we get a formalization of the well-ordering property. But we do not have anything like the same understanding of the power set of the set of natural numbers. First-Order Logic in Artificial intelligence. They differ on the interpretation of the phrase “for every set of objects.” Does this have some fixed meaning to which we can refer, or do we need to consider the variety of meanings the phrase might have? For comparison, let V²(=) be the set of valid sentences in the second-order language of equality. The fact that we can express the power-set operation in second-order logic (and can iterate the procedure) gives second-order logic some large part of the expressiveness of set theory. Where ρ(0, 1, +, ×, <) is the second-order sentence that characterizes the ordered field of real numbers up to isomorphism, the sentence. That is, suppose φ is a formula in which only the variable u occurs free. How do we match our syntax trees to our semantics? As the foregoing example shows, in a second-order language for arithmetic, we can say that the natural numbers are well ordered. Second-order logic is an extension of first-order logic where, in addition to quantifiers such as “for every object (in the universe of discourse),” one has quantifiers such as “for every property of objects (in the universe of discourse).” This augmentation of the language increases its expressive strength, without adding new non-logical symbols, such as new predicate symbols. defeasible reasoning), i.e., a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusion(s) based on further evidence. ARTIFICIAL INTELLIGENCE – Logic in AI - J.-J.Ch. ∀x ∀y (Sx & Sy & ∀t(Etx ↔ Ety) → x = y) (extensionality) George Boolos suggested the example, “There are some critics who admire only each other.” This sentence asserts the existence of a set of individuals having a certain property; Consider a structure M = (A, R, ...) consisting of a non-empty set A serving as the universe of discourse, and some relations and functions on A interpreting the non-logical symbols. The name derives from the fact that it is possible to identify real numbers with sets of natural numbers. A non-monotonic logic is a formal logic whose consequence relation is not monotonic. (There is a non-deterministic Turing machine M and a polynomial p such that whenever (V, E), suitably encoded, is given to M, then if (V, E) is three-colorable then some computation of M will accept the graph within p(n) steps, where n measures the size of (V, E), and if (V, E) is not three-colorable then no computation of M will ever accept the graph.). ], computability and complexity | plural quantification | type theory, From Wikibooks, open books for an open world, https://en.wikibooks.org/w/index.php?title=Artificial_Intelligence/Logic/Representation/Second-order_logic&oldid=3222328. property. If P is a new one-place predicate symbol, then, ∃x Px → ∃x(Px & ∀y(Py → (y = x v x < y))), expresses the idea that P is true of some smallest number, if it is true of any numbers at all. We can conclude that not all issues involving second-order logic are necessarily settled in ZFC. The sentence π is categorical; its only model is (N, 0, S, +, ×), up to isomorphism. We know that the well-ordering property is not expressible by any first-order sentence, because the non-standard models of the (first-order) theory of (N; 0, S, <, +, ×) are never well ordered. The statement that the graph can be properly colored with three colors can be expressed by a second-order sentence: there exist subsets R, G, B that partition V in such a way that two vertices connected by an edge are never the same color. This page was last edited on 26 May 2017, at 00:28. A sentence that is valid in the standard semantics is true in those general structures for which the 1-place relation universe is the full power set of the individual universe, and so forth. ⊨ ∀P φ[s] iff for every k-ary relation Q in the k-place relation universe, we have M ⊨ φ [s′] where, ⊨ ∀F φ[s] iff for every k-place function G in the k-place function universe, we have M ⊨ φ [s′] where. Then it is reasonable to restrict attention to general pre-structures that are closed under definability. And to a Turing machine we can effectively assign an elementary sentence having models of every finite size iff the machine never halts.) The models of this are called real-closed ordered fields. According to the other scheme, third-order logic already allows quantification of super-predicate symbols.). (The sentence ∃Pφ(P) is true in every non-zero cardinality iff the elementary sentence φ(P) has models of every finite size, a co-c.e. For example, a finite graph can be regarded as a pair (V, E) consisting of a non-empty vertex set V and a symmetric edge relation E on V. The statement that the graph can be properly colored with three colors can be expressed by a second-order sentence: there exist subsets R, G, B that partition V in such a way that two vertices connected by an edge are never the same color. Now we can quantify away all the non-logical symbols; a sentence φ(P) is valid iff the sentence ∀P φ(P) is valid. The foregoing examples are drawn from mathematical situations. Here the second-order quantifier “∃R” expresses the existence of some binary relation on the universe. One might ask what other structures might be second-order characterizable. The effect of this schema is well known; it assures that any definable set that contains 0 and is closed under successor must contain everything. We can obtain a stronger theory by restricting attention to the models of analysis that differ from the usual model in only the second of the two ways described previously. As axioms, we can take the usual Peano postulates, including the second-order induction axiom. (The reader is to be cautioned that there are in the literature two different ways of counting the order. (Here φ might contain quantifiers over predicate variables, so that even impredicative comprehension axioms are to be true. According to one scheme, third-order logic allows super-predicate symbols to occur free, and fourth-order logic allows them to be quantified. By a general pre-structure for a second-order language we mean a structure in the usual sense (a universe of discourse plus interpretations for the non-logical symbols) together with the additional sets: For a general pre-structure M, there is a natural way to define what it means for a second-order formula φ to be satisfied in a structure M under an assignment s of objects to the free variables in φ, which again will be written M ⊨ φ[s]. Of course, there can be only countably many such structures, up to isomorphism, because each one needs a sentence. For another example, we can (using choice) say that the universe of discourse is infinite by saying that there is a transitive relation on the universe such that every element bears the relation to something, but not to itself: ∃R[∀x ∀y ∀z(Rxy & Ryz → Rxz) & ∀x[¬Rxx & ∃y Rxy]. A key feature of the “standard semantics” discussed in the previous section is that, for a one-place predicate variable X, the quantifier ∀X ranges over the entire power set of the universe of discourse. In the case of a second-order sentence σ (i.e., a formula with no free variable), the assignment s is no longer relevant, and we may speak unambiguously of the truth or falsity of σ in the structure M (that is, we can say that M is or is not a model of σ). Obviously, this concept can be generalized to the situation where a k-ary relation is defined from any particular number of parameters. Artificial Intelligence and. FIRST-ORDER LOGIC meaning & explanation. (Here we can assign Gödel numbers and view V¹(P) as a set of natural numbers, or equivalently we can view it directly as a set of words over a finite alphabet.) Introduction The discipline of artificial intelligence (AI) studies the question of how artifacts can be ascribed or endowed with intelligence. We can form the sentence ∀x(x < Sx) expressing the fact that each number is smaller than the next one, for example. The independence of the continuum hypothesis illustrates one such obscurity. A sentence in the language of equality is determined up to logical equivalence by its spectrum. Finally, we can prefix universal quantifiers to obtain a sentence ψ in the language of equality. They differ on the interpretation of the phrase “for every set of objects.” Does this have some fixed meaning to which we can refer, or do we need to consider the variety of meanings the phrase might have? [existential second-order quantifiers] [first-order formula]. It does not contain every true Π-0-1 sentence. We can also find general models of the Peano postulates in which the universe of sets is less than the full power set of the individual universe (i.e., general models that are not absolute). 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