = (1 - (1-a)*(1-b)), etc.
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This article was woefully short, so I am going to work on expanding it. I don't want to repeat all of
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with arguments propositional variables or propositional constants (simple propositions).
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like p, q. They have the same notation, with small letters.--
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The development has become too unwieldy so I've moved it to
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I agree. I changed above what I had written before. Bill
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And then pg 5 adds more operators to the above, stating:
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