4766:
1515:, so this second approach is the same as only studying interpretations that happen to be normal models. The advantage of this approach is that the axioms related to equality are automatically satisfied by every normal model, and so they do not need to be explicitly included in first-order theories when equality is treated this way. This second approach is sometimes called
2495:
2418:
1547:
of variables. The idea is different sorts of variables represent different types of objects. Every sort of variable can be quantified; thus an interpretation for a many-sorted language has a separate domain for each of the sorts of variables to range over (there is an infinite collection of variables
1458:
fails in any structure with an empty domain. Thus the proof theory of first-order logic becomes more complicated when empty structures are permitted. However, the gain in allowing them is negligible, as both the intended interpretations and the interesting interpretations of the theories people study
899:
Given any truth assignment for a set of propositional symbols, there is a unique extension to an interpretation for all the propositional formulas built up from those variables. This extended interpretation is defined inductively, using the truth-table definitions of the logical connectives discussed
1595:
looks much the same as a formal language for first-order logic. The difference is that there are now many different types of variables. Some variables correspond to elements of the domain, as in first-order logic. Other variables correspond to objects of higher type: subsets of the domain, functions
1462:
Empty relations do not cause any problem for first-order interpretations, because there is no similar notion of passing a relation symbol across a logical connective, enlarging its scope in the process. Thus it is acceptable for relation symbols to be interpreted as being identically false. However,
1139:
is an element of the domain. There are two ways of handling this technical issue. The first is to pass to a larger language in which each element of the domain is named by a constant symbol. The second is to add to the interpretation a function that assigns each variable to an element of the domain.
1668:
Many formal languages are associated with a particular interpretation that is used to motivate them. For example, the first-order signature for set theory includes only one binary relation, ∈, which is intended to represent set membership, and the domain of discourse in a first-order theory of the
1603:
require that, once the domain of discourse is satisfied, the higher-order variables range over all possible elements of the correct type (all subsets of the domain, all functions from the domain to itself, etc.). Thus the specification of a full interpretation is the same as the specification of a
947:
Given a signature σ, the corresponding formal language is known as the set of σ-formulas. Each σ-formula is built up out of atomic formulas by means of logical connectives; atomic formulas are built from terms using predicate symbols. The formal definition of the set of σ-formulas proceeds in the
582:
So under a given interpretation of all the sentence letters Φ and Ψ (i.e., after assigning a truth-value to each sentence letter), we can determine the truth-values of all formulas that have them as constituents, as a function of the logical connectives. The following table shows how this kind of
948:
other direction: first, terms are assembled from the constant and function symbols together with the variables. Then, terms can be combined into an atomic formula using a predicate symbol (relation symbol) from the signature or the special predicate symbol "=" for equality (see the section "
1474:
The first approach is to treat equality as no different than any other binary relation. In this case, if an equality symbol is included in the signature, it is usually necessary to add various axioms about equality to axiom systems (for example, the substitution axiom saying that if
515:, no sentence can be made both true and false by the same interpretation, although this is not true of glut logics such as LP. Even in classical logic, however, it is possible that the truth value of the same sentence can be different under different interpretations. A sentence is
550:
The truth-functional connectives enable compound sentences to be built up from simpler sentences. In this way, the truth value of the compound sentence is defined as a certain truth function of the truth values of the simpler sentences. The connectives are usually taken to be
2343:
1530:
interpretation on a subset of the original domain. Thus there is little additional generality in studying non-normal models. Second, if non-normal models are considered, then every consistent theory has an infinite model; this affects the statements of results such as the
583:
thing looks. The first two columns show the truth-values of the sentence letters as determined by the four possible interpretations. The other columns show the truth-values of formulas built from these sentence letters, with truth-values determined recursively.
485:
Logical constants are always given the same meaning by every interpretation of the standard kind, so that only the meanings of the non-logical symbols are changed. Logical constants include quantifier symbols ∀ ("all") and ∃ ("some"), symbols for
2280:
2426:
2349:
1608:, which are essentially multi-sorted first-order semantics, require the interpretation to specify a separate domain for each type of higher-order variable to range over. Thus an interpretation in Henkin semantics includes a domain
1335:
1510:
The second approach is to treat the equality relation symbol as a logical constant that must be interpreted by the real equality relation in any interpretation. An interpretation that interprets equality this way is known as a
2179:
908:
Unlike propositional logic, where every language is the same apart from a choice of a different set of propositional variables, there are many different first-order languages. Each first-order language is defined by a
924:
is also assigned. The alphabet for the formal language consists of logical constants, the equality relation symbol =, all the symbols from the signature, and an additional infinite set of symbols known as variables.
801:
in a formal language for propositional logic are the propositional symbols, which are often denoted by capital letters. To make the formal language precise, a specific set of propositional symbols must be fixed.
1341:
is not a free variable of φ, are logically valid. This equivalence holds in every interpretation with a nonempty domain, but does not always hold when empty domains are permitted. For example, the equivalence
1632:
The interpretations of propositional logic and predicate logic described above are not the only possible interpretations. In particular, there are other types of interpretations that are used in the study of
1147:
in first-order logic, which must then also be interpreted. A propositional variable can stand on its own as an atomic formula. The interpretation of a propositional variable is one of the two truth values
2286:
1994:
773:
correspondence between certain elementary statements of the theory, and certain statements related to the subject matter. If every elementary statement in the theory has a correspondent it is called a
1853:
1272:
As stated above, a first-order interpretation is usually required to specify a nonempty set as the domain of discourse. The reason for this requirement is to guarantee that equivalences such as
1162:, they do not associate each predicate symbol with a property (or relation), but rather with the extension of that property (or relation). In other words, these first-order interpretations are
1906:(or in other contexts: a non-intended arbitrary interpretation used to clarify such an intended factually-true descriptive interpretation.) All models are interpretations that have the same
259:
547:
that represent truth functions — functions that take truth values as arguments and return truth values as outputs (in other words, these are operations on truth values of sentences).
436:
1140:
Then the T-schema can quantify over variations of the original interpretation in which this variable assignment function is changed, instead of quantifying over substitution instances.
1053:, which is a definition of first-order semantics developed by Alfred Tarski. The T-schema interprets the logical connectives using truth tables, as discussed above. Thus, for example,
1522:
There are a few other reasons to restrict study of first-order logic to normal models. First, it is known that any first-order interpretation in which equality is interpreted by an
2134:
2094:
2062:
2030:
1956:
1596:
from the domain, functions that take a subset of the domain and return a function from the domain to subsets of the domain, etc. All of these types of variables can be quantified.
1805:
1770:
2755:
The extension of a property (also called an attribute) is a set of individuals, so a property is a unary relation. E.g. The properties "yellow" and "prime" are unary relations.
1456:
3145:
1254:
1200:
460:
370:
283:
221:
2223:
1548:
of each of the different sorts). Function and relation symbols, in addition to having arities, are specified so that each of their arguments must come from a certain sort.
498:
Many of the commonly studied interpretations associate each sentence in a formal language with a single truth value, either True or False. These interpretations are called
2235:
1275:
177:
over which the language is defined. To distinguish the strings of symbols that are in a formal language from arbitrary strings of symbols, the former are sometimes called
390:
323:
303:
932:, there are constant symbols 0 and 1, two binary function symbols + and ·, and no binary relation symbols. (Here the equality relation is taken as a logical constant.)
3820:
410:
343:
470:
In the specific cases of propositional logic and predicate logic, the formal languages considered have alphabets that are divided into two sets: the logical symbols (
2199:
112:
is to stand for tall and 'a' for
Abraham Lincoln. Nor does logical interpretation have anything to say about logical connectives like 'and', 'or' and 'not'. Though
1873:
2490:{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare \ \blacksquare }
2413:{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare \ \blacksquare }
1555:. There are two sorts; points and lines. There is an equality relation symbol for points, an equality relation symbol for lines, and a binary incidence relation
3903:
3044:
913:. The signature consists of a set of non-logical symbols and an identification of each of these symbols as either a constant symbol, a function symbol, or a
183:(wff). The essential feature of a formal language is that its syntax can be defined without reference to interpretation. For example, we can determine that (
502:; they include the usual interpretations of propositional and first-order logic. The sentences that are made true by a particular assignment are said to be
2497:" can be interpreted as meaning "One plus three equals four." A different interpretation would be to read it backwards as "Four minus three equals one."
1499:) holds as well). This approach to equality is most useful when studying signatures that do not include the equality relation, such as the signature for
1559:
which takes one point variable and one line variable. The intended interpretation of this language has the point variables range over all points on the
69:
terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called
939:, as consisting of individual symbols a, b, and c; predicate symbols F, G, H, I and J; variables x, y, z; no function letters; no sentential symbols.
2143:
555:, meaning that the meaning of the connectives is always the same, independent of what interpretations are given to the other symbols in a formula.
4217:
2505:
There are other uses of the term "interpretation" that are commonly used, which do not refer to the assignment of meanings to formal languages.
735:
is True. Now the only other possible interpretation of Φ makes it False, and if so, ¬Φ is made True by the negation function. That would make
4375:
2955:
2895:
4870:
3163:
4946:
4230:
3553:
1045:
The information specified in the interpretation provides enough information to give a truth value to any atomic formula, after each of its
4802:
2609:
1049:, if any, has been replaced by an element of the domain. The truth value of an arbitrary sentence is then defined inductively using the
3815:
4235:
4225:
3962:
3168:
2600:
4865:
3713:
3159:
4910:
4371:
2927:
2741:
1471:
The equality relation is often treated specially in first order logic and other predicate logics. There are two general approaches.
2338:{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare }
727:: (Φ ∨ ¬Φ). If our interpretation function makes Φ True, then ¬Φ is made False by the negation connective. Since the disjunct Φ of
4468:
4212:
3037:
4895:
3773:
3466:
3207:
2629:
1961:
1532:
952:
below). Finally, the formulas of the language are assembled from atomic formulas using the logical connectives and quantifiers.
4956:
31:
1810:
529:
if it is satisfied by every interpretation (if φ is satisfied by every interpretation that satisfies ψ then φ is said to be a
4729:
4431:
4194:
4189:
4014:
3435:
3119:
2788:
1035:
910:
4724:
4507:
4424:
4137:
4068:
3945:
3187:
2779:
2619:
2553:
1734:
124:
96:
of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate
3795:
88:
analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a
4649:
4475:
4161:
3394:
490:∧ ("and"), ∨ ("or"), ¬ ("not"), parentheses and other grouping symbols, and (in many treatments) the equality symbol =.
62:
4920:
3800:
116:
may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.
4132:
3871:
3129:
3030:
2765:
1345:
760:
173:
128:
93:
4915:
4527:
4522:
226:
805:
The standard kind of interpretation in this setting is a function that maps each propositional symbol to one of the
4853:
4456:
4046:
3440:
3408:
3099:
2971:
3173:
415:
4941:
4746:
4695:
4592:
4090:
4051:
3528:
4838:
4587:
3202:
793:
consists of formulas built up from propositional symbols (also called sentential symbols, sentential variables,
4517:
4056:
3908:
3891:
3614:
3094:
817:
function. In many presentations, it is literally a truth value that is assigned, but some presentations assign
104:} (for "Abraham Lincoln"). All our interpretation does is assign the extension {a} to the non-logical constant
70:
2948:
The
Concept and the Role of the Model in Mathematics and Natural and Social Sciences (Colloquium proceedings)
4858:
4795:
4419:
4396:
4357:
4243:
4184:
3830:
3750:
3594:
3538:
3151:
2729:
2614:
2568:
1527:
1504:
1463:
the interpretation of a function symbol must always assign a well-defined and total function to the symbol.
961:
828:
distinct propositional variables there are 2 distinct possible interpretations. For any particular variable
4709:
4436:
4381:
4274:
4120:
4105:
4078:
4029:
3913:
3848:
3673:
3639:
3634:
3508:
3339:
3316:
2202:
1507:
in which there is only an equality relation for numbers, but not an equality relation for set of numbers.
1167:
1163:
1144:
794:
89:
4951:
4639:
4492:
4284:
4002:
3738:
3644:
3503:
3488:
3369:
3344:
1779:
1744:
1720:
4765:
1645:
2107:
2067:
2035:
2003:
1929:
1684:, it consists of the natural numbers with their ordinary arithmetical operations. All models that are
4885:
4880:
4832:
4612:
4574:
4451:
4255:
4095:
4019:
3997:
3825:
3783:
3682:
3649:
3513:
3301:
3212:
2275:{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare }
1915:
1876:
1723:
1708:
1649:
1638:
1523:
132:
42:
1235:
1181:
966:
To ascribe meaning to all sentences of a first-order language, the following information is needed.
441:
351:
264:
202:
4826:
4741:
4632:
4617:
4597:
4554:
4441:
4391:
4317:
4262:
4199:
3992:
3987:
3935:
3703:
3692:
3364:
3264:
3192:
3183:
3179:
3114:
3109:
2208:
1907:
1634:
971:
790:
531:
179:
159:
77:
4961:
4788:
4770:
4539:
4502:
4487:
4480:
4463:
4249:
4115:
4041:
4024:
3977:
3790:
3699:
3533:
3518:
3478:
3430:
3415:
3403:
3359:
3334:
3104:
3053:
2869:
2853:
2816:
2800:
1887:
1738:
1716:
1624:, etc. The relationship between these two semantics is an important topic in higher order logic.
1592:
1552:
1085:
929:
798:
544:
487:
4267:
3723:
375:
308:
288:
4705:
4512:
4322:
4312:
4204:
4085:
3920:
3896:
3677:
3661:
3566:
3543:
3420:
3389:
3354:
3249:
3084:
2951:
2923:
2891:
2737:
1891:
743:
s disjuncts, ¬Φ, would be true under this interpretation. Since these two interpretations for
395:
328:
4848:
4719:
4714:
4607:
4564:
4386:
4347:
4342:
4327:
4153:
4110:
4007:
3805:
3755:
3329:
3291:
2943:
2845:
2792:
2733:
2722:
2690:
2595:
1911:
1886:
Most formal systems have many more models than they were intended to have (the existence of
1712:
1700:
1681:
1677:
914:
552:
471:
2865:
2812:
2184:
4815:
4700:
4690:
4644:
4627:
4582:
4544:
4446:
4366:
4173:
4100:
4073:
4061:
3967:
3881:
3855:
3810:
3778:
3579:
3381:
3324:
3274:
3239:
3197:
3012:
2861:
2808:
1858:
1560:
512:
144:
81:
50:
46:
1699:
While the intended interpretation can have no explicit indication in the strictly formal
1535:, which are usually stated under the assumption that only normal models are considered.
1519:, but many authors adopt it for the general study of first-order logic without comment.
4905:
4685:
4664:
4622:
4602:
4497:
4352:
3950:
3940:
3930:
3925:
3859:
3733:
3609:
3498:
3493:
3471:
3072:
2992:
2624:
1704:
918:
17:
4935:
4659:
4337:
3844:
3629:
3619:
3589:
3574:
3244:
3017:
2916:
2911:
2833:
2685:
2668:
1330:{\displaystyle (\phi \lor \exists x\psi )\leftrightarrow \exists x(\phi \lor \psi ),}
1213:
Individual constants: a: The white King, b: The black Queen, c: The white King's pawn
1046:
2873:
2820:
2032:
which is at least 6 symbols long, and which is not infinitely long, is a formula of
478:
symbols have the same meaning regardless of the subject matter being studied, while
4559:
4406:
4307:
4299:
4179:
4127:
4036:
3972:
3955:
3886:
3745:
3604:
3306:
3089:
3007:
2997:
2717:
2639:
2604:
2509:
2101:
1880:
1773:
1693:
1689:
1653:
2777:
Hailperin, Theodore (1953), "Quantification theory and empty individual-domains",
1563:, the line variable range over all lines on the plane, and the incidence relation
1599:
There are two kinds of interpretations commonly employed for higher-order logic.
1135:) mentioned above is not a formula in the original formal language of φ, because
127:
in a language. If a given interpretation assigns the value True to a sentence or
4900:
4843:
4780:
4669:
4549:
3728:
3718:
3665:
3349:
3269:
3254:
3134:
3079:
2644:
2634:
1657:
818:
806:
770:
517:
504:
120:
85:
54:
3002:
2174:{\displaystyle \blacksquare \ \bigstar \ast \blacklozenge \ \blacksquare \ast }
723:
Now it is easier to see what makes a formula logically valid. Take the formula
3599:
3454:
3425:
3231:
1685:
1500:
1159:
1100:
is true under an interpretation exactly when every substitution instance of φ(
1688:
to the one just given are also called standard; these models all satisfy the
1543:
A generalization of first order logic considers languages with more than one
769:
is the relationship between a theory and some subject matter when there is a
191:) is a well-formed formula even without knowing whether it is true or false.
4875:
4811:
4751:
4654:
3707:
3624:
3584:
3548:
3484:
3296:
3286:
3259:
751:
comes out True for both, we say that it is logically valid or tautologous.
2976:
Metalogic: An
Introduction to the Metatheory of Standard First Order Logic
2886:
Roland Müller (2009). "The Notion of a Model". In
Anthonie Meijers (ed.).
2540:
and these functions and relations. In some settings, it is not the domain
4736:
4534:
3982:
3687:
3281:
1719:
are chosen so that their counterparts in the intended interpretation are
1526:
and satisfies the substitution axioms for equality can be cut down to an
1050:
119:
An interpretation often (but not always) provides a way to determine the
474:) and the non-logical symbols. The idea behind this terminology is that
4332:
3124:
2857:
2804:
1895:
543:
Some of the logical symbols of a language (other than quantifiers) are
1158:
Because the first-order interpretations described here are defined in
3022:
1108:
is replaced by some element of the domain, is satisfied. The formula
66:
2849:
2796:
1694:
non-standard models of the (first-order version of the) Peano axioms
3876:
3222:
3067:
2649:
1731:
1727:
1260:
the following are true sentences: F(a), G(c), H(b), I(a), J(b, c),
921:
558:
This is how we define logical connectives in propositional logic:
482:
symbols change in meaning depending on the area of investigation.
58:
1696:, which contain elements not correlated with any natural number.
1131:
Strictly speaking, a substitution instance such as the formula φ(
521:
if it is true under at least one interpretation; otherwise it is
171:. The inventory from which these letters are taken is called the
2890:. Handbook of the Philosophy of Science. Vol. 9. Elsevier.
4784:
3026:
1060:
This leaves the issue of how to interpret formulas of the form
575:(Φ → Ψ) is True iff ¬Φ is True or Ψ is True (or both are True).
1669:
natural numbers is intended to be the set of natural numbers.
572:(Φ ∨ Ψ) is True iff Φ is True or Ψ is True (or both are True).
563:
856:
there are 2=4 possible interpretations: 1) both are assigned
2117:
2113:
2077:
2073:
2045:
2041:
2013:
2009:
1939:
1935:
1834:
1817:
1786:
1751:
1241:
1187:
447:
357:
270:
208:
1989:{\displaystyle \{\blacksquare ,\bigstar ,\blacklozenge \}}
1644:
Interpretations used to study non-classical logic include
832:, for example, there are 2=2 possible interpretations: 1)
747:
are the only possible logical interpretations, and since
149:
A formal language consists of a possibly infinite set of
27:
Assignment of meaning to the symbols of a formal language
1848:{\displaystyle {\mathcal {I}}_{i}\to {\mathcal {I}}_{j}}
1057:
is satisfied if and only if both φ and ψ are satisfied.
578:(Φ ↔ Ψ) is True iff (Φ → Ψ) is True and (Ψ → Φ) is True.
2673:
1715:
must permit expression of the concepts to be modeled;
494:
General properties of truth-functional interpretations
2429:
2352:
2289:
2238:
2211:
2187:
2146:
2110:
2070:
2038:
2006:
1964:
1932:
1926:
Given a simple formal system (we shall call this one
1861:
1813:
1782:
1747:
1348:
1278:
1238:
1184:
1088:
for these quantifiers. The idea is that the sentence
444:
418:
398:
378:
354:
331:
311:
291:
267:
229:
205:
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3843:
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3380:
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3230:
3221:
3143:
3060:
2918:
Introduction to
Symbolic Logic and its Applications
917:. In the case of function and predicate symbols, a
2915:
2721:
2689:
2489:
2412:
2337:
2274:
2217:
2193:
2173:
2128:
2088:
2056:
2024:
1988:
1958:) whose alphabet α consists only of three symbols
1950:
1904:intended factually-true descriptive interpretation
1867:
1847:
1799:
1764:
1450:
1329:
1248:
1194:
1034:An object carrying this information is known as a
454:
430:
404:
384:
364:
337:
317:
297:
277:
253:
215:
2993:Stanford Enc. Phil: Classical Logic, 4. Semantics
2888:Philosophy of technology and engineering sciences
1263:the following are false sentences: J(a, c), G(a).
1898:to be a model of our science, to speak about an
1890:is an example). When we speak about 'models' in
2836:(1954), "Quantification and the empty domain",
1879:, as usual. These requirements ensure that all
1551:One example of many-sorted logic is for planar
977:, usually required to be non-empty (see below).
254:{\displaystyle \alpha =\{\triangle ,\square \}}
2844:(3), Association for Symbolic Logic: 177–179,
2229:A formal proof can be constructed as follows:
1120:is satisfied if there is at least one element
935:Again, we might define a first-order language
438:would denote 101 under this interpretation of
4796:
3038:
3013:mathworld.wolfram.com: Propositional Calculus
1042:-structure (of language L), or as a "model".
8:
2707:Sometimes called the "universe of discourse"
2548:modulo an equivalence relation definable in
1983:
1965:
1026:as its interpretation (that is, a subset of
809:true and false. This function is known as a
569:(Φ ∧ Ψ) is True iff Φ is True and Ψ is True.
431:{\displaystyle \triangle \square \triangle }
248:
236:
76:The most commonly studied formal logics are
2536:is isomorphic to the structure with domain
2528:, and definable relations and functions on
1003:as its interpretation (that is, a function
4803:
4789:
4781:
3864:
3459:
3227:
3045:
3031:
3023:
1996:and whose formation rule for formulas is:
1902:. A model in the empirical sciences is an
1672:The intended interpretation is called the
949:
108:, and does not make a claim about whether
100:(for "tall") and assign it the extension {
2428:
2351:
2288:
2237:
2210:
2186:
2145:
2112:
2111:
2109:
2072:
2071:
2069:
2040:
2039:
2037:
2008:
2007:
2005:
1963:
1934:
1933:
1931:
1860:
1839:
1833:
1832:
1822:
1816:
1815:
1812:
1791:
1785:
1784:
1781:
1756:
1750:
1749:
1746:
1703:, it naturally affects the choice of the
1347:
1277:
1240:
1239:
1237:
1186:
1185:
1183:
980:For every constant symbol, an element of
956:Interpretations of a first-order language
446:
445:
443:
417:
397:
377:
356:
355:
353:
330:
310:
290:
269:
268:
266:
228:
207:
206:
204:
3018:mathworld.wolfram.com: First Order Logic
1711:of the syntactical system. For example,
585:
2661:
2423:In this example the theorem produced "
1174:Example of a first-order interpretation
785:Interpretations for propositional logic
1855:turns out to be a true sentence, with
943:Formal languages for first-order logic
372:could assign the decimal digit '1' to
305:and is composed solely of the symbols
3008:mathworld.wolfram.com: Interpretation
2998:mathworld.wolfram.com: FormalLanguage
1660:is also studied using Kripke models.
1038:(of signature σ), or σ-structure, or
7:
2563:is said to interpret another theory
1883:sentences also come out to be true.
1641:), and in the study of modal logic.
1084:. The domain of discourse forms the
797:) and logical connectives. The only
2675:2nd ed. Cambridge University Press.
2610:Formal semantics (natural language)
1741:must be such that, if the sentence
731:is True under that interpretation,
2552:. For additional information, see
2205:standing for a finite string of "
1800:{\displaystyle {\mathcal {I}}_{i}}
1765:{\displaystyle {\mathcal {I}}_{j}}
1412:
1403:
1376:
1352:
1303:
1288:
425:
419:
379:
312:
292:
239:
25:
3003:mathworld.wolfram.com: Connective
2978:. University of California Press.
2692:Foundations of Mathematical Logic
2516:is said to interpret a structure
1616:, a collection of functions from
223:can be defined with the alphabet
131:, the interpretation is called a
4764:
2922:. New York: Dover publications.
2724:Elementary Logic, Second Edition
2501:Other concepts of interpretation
2129:{\displaystyle {\mathcal {FS'}}}
2089:{\displaystyle {\mathcal {FS'}}}
2057:{\displaystyle {\mathcal {FS'}}}
2025:{\displaystyle {\mathcal {FS'}}}
1951:{\displaystyle {\mathcal {FS'}}}
1451:{\displaystyle \equiv \exists x}
928:For example, in the language of
53:. Many formal languages used in
4871:Gödel's incompleteness theorems
2520:if there is a definable subset
2064:. Nothing else is a formula of
1910:as the intended one, but other
1517:first order logic with equality
1206:described above is as follows.
32:Interpretation (disambiguation)
2789:Association for Symbolic Logic
1862:
1828:
1445:
1430:
1418:
1409:
1397:
1394:
1382:
1370:
1358:
1349:
1321:
1309:
1300:
1297:
1279:
1249:{\displaystyle {\mathcal {I}}}
1195:{\displaystyle {\mathcal {I}}}
455:{\displaystyle {\mathcal {W}}}
365:{\displaystyle {\mathcal {W}}}
278:{\displaystyle {\mathcal {W}}}
216:{\displaystyle {\mathcal {W}}}
1:
4725:History of mathematical logic
2838:The Journal of Symbolic Logic
2780:The Journal of Symbolic Logic
2620:Interpretation (model theory)
2554:Interpretation (model theory)
2218:{\displaystyle \blacksquare }
1628:Non-classical interpretations
1612:, a collection of subsets of
1587:Higher-order predicate logics
1575:) holds if and only if point
1539:Many-sorted first-order logic
1178:An example of interpretation
525:. A sentence φ is said to be
348:A possible interpretation of
4866:Gödel's completeness theorem
4650:Primitive recursive function
1680:in 1960). In the context of
1604:first-order interpretation.
1593:higher-order predicate logic
1268:Non-empty domain requirement
545:truth-functional connectives
163:) built from a fixed set of
135:of that sentence or theory.
63:theoretical computer science
4947:Interpretation (philosophy)
2766:Extension (predicate logic)
777:, otherwise it is called a
761:Theory (mathematical logic)
261:, and with a word being in
4978:
4854:Foundations of mathematics
3714:Schröder–Bernstein theorem
3441:Monadic predicate calculus
3100:Foundations of mathematics
2000:'Any string of symbols of
1124:of the domain such that φ(
1018:-ary predicate symbol, an
959:
767:interpretation of a theory
758:
755:Interpretation of a theory
385:{\displaystyle \triangle }
318:{\displaystyle \triangle }
298:{\displaystyle \triangle }
142:
29:
4822:
4760:
4747:Philosophy of mathematics
4696:Automated theorem proving
3867:
3821:Von Neumann–Bernays–Gödel
3462:
2544:that is used, but rather
991:-ary function symbol, an
739:True again, since one of
4896:Löwenheim–Skolem theorem
2630:Löwenheim–Skolem theorem
2569:extension by definitions
1664:Intended interpretations
1533:Löwenheim–Skolem theorem
1459:have non-empty domains.
1228:J(x, y): x can capture y
1143:Some authors also admit
789:The formal language for
405:{\displaystyle \square }
338:{\displaystyle \square }
4921:Use–mention distinction
4397:Self-verifying theories
4218:Tarski's axiomatization
3169:Tarski's undefinability
3164:incompleteness theorems
2730:Oxford University Press
2615:Herbrand interpretation
1737:in the interpretation;
1528:elementarily equivalent
1505:second-order arithmetic
1145:propositional variables
962:Interpretation function
860:, 2) both are assigned
795:propositional variables
4957:Philosophy of language
4916:Type–token distinction
4771:Mathematics portal
4382:Proof of impossibility
4030:propositional variable
3340:Propositional calculus
2491:
2414:
2339:
2276:
2219:
2203:metasyntactic variable
2195:
2175:
2130:
2090:
2058:
2026:
1990:
1952:
1894:, we mean, if we want
1869:
1849:
1801:
1766:
1676:(a term introduced by
1591:A formal language for
1452:
1331:
1250:
1232:In the interpretation
1196:
984:as its interpretation.
950:Interpreting equality"
779:partial interpretation
456:
432:
406:
386:
366:
339:
319:
299:
279:
255:
217:
65:are defined in solely
18:Logical interpretation
4640:Kolmogorov complexity
4593:Computably enumerable
4493:Model complete theory
4285:Principia Mathematica
3345:Propositional formula
3174:Banach–Tarski paradox
2567:if there is a finite
2492:
2415:
2340:
2277:
2220:
2196:
2194:{\displaystyle \ast }
2176:
2131:
2091:
2059:
2027:
1991:
1953:
1916:non-logical constants
1870:
1850:
1802:
1767:
1724:declarative sentences
1650:Boolean-valued models
1503:or the signature for
1467:Interpreting equality
1453:
1332:
1251:
1197:
457:
433:
407:
387:
367:
340:
320:
300:
280:
256:
218:
4839:Church–Turing thesis
4833:Entscheidungsproblem
4588:Church–Turing thesis
4575:Computability theory
3784:continuum hypothesis
3302:Square of opposition
3160:Gödel's completeness
2427:
2350:
2287:
2236:
2209:
2185:
2144:
2108:
2068:
2036:
2004:
1962:
1930:
1868:{\displaystyle \to }
1859:
1811:
1780:
1745:
1730:need to come out as
1709:transformation rules
1639:intuitionistic logic
1524:equivalence relation
1346:
1276:
1236:
1182:
824:For a language with
587:Logical connectives
508:by that assignment.
442:
416:
396:
376:
352:
329:
309:
289:
265:
227:
203:
41:is an assignment of
30:For other uses, see
4742:Mathematical object
4633:P versus NP problem
4598:Computable function
4392:Reverse mathematics
4318:Logical consequence
4195:primitive recursive
4190:elementary function
3963:Free/bound variable
3816:Tarski–Grothendieck
3335:Logical connectives
3265:Logical equivalence
3115:Logical consequence
1908:domain of discourse
1888:non-standard models
1728:primitive sentences
1717:sentential formulas
1635:non-classical logic
1210:Domain: A chess set
995:-ary function from
972:domain of discourse
799:non-logical symbols
791:propositional logic
775:full interpretation
588:
539:Logical connectives
532:logical consequence
488:logical connectives
180:well-formed formulæ
78:propositional logic
4540:Transfer principle
4503:Semantics of logic
4488:Categorical theory
4464:Non-standard model
3978:Logical connective
3105:Information theory
3054:Mathematical logic
2946:, ed. (Jan 1960).
2487:
2410:
2335:
2272:
2215:
2191:
2171:
2126:
2086:
2054:
2022:
1986:
1948:
1892:empirical sciences
1865:
1845:
1797:
1762:
1739:rules of inference
1646:topological models
1553:Euclidean geometry
1448:
1327:
1246:
1216:F(x): x is a piece
1192:
586:
452:
428:
402:
382:
362:
335:
315:
295:
285:if it begins with
275:
251:
213:
199:A formal language
153:(variously called
92:that provides the
4929:
4928:
4778:
4777:
4710:Abstract category
4513:Theories of truth
4323:Rule of inference
4313:Natural deduction
4294:
4293:
3839:
3838:
3544:Cartesian product
3449:
3448:
3355:Many-valued logic
3330:Boolean functions
3213:Russell's paradox
3188:diagonal argument
3085:First-order logic
2957:978-94-010-3669-6
2897:978-0-444-51667-1
2483:
2477:
2471:
2465:
2459:
2453:
2447:
2441:
2435:
2406:
2400:
2394:
2388:
2382:
2376:
2370:
2364:
2358:
2331:
2325:
2319:
2313:
2307:
2301:
2295:
2268:
2262:
2256:
2250:
2244:
2164:
2152:
1701:syntactical rules
1692:. There are also
1219:G(x): x is a pawn
1022:-ary relation on
904:First-order logic
721:
720:
553:logical constants
472:logical constants
466:Logical constants
16:(Redirected from
4969:
4942:Formal languages
4849:Effective method
4827:Cantor's theorem
4805:
4798:
4791:
4782:
4769:
4768:
4720:History of logic
4715:Category of sets
4608:Decision problem
4387:Ordinal analysis
4328:Sequent calculus
4226:Boolean algebras
4166:
4165:
4140:
4111:logical/constant
3865:
3851:
3774:Zermelo–Fraenkel
3525:Set operations:
3460:
3397:
3228:
3208:Löwenheim–Skolem
3095:Formal semantics
3047:
3040:
3033:
3024:
2980:
2979:
2968:
2962:
2961:
2944:Hans Freudenthal
2940:
2934:
2933:
2921:
2908:
2902:
2901:
2883:
2877:
2876:
2830:
2824:
2823:
2774:
2768:
2762:
2756:
2753:
2747:
2746:
2727:
2714:
2708:
2705:
2699:
2697:
2695:
2682:
2676:
2666:
2596:Conceptual model
2582:is contained in
2496:
2494:
2493:
2488:
2481:
2475:
2469:
2463:
2457:
2451:
2445:
2439:
2433:
2419:
2417:
2416:
2411:
2404:
2398:
2392:
2386:
2380:
2374:
2368:
2362:
2356:
2344:
2342:
2341:
2336:
2329:
2323:
2317:
2311:
2305:
2299:
2293:
2281:
2279:
2278:
2273:
2266:
2260:
2254:
2248:
2242:
2224:
2222:
2221:
2216:
2200:
2198:
2197:
2192:
2180:
2178:
2177:
2172:
2162:
2150:
2135:
2133:
2132:
2127:
2125:
2124:
2123:
2095:
2093:
2092:
2087:
2085:
2084:
2083:
2063:
2061:
2060:
2055:
2053:
2052:
2051:
2031:
2029:
2028:
2023:
2021:
2020:
2019:
1995:
1993:
1992:
1987:
1957:
1955:
1954:
1949:
1947:
1946:
1945:
1874:
1872:
1871:
1866:
1854:
1852:
1851:
1846:
1844:
1843:
1838:
1837:
1827:
1826:
1821:
1820:
1806:
1804:
1803:
1798:
1796:
1795:
1790:
1789:
1776:from a sentence
1771:
1769:
1768:
1763:
1761:
1760:
1755:
1754:
1682:Peano arithmetic
1678:Abraham Robinson
1606:Henkin semantics
1457:
1455:
1454:
1449:
1336:
1334:
1333:
1328:
1255:
1253:
1252:
1247:
1245:
1244:
1225:I(x): x is white
1222:H(x): x is black
1202:of the language
1201:
1199:
1198:
1193:
1191:
1190:
1128:) is satisfied.
1119:
1099:
1083:
1071:
1056:
915:predicate symbol
811:truth assignment
589:
500:truth functional
461:
459:
458:
453:
451:
450:
437:
435:
434:
429:
411:
409:
408:
403:
391:
389:
388:
383:
371:
369:
368:
363:
361:
360:
344:
342:
341:
336:
324:
322:
321:
316:
304:
302:
301:
296:
284:
282:
281:
276:
274:
273:
260:
258:
257:
252:
222:
220:
219:
214:
212:
211:
139:Formal languages
71:formal semantics
21:
4977:
4976:
4972:
4971:
4970:
4968:
4967:
4966:
4932:
4931:
4930:
4925:
4818:
4816:metamathematics
4809:
4779:
4774:
4763:
4756:
4701:Category theory
4691:Algebraic logic
4674:
4645:Lambda calculus
4583:Church encoding
4569:
4545:Truth predicate
4401:
4367:Complete theory
4290:
4159:
4155:
4151:
4146:
4138:
3858: and
3854:
3849:
3835:
3811:New Foundations
3779:axiom of choice
3762:
3724:Gödel numbering
3664: and
3656:
3560:
3445:
3395:
3376:
3325:Boolean algebra
3311:
3275:Equiconsistency
3240:Classical logic
3217:
3198:Halting problem
3186: and
3162: and
3150: and
3149:
3144:Theorems (
3139:
3056:
3051:
2989:
2984:
2983:
2972:Geoffrey Hunter
2970:
2969:
2965:
2958:
2942:
2941:
2937:
2930:
2910:
2909:
2905:
2898:
2885:
2884:
2880:
2850:10.2307/2268615
2832:
2831:
2827:
2797:10.2307/2267402
2776:
2775:
2771:
2763:
2759:
2754:
2750:
2744:
2716:
2715:
2711:
2706:
2702:
2684:
2683:
2679:
2667:
2663:
2658:
2592:
2503:
2425:
2424:
2348:
2347:
2285:
2284:
2234:
2233:
2207:
2206:
2183:
2182:
2142:
2141:
2116:
2106:
2105:
2076:
2066:
2065:
2044:
2034:
2033:
2012:
2002:
2001:
1960:
1959:
1938:
1928:
1927:
1924:
1857:
1856:
1831:
1814:
1809:
1808:
1783:
1778:
1777:
1748:
1743:
1742:
1713:primitive signs
1666:
1630:
1589:
1561:Euclidean plane
1541:
1469:
1344:
1343:
1274:
1273:
1270:
1234:
1233:
1180:
1179:
1176:
1109:
1089:
1073:
1061:
1054:
964:
958:
945:
906:
848:. For the pair
787:
763:
757:
541:
527:logically valid
513:classical logic
496:
468:
440:
439:
414:
413:
394:
393:
374:
373:
350:
349:
327:
326:
307:
306:
287:
286:
263:
262:
225:
224:
201:
200:
197:
147:
145:Formal language
141:
82:predicate logic
51:formal language
35:
28:
23:
22:
15:
12:
11:
5:
4975:
4973:
4965:
4964:
4959:
4954:
4949:
4944:
4934:
4933:
4927:
4926:
4924:
4923:
4918:
4913:
4908:
4906:Satisfiability
4903:
4898:
4893:
4891:Interpretation
4888:
4883:
4878:
4873:
4868:
4863:
4862:
4861:
4851:
4846:
4841:
4836:
4829:
4823:
4820:
4819:
4810:
4808:
4807:
4800:
4793:
4785:
4776:
4775:
4761:
4758:
4757:
4755:
4754:
4749:
4744:
4739:
4734:
4733:
4732:
4722:
4717:
4712:
4703:
4698:
4693:
4688:
4686:Abstract logic
4682:
4680:
4676:
4675:
4673:
4672:
4667:
4665:Turing machine
4662:
4657:
4652:
4647:
4642:
4637:
4636:
4635:
4630:
4625:
4620:
4615:
4605:
4603:Computable set
4600:
4595:
4590:
4585:
4579:
4577:
4571:
4570:
4568:
4567:
4562:
4557:
4552:
4547:
4542:
4537:
4532:
4531:
4530:
4525:
4520:
4510:
4505:
4500:
4498:Satisfiability
4495:
4490:
4485:
4484:
4483:
4473:
4472:
4471:
4461:
4460:
4459:
4454:
4449:
4444:
4439:
4429:
4428:
4427:
4422:
4415:Interpretation
4411:
4409:
4403:
4402:
4400:
4399:
4394:
4389:
4384:
4379:
4369:
4364:
4363:
4362:
4361:
4360:
4350:
4345:
4335:
4330:
4325:
4320:
4315:
4310:
4304:
4302:
4296:
4295:
4292:
4291:
4289:
4288:
4280:
4279:
4278:
4277:
4272:
4271:
4270:
4265:
4260:
4240:
4239:
4238:
4236:minimal axioms
4233:
4222:
4221:
4220:
4209:
4208:
4207:
4202:
4197:
4192:
4187:
4182:
4169:
4167:
4148:
4147:
4145:
4144:
4143:
4142:
4130:
4125:
4124:
4123:
4118:
4113:
4108:
4098:
4093:
4088:
4083:
4082:
4081:
4076:
4066:
4065:
4064:
4059:
4054:
4049:
4039:
4034:
4033:
4032:
4027:
4022:
4012:
4011:
4010:
4005:
4000:
3995:
3990:
3985:
3975:
3970:
3965:
3960:
3959:
3958:
3953:
3948:
3943:
3933:
3928:
3926:Formation rule
3923:
3918:
3917:
3916:
3911:
3901:
3900:
3899:
3889:
3884:
3879:
3874:
3868:
3862:
3845:Formal systems
3841:
3840:
3837:
3836:
3834:
3833:
3828:
3823:
3818:
3813:
3808:
3803:
3798:
3793:
3788:
3787:
3786:
3781:
3770:
3768:
3764:
3763:
3761:
3760:
3759:
3758:
3748:
3743:
3742:
3741:
3734:Large cardinal
3731:
3726:
3721:
3716:
3711:
3697:
3696:
3695:
3690:
3685:
3670:
3668:
3658:
3657:
3655:
3654:
3653:
3652:
3647:
3642:
3632:
3627:
3622:
3617:
3612:
3607:
3602:
3597:
3592:
3587:
3582:
3577:
3571:
3569:
3562:
3561:
3559:
3558:
3557:
3556:
3551:
3546:
3541:
3536:
3531:
3523:
3522:
3521:
3516:
3506:
3501:
3499:Extensionality
3496:
3494:Ordinal number
3491:
3481:
3476:
3475:
3474:
3463:
3457:
3451:
3450:
3447:
3446:
3444:
3443:
3438:
3433:
3428:
3423:
3418:
3413:
3412:
3411:
3401:
3400:
3399:
3386:
3384:
3378:
3377:
3375:
3374:
3373:
3372:
3367:
3362:
3352:
3347:
3342:
3337:
3332:
3327:
3321:
3319:
3313:
3312:
3310:
3309:
3304:
3299:
3294:
3289:
3284:
3279:
3278:
3277:
3267:
3262:
3257:
3252:
3247:
3242:
3236:
3234:
3225:
3219:
3218:
3216:
3215:
3210:
3205:
3200:
3195:
3190:
3178:Cantor's
3176:
3171:
3166:
3156:
3154:
3141:
3140:
3138:
3137:
3132:
3127:
3122:
3117:
3112:
3107:
3102:
3097:
3092:
3087:
3082:
3077:
3076:
3075:
3064:
3062:
3058:
3057:
3052:
3050:
3049:
3042:
3035:
3027:
3021:
3020:
3015:
3010:
3005:
3000:
2995:
2988:
2987:External links
2985:
2982:
2981:
2963:
2956:
2935:
2928:
2903:
2896:
2878:
2825:
2769:
2757:
2748:
2742:
2709:
2700:
2696:. Mcgraw Hill.
2677:
2669:Priest, Graham
2660:
2659:
2657:
2654:
2653:
2652:
2647:
2642:
2637:
2632:
2627:
2625:Logical system
2622:
2617:
2612:
2607:
2601:Free variables
2598:
2591:
2588:
2512:, a structure
2502:
2499:
2486:
2480:
2474:
2468:
2462:
2456:
2450:
2444:
2438:
2432:
2421:
2420:
2409:
2403:
2397:
2391:
2385:
2379:
2373:
2367:
2361:
2355:
2345:
2334:
2328:
2322:
2316:
2310:
2304:
2298:
2292:
2282:
2271:
2265:
2259:
2253:
2247:
2241:
2227:
2226:
2214:
2190:
2170:
2167:
2161:
2158:
2155:
2149:
2122:
2119:
2115:
2098:
2097:
2082:
2079:
2075:
2050:
2047:
2043:
2018:
2015:
2011:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1944:
1941:
1937:
1923:
1920:
1900:intended model
1864:
1842:
1836:
1830:
1825:
1819:
1794:
1788:
1759:
1753:
1674:standard model
1665:
1662:
1629:
1626:
1601:Full semantics
1588:
1585:
1540:
1537:
1468:
1465:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1269:
1266:
1265:
1264:
1261:
1243:
1230:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1189:
1175:
1172:
1047:free variables
1032:
1031:
1012:
985:
978:
957:
954:
944:
941:
919:natural number
905:
902:
786:
783:
759:Main article:
756:
753:
719:
718:
715:
712:
709:
706:
703:
700:
697:
693:
692:
689:
686:
683:
680:
677:
674:
671:
667:
666:
663:
660:
657:
654:
651:
648:
645:
641:
640:
637:
634:
631:
628:
625:
622:
619:
615:
614:
611:
608:
605:
602:
599:
596:
593:
592:Interpretation
580:
579:
576:
573:
570:
567:
540:
537:
495:
492:
467:
464:
449:
427:
424:
421:
401:
381:
359:
334:
314:
294:
272:
250:
247:
244:
241:
238:
235:
232:
210:
196:
193:
143:Main article:
140:
137:
39:interpretation
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4974:
4963:
4960:
4958:
4955:
4953:
4950:
4948:
4945:
4943:
4940:
4939:
4937:
4922:
4919:
4917:
4914:
4912:
4909:
4907:
4904:
4902:
4899:
4897:
4894:
4892:
4889:
4887:
4884:
4882:
4879:
4877:
4874:
4872:
4869:
4867:
4864:
4860:
4857:
4856:
4855:
4852:
4850:
4847:
4845:
4842:
4840:
4837:
4835:
4834:
4830:
4828:
4825:
4824:
4821:
4817:
4813:
4806:
4801:
4799:
4794:
4792:
4787:
4786:
4783:
4773:
4772:
4767:
4759:
4753:
4750:
4748:
4745:
4743:
4740:
4738:
4735:
4731:
4728:
4727:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4707:
4704:
4702:
4699:
4697:
4694:
4692:
4689:
4687:
4684:
4683:
4681:
4677:
4671:
4668:
4666:
4663:
4661:
4660:Recursive set
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4611:
4610:
4609:
4606:
4604:
4601:
4599:
4596:
4594:
4591:
4589:
4586:
4584:
4581:
4580:
4578:
4576:
4572:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4529:
4526:
4524:
4521:
4519:
4516:
4515:
4514:
4511:
4509:
4506:
4504:
4501:
4499:
4496:
4494:
4491:
4489:
4486:
4482:
4479:
4478:
4477:
4474:
4470:
4469:of arithmetic
4467:
4466:
4465:
4462:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4435:
4434:
4433:
4430:
4426:
4423:
4421:
4418:
4417:
4416:
4413:
4412:
4410:
4408:
4404:
4398:
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4377:
4376:from ZFC
4373:
4370:
4368:
4365:
4359:
4356:
4355:
4354:
4351:
4349:
4346:
4344:
4341:
4340:
4339:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4305:
4303:
4301:
4297:
4287:
4286:
4282:
4281:
4276:
4275:non-Euclidean
4273:
4269:
4266:
4264:
4261:
4259:
4258:
4254:
4253:
4251:
4248:
4247:
4245:
4241:
4237:
4234:
4232:
4229:
4228:
4227:
4223:
4219:
4216:
4215:
4214:
4210:
4206:
4203:
4201:
4198:
4196:
4193:
4191:
4188:
4186:
4183:
4181:
4178:
4177:
4175:
4171:
4170:
4168:
4163:
4157:
4152:Example
4149:
4141:
4136:
4135:
4134:
4131:
4129:
4126:
4122:
4119:
4117:
4114:
4112:
4109:
4107:
4104:
4103:
4102:
4099:
4097:
4094:
4092:
4089:
4087:
4084:
4080:
4077:
4075:
4072:
4071:
4070:
4067:
4063:
4060:
4058:
4055:
4053:
4050:
4048:
4045:
4044:
4043:
4040:
4038:
4035:
4031:
4028:
4026:
4023:
4021:
4018:
4017:
4016:
4013:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3980:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3957:
3954:
3952:
3949:
3947:
3944:
3942:
3939:
3938:
3937:
3934:
3932:
3929:
3927:
3924:
3922:
3919:
3915:
3912:
3910:
3909:by definition
3907:
3906:
3905:
3902:
3898:
3895:
3894:
3893:
3890:
3888:
3885:
3883:
3880:
3878:
3875:
3873:
3870:
3869:
3866:
3863:
3861:
3857:
3852:
3846:
3842:
3832:
3829:
3827:
3824:
3822:
3819:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3796:Kripke–Platek
3794:
3792:
3789:
3785:
3782:
3780:
3777:
3776:
3775:
3772:
3771:
3769:
3765:
3757:
3754:
3753:
3752:
3749:
3747:
3744:
3740:
3737:
3736:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3709:
3705:
3701:
3698:
3694:
3691:
3689:
3686:
3684:
3681:
3680:
3679:
3675:
3672:
3671:
3669:
3667:
3663:
3659:
3651:
3648:
3646:
3643:
3641:
3640:constructible
3638:
3637:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3578:
3576:
3573:
3572:
3570:
3568:
3563:
3555:
3552:
3550:
3547:
3545:
3542:
3540:
3537:
3535:
3532:
3530:
3527:
3526:
3524:
3520:
3517:
3515:
3512:
3511:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3486:
3482:
3480:
3477:
3473:
3470:
3469:
3468:
3465:
3464:
3461:
3458:
3456:
3452:
3442:
3439:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3419:
3417:
3414:
3410:
3407:
3406:
3405:
3402:
3398:
3393:
3392:
3391:
3388:
3387:
3385:
3383:
3379:
3371:
3368:
3366:
3363:
3361:
3358:
3357:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3322:
3320:
3318:
3317:Propositional
3314:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3276:
3273:
3272:
3271:
3268:
3266:
3263:
3261:
3258:
3256:
3253:
3251:
3248:
3246:
3245:Logical truth
3243:
3241:
3238:
3237:
3235:
3233:
3229:
3226:
3224:
3220:
3214:
3211:
3209:
3206:
3204:
3201:
3199:
3196:
3194:
3191:
3189:
3185:
3181:
3177:
3175:
3172:
3170:
3167:
3165:
3161:
3158:
3157:
3155:
3153:
3147:
3142:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3081:
3078:
3074:
3071:
3070:
3069:
3066:
3065:
3063:
3059:
3055:
3048:
3043:
3041:
3036:
3034:
3029:
3028:
3025:
3019:
3016:
3014:
3011:
3009:
3006:
3004:
3001:
2999:
2996:
2994:
2991:
2990:
2986:
2977:
2973:
2967:
2964:
2959:
2953:
2949:
2945:
2939:
2936:
2931:
2929:9780486604534
2925:
2920:
2919:
2913:
2912:Rudolf Carnap
2907:
2904:
2899:
2893:
2889:
2882:
2879:
2875:
2871:
2867:
2863:
2859:
2855:
2851:
2847:
2843:
2839:
2835:
2829:
2826:
2822:
2818:
2814:
2810:
2806:
2802:
2798:
2794:
2790:
2786:
2782:
2781:
2773:
2770:
2767:
2761:
2758:
2752:
2749:
2745:
2743:0-19-501491-X
2739:
2735:
2731:
2726:
2725:
2719:
2718:Mates, Benson
2713:
2710:
2704:
2701:
2694:
2693:
2687:
2686:Haskell Curry
2681:
2678:
2674:
2670:
2665:
2662:
2655:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2602:
2599:
2597:
2594:
2593:
2589:
2587:
2585:
2581:
2577:
2573:
2570:
2566:
2562:
2557:
2555:
2551:
2547:
2543:
2539:
2535:
2531:
2527:
2523:
2519:
2515:
2511:
2506:
2500:
2498:
2484:
2478:
2472:
2466:
2460:
2454:
2448:
2442:
2436:
2430:
2407:
2401:
2395:
2389:
2383:
2377:
2371:
2365:
2359:
2353:
2346:
2332:
2326:
2320:
2314:
2308:
2302:
2296:
2290:
2283:
2269:
2263:
2257:
2251:
2245:
2239:
2232:
2231:
2230:
2212:
2204:
2188:
2168:
2165:
2159:
2156:
2153:
2147:
2139:
2138:
2137:
2120:
2103:
2080:
2048:
2016:
1999:
1998:
1997:
1980:
1977:
1974:
1971:
1968:
1942:
1921:
1919:
1917:
1913:
1909:
1905:
1901:
1897:
1893:
1889:
1884:
1882:
1878:
1840:
1823:
1792:
1775:
1757:
1740:
1736:
1733:
1729:
1725:
1722:
1718:
1714:
1710:
1706:
1702:
1697:
1695:
1691:
1687:
1683:
1679:
1675:
1670:
1663:
1661:
1659:
1655:
1654:Kripke models
1651:
1647:
1642:
1640:
1636:
1627:
1625:
1623:
1619:
1615:
1611:
1607:
1602:
1597:
1594:
1586:
1584:
1582:
1578:
1574:
1570:
1566:
1562:
1558:
1554:
1549:
1546:
1538:
1536:
1534:
1529:
1525:
1520:
1518:
1514:
1508:
1506:
1502:
1498:
1494:
1491:) holds then
1490:
1486:
1482:
1478:
1472:
1466:
1464:
1460:
1442:
1439:
1436:
1433:
1427:
1424:
1421:
1415:
1406:
1400:
1391:
1388:
1385:
1379:
1373:
1367:
1364:
1361:
1355:
1340:
1324:
1318:
1315:
1312:
1306:
1294:
1291:
1285:
1282:
1267:
1262:
1259:
1258:
1257:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1208:
1207:
1205:
1173:
1171:
1169:
1165:
1161:
1156:
1155:
1151:
1146:
1141:
1138:
1134:
1129:
1127:
1123:
1117:
1113:
1107:
1103:
1097:
1093:
1087:
1081:
1077:
1069:
1065:
1058:
1052:
1048:
1043:
1041:
1037:
1029:
1025:
1021:
1017:
1013:
1010:
1007: →
1006:
1002:
998:
994:
990:
986:
983:
979:
976:
973:
969:
968:
967:
963:
955:
953:
951:
942:
940:
938:
933:
931:
926:
923:
920:
916:
912:
903:
901:
897:
895:
891:
887:
883:
879:
875:
871:
867:
863:
859:
855:
851:
847:
843:
839:
835:
831:
827:
822:
820:
816:
812:
808:
803:
800:
796:
792:
784:
782:
780:
776:
772:
768:
762:
754:
752:
750:
746:
742:
738:
734:
730:
726:
716:
713:
710:
707:
704:
701:
698:
695:
694:
690:
687:
684:
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678:
675:
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669:
668:
664:
661:
658:
655:
652:
649:
646:
643:
642:
638:
635:
632:
629:
626:
623:
620:
617:
616:
612:
609:
606:
603:
600:
597:
594:
591:
590:
584:
577:
574:
571:
568:
565:
561:
560:
559:
556:
554:
548:
546:
538:
536:
534:
533:
528:
524:
520:
519:
514:
509:
507:
506:
501:
493:
491:
489:
483:
481:
477:
473:
465:
463:
422:
399:
346:
332:
245:
242:
233:
230:
194:
192:
190:
186:
182:
181:
176:
175:
170:
166:
162:
161:
156:
152:
146:
138:
136:
134:
130:
126:
122:
117:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
74:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
4952:Model theory
4911:Independence
4890:
4886:Decidability
4881:Completeness
4831:
4762:
4560:Ultraproduct
4414:
4407:Model theory
4372:Independence
4308:Formal proof
4300:Proof theory
4283:
4256:
4213:real numbers
4185:second-order
4096:Substitution
3973:Metalanguage
3914:conservative
3887:Axiom schema
3831:Constructive
3801:Morse–Kelley
3767:Set theories
3746:Aleph number
3739:inaccessible
3645:Grothendieck
3529:intersection
3416:Higher-order
3404:Second-order
3350:Truth tables
3307:Venn diagram
3090:Formal proof
2975:
2966:
2950:. Springer.
2947:
2938:
2917:
2906:
2887:
2881:
2841:
2837:
2834:Quine, W. V.
2828:
2784:
2778:
2772:
2760:
2751:
2728:, New York:
2723:
2712:
2703:
2691:
2680:
2672:
2664:
2640:Model theory
2605:Name binding
2583:
2579:
2575:
2571:
2564:
2560:
2558:
2549:
2545:
2541:
2537:
2533:
2532:, such that
2529:
2525:
2521:
2517:
2513:
2510:model theory
2507:
2504:
2422:
2228:
2102:axiom schema
2099:
1925:
1903:
1899:
1885:
1772:is directly
1698:
1690:Peano axioms
1673:
1671:
1667:
1643:
1631:
1621:
1617:
1613:
1609:
1605:
1600:
1598:
1590:
1580:
1576:
1572:
1568:
1564:
1556:
1550:
1544:
1542:
1521:
1516:
1513:normal model
1512:
1509:
1496:
1492:
1488:
1484:
1480:
1476:
1473:
1470:
1461:
1338:
1271:
1231:
1203:
1177:
1157:
1153:
1149:
1142:
1136:
1132:
1130:
1125:
1121:
1115:
1111:
1105:
1101:
1095:
1091:
1079:
1075:
1067:
1063:
1059:
1044:
1039:
1033:
1027:
1023:
1019:
1015:
1008:
1004:
1000:
996:
992:
988:
981:
974:
965:
946:
936:
934:
927:
907:
898:
893:
892:is assigned
889:
885:
884:is assigned
881:
877:
876:is assigned
873:
869:
868:is assigned
865:
861:
857:
853:
849:
845:
844:is assigned
841:
837:
836:is assigned
833:
829:
825:
823:
819:truthbearers
814:
810:
807:truth values
804:
788:
778:
774:
766:
764:
748:
744:
740:
736:
732:
728:
724:
722:
581:
557:
549:
542:
530:
526:
523:inconsistent
522:
516:
510:
503:
499:
497:
484:
479:
475:
469:
347:
198:
188:
184:
178:
172:
168:
164:
158:
154:
150:
148:
121:truth values
118:
113:
109:
105:
101:
97:
75:
38:
36:
4901:Metatheorem
4859:of geometry
4844:Consistency
4670:Type theory
4618:undecidable
4550:Truth value
4437:equivalence
4116:non-logical
3729:Enumeration
3719:Isomorphism
3666:cardinality
3650:Von Neumann
3615:Ultrafilter
3580:Uncountable
3514:equivalence
3431:Quantifiers
3421:Fixed-point
3390:First-order
3270:Consistency
3255:Proposition
3232:Traditional
3203:Lindström's
3193:Compactness
3135:Type theory
3080:Cardinality
2791:: 197–200,
2732:, pp.
2645:Satisfiable
2635:Modal logic
2181:" (where "
2100:The single
1912:assignments
1877:implication
1658:Modal logic
1579:is on line
1168:intensional
1164:extensional
771:many-to-one
566:Φ is False.
562:¬Φ is True
480:non-logical
392:and '0' to
55:mathematics
4936:Categories
4481:elementary
4174:arithmetic
4042:Quantifier
4020:functional
3892:Expression
3610:Transitive
3554:identities
3539:complement
3472:hereditary
3455:Set theory
2698:Here: p.48
2656:References
2578:such that
1721:meaningful
1686:isomorphic
1501:set theory
1160:set theory
1014:For every
987:For every
960:See also:
518:consistent
84:and their
4962:Semantics
4876:Soundness
4812:Metalogic
4752:Supertask
4655:Recursion
4613:decidable
4447:saturated
4425:of models
4348:deductive
4343:axiomatic
4263:Hilbert's
4250:Euclidean
4231:canonical
4154:axiomatic
4086:Signature
4015:Predicate
3904:Extension
3826:Ackermann
3751:Operation
3630:Universal
3620:Recursive
3595:Singleton
3590:Inhabited
3575:Countable
3565:Types of
3549:power set
3519:partition
3436:Predicate
3382:Predicate
3297:Syllogism
3287:Soundness
3260:Inference
3250:Tautology
3152:paradoxes
2764:see also
2559:A theory
2485:◼
2479:◼
2473:◼
2467:◼
2461:⧫
2455:◼
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2240:◼
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2189:∗
2169:∗
2166:◼
2160:⧫
2157:∗
2154:★
2148:◼
1981:⧫
1975:★
1969:◼
1863:→
1829:→
1774:derivable
1735:sentences
1705:formation
1637:(such as
1434:∨
1413:∀
1404:∃
1401:≡
1377:∃
1374:∨
1353:∀
1319:ψ
1316:∨
1313:ϕ
1304:∃
1301:↔
1295:ψ
1289:∃
1286:∨
1283:ϕ
1104:), where
1036:structure
911:signature
821:instead.
815:valuation
505:satisfied
426:△
423:◻
420:△
400:◻
380:△
333:◻
313:△
293:△
246:◻
240:△
231:α
151:sentences
125:sentences
94:extension
67:syntactic
4737:Logicism
4730:timeline
4706:Concrete
4565:Validity
4535:T-schema
4528:Kripke's
4523:Tarski's
4518:semantic
4508:Strength
4457:submodel
4452:spectrum
4420:function
4268:Tarski's
4257:Elements
4244:geometry
4200:Robinson
4121:variable
4106:function
4079:spectrum
4069:Sentence
4025:variable
3968:Language
3921:Relation
3882:Automata
3872:Alphabet
3856:language
3710:-jection
3688:codomain
3674:Function
3635:Universe
3605:Infinite
3509:Relation
3292:Validity
3282:Argument
3180:theorem,
2974:(1992).
2914:(1958).
2874:27053902
2821:40988137
2720:(1972),
2688:(1963).
2671:, 2008.
2590:See also
2121:′
2081:′
2049:′
2017:′
1943:′
1881:provable
1875:meaning
1051:T-schema
880:, or 4)
840:, or 2)
613:(Φ ↔ Ψ)
174:alphabet
160:formulas
90:function
4679:Related
4476:Diagram
4374: (
4353:Hilbert
4338:Systems
4333:Theorem
4211:of the
4156:systems
3936:Formula
3931:Grammar
3847: (
3791:General
3504:Forcing
3489:Element
3409:Monadic
3184:paradox
3125:Theorem
3061:General
2866:0064715
2858:2268615
2813:0057820
2805:2267402
2201:" is a
1922:Example
1896:reality
1807:, then
900:above.
610:(Φ → Ψ)
607:(Φ ∨ Ψ)
604:(Φ ∧ Ψ)
535:of ψ).
476:logical
412:. Then
195:Example
169:symbols
165:letters
47:symbols
45:to the
43:meaning
4442:finite
4205:Skolem
4158:
4133:Theory
4101:Symbol
4091:String
4074:atomic
3951:ground
3946:closed
3941:atomic
3897:ground
3860:syntax
3756:binary
3683:domain
3600:Finite
3365:finite
3223:Logics
3182:
3130:Theory
2954:
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1256:of L:
1154:false.
129:theory
61:, and
4432:Model
4180:Peano
4037:Proof
3877:Arity
3806:Naive
3693:image
3625:Fuzzy
3585:Empty
3534:union
3479:Class
3120:Model
3110:Lemma
3068:Axiom
2870:S2CID
2854:JSTOR
2817:S2CID
2801:JSTOR
2787:(3),
2650:Truth
2574:′ of
1086:range
1055:φ ∧ ψ
930:rings
922:arity
864:, 3)
155:words
133:model
86:modal
59:logic
49:of a
4814:and
4555:Type
4358:list
4162:list
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4062:rank
3956:open
3850:list
3662:Maps
3567:sets
3426:Free
3396:list
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3073:list
2952:ISBN
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2603:and
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2136:is:
1914:for
1732:true
1707:and
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1166:not
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3678:Map
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564:iff
511:In
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970:A
896:.
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691:F
670:#3
665:F
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601:¬Φ
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2534:B
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2118:S
2114:F
2078:S
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2014:S
2010:F
1984:}
1978:,
1972:,
1966:{
1940:S
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1040:L
1028:D
1024:D
1020:n
1016:n
1009:D
1005:D
1001:D
997:D
993:n
989:n
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975:D
937:L
894:T
890:b
886:F
882:a
878:F
874:b
870:T
866:a
862:F
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850:a
846:F
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838:T
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830:a
826:n
749:F
745:F
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737:F
733:F
729:F
725:F
714:T
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705:T
702:F
699:F
688:T
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679:T
676:T
673:F
662:F
659:T
656:F
653:F
650:F
647:T
636:T
633:T
630:T
627:F
624:T
621:T
598:Ψ
595:Φ
448:W
358:W
271:W
249:}
243:,
237:{
234:=
209:W
189:Q
185:P
110:T
106:T
102:a
98:T
34:.
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