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st 1 -order Predicate Logic (FOL) Now a “real logic”; Think and concentrate ! First-Order Predicate Logic 1 Simple arguments, where propositional logic does not suffice All monkeys like bananas. Judy is a monkey. Judy likes bananas. From the viewpoint of Propositional Logic (PL) the above are simple (atomic) sentences: p, q, r, and p, q does not entail r. All students are clever Charles is not clever Charles is not a student What are the valid schemata of these arguments? First-Order Predicate Logic 2 Logical form (scheme) of an argument The schemata of the arguments above remind valid schemata of PL: p q, p |= q (modus ponens) or p q, q |= p (modus tollens) But, in PL we cannot refine the analyses of simple sentences. Let us reformulate them: 1. Every individual, if it is a Monkey, then it likes Bananas 2. Judy is an individual with the property of being a Monkey 3. Judy is an individual that likes Bananas x [M(x) B(x)], M(J) |= B(J), where x is an individual variable, M, B are predicate symbols, J is a functional symbol It is again a schema: For M, B, J we can substitute other properties and individual, respectively. For instance, man for M, mortal for B and Charles for J. M, B, J are here only symbols which stand for properties and individuals First-Order Predicate Logic 3 Formal language of FOL (firstorder predicate logic) Alphabet Logical symbols individual variables: x, y, z, ... Symbols for truth-connectives: , , , , Symbols for quantifiers: , Special symbols Predicate: Pn, Qn, ... Functional: fn, gn, hn, ... n – arity = number of arguments -- „ -- Auxiliary symbols: (, ), [, ], {, }, ... First-Order Predicate Logic 4 Formal language of FOL Grammar terms: each symbol for a variable x, y, ... is a term ii. if t1,…,tn (n 0) are terms and if f is an n-ary functional symbol, then the expression f(t1,…,tn) is a term; If n = 0, then we talk about individual constant (denoted a, b, c, …) iii. only expressions due to i. and ii. are terms i. First-Order Predicate Logic 5 Formal language of FOL Grammar atomic formulas: If P is an n-ary predicate symbol and if t1,…,tn are terms, then P(t1,…,tn) is an atomic formula (composed) formulas: each atomic formula is a formula if A is a formula, then A is a formula if A and B are formulas, then (A B), (A B), (A B), (A B) are formulas if x is a variable and A a formula, then x A and x A are formulas First-Order Predicate Logic 6 Formal language of FOL 1st-order We can quantify only over individual variables We cannot quantify over properties or functions Example: Leibniz’s definition of identity: If two individuals have all the properties identical, then it is one and the same individual P [ P(x) = P(y)] (x = y) here we need a 2nd-order language, because we quantify over properties First-Order Predicate Logic 7 Example: the language of arithmetic We need special functional symbols: 0-ary symbol: 0 (the constant zero) – constant is a 0-ary functional symbol unary symbol: s (the successor function) binary symbols: + and (functions of adding and multiplying) Examples of terms (using infix notation for + and ): 0, s(x), s(s(x)), (x + y) s(s(0)), etc. Formulas are, e.g.: (= is here a special predicate symbol): s(0) = (0 x) + s(0), x (y = x z), x [(x = y) y (x = s(y))] First-Order Predicate Logic 8 Transforming natural language into the language of FOL “all”, “every”, “none”, “nobody”, “any”, ... “somebody”, “something”, “some”, “there is”, ... A sentence often needs to be reformulated (in an equivalent way) No student is retired (For any student it holds that he is not retired): x [S(x) R(x)] But: Not all students are retired (It is not true that any student is retired): x [S(x) R(x)] x [S(x) R(x)] First-Order Predicate Logic 9 Transforming natural language into the language of FOL An auxiliary rule: + , + (almost always) x [P(x) Q(x)] x [P(x) Q(x)] It is not true that all P’s are Q’s Some P’s are not Q’s x [P(x) Q(x)] x [P(x) Q(x)] It is not true that some P’s are Q’s No P is a Q de Morgan laws in FOL First-Order Predicate Logic 10 Transforming natural language into the language of FOL The lift is used only by employees x [L(x) E(x)] All employees use the lift x [E(x) L(x)] Mary likes only the winners: Hence, for all individuals it holds that if Mary likes him then he must be a winner: x [L(m, x) W(x)], “to like” is a binary relation, not a property !!! First-Order Predicate Logic 11 Transforming natural language into the language of FOL Everybody loves somebody sometimes x y t L(x, y, t) Everybody loves somebody sometimes but Hitler doesn’t like anybody x y t L(x, y, t) z L’(h, z) Everybody loves nobody – ambiguous Nobody loves anybody – ambiguous; Everybody dislikes anybody: x y L’(x, y) x y L’(x, y) First-Order Predicate Logic 12 Free, bound variables x y P(x, y, t) x Q(y, x) bound, free free, bound Formula with clear variables: each variable has only free occurrences, or only bound occurrences; each quantifier “has its own variables”. For instance, the above formula does not have clear variables: x in the second conjunct is another variable than the x in the first conjunct, similarly for y. Clear formula: x y P(x, y, t) z Q(u, z) First-Order Predicate Logic 13 Substitution of terms for variables Ax/t arises from A by a correct (i.e., collisionless) substitution of a term t for the variable x. There are two rules for a correct substitution: We can substitute a term t only for free occurrences of a variable x in a formula A, and we have to substitute for all the free occurrences. No individual variable that occurrs in the term t can become bound in A (in such a case the term t is not substitutable for x in the formula A). First-Order Predicate Logic 14 Substitution, example A(x): P(x) y Q(x, y), term t = f(y) After executing the substitution A(x/f(y)), we obtain: P(f(y)) y Q(f(y), y). The term f(y) is not substitutable for x in A We’d change the sense of the formula First-Order Predicate Logic 15 Semantics of FOL !!! P(x) y Q(x, y) – is this formula true? A non-reasonable question; For, we do not know what the symbols P, Q mean, what they stand for. They are only symbols which can stand for any predicate (property). P(x) P(x) – is this formula true? YES, it is; and it is always so, in all the circumstances. It is necessarily true. First-Order Predicate Logic 16 Semantics of FOL !!! x P(x, f(x)) we have to specify first, x P(x , f(x)) how to understand these formulas: 1) What do they talk about; we have to choose the universe of discourse: any non-empty set U 2) What does the symbol P denote; it is binary, with two arguments; it has to denote a binary relation R U U 3) What does the symbol f denote; it is an unary, oneargument symbol; it has to denote a function F U U, denoted F: U U First-Order Predicate Logic 17 Semantics of FOL !!! A: x P(x, f(x)) we have to specify B: x P(x , f(x)) how to understand these formulas: 1) Let U = N (the set of natural numbers) 2) let P denote the relation < (i.e., the set of pairs, where the first element is strictly less than the second one: {0,1, 0,2, …,1,2, …}) 3) Let f denote the function second power x2, i.e., the set of pairs where the second element is the power of the first one: {0,0, 1,1, 2,4, …,5,25, …} Now we can evaluate the truth values of the formulas A, B First-Order Predicate Logic 18 Semantics of FOL !!! A: x P(x, f(x)) B: x P(x , f(x)) We evaluate “from the inside”: First evaluate the term f(x). Each term denotes an element of the universe. Which one? It depends on the valuation e of the variable x. Let e(x) = 0, then f(x) = x2 = 0. Let e(x) = 1, then f(x) = x2 = 1, Let e(x) = 2, then f(x) = x2 = 4, etc. Now by evaluating P(x , f(x)) we have to obtain a truth value: e(x) = 0, 0 is not < 0 False e(x) = 1, 1 is not < 1 False, e(x) = 2, 2 is < 4 True. First-Order Predicate Logic 19 Semantics of FOL !!! A: x P(x, f(x)) B: x P(x , f(x)) The formula P(x , f(x)) is in the given interpretation True for some valuations of the variable x, and False for other valuations. The meaning of x (x): the formula is true for all (some) valuations of x Formula A: False in our interpretation I: |I A Formula B: True in the interpretation I: |=I B First-Order Predicate Logic 20 Model of a formula, interpretation A: x P(x, f(x)) B: x P(x , f(x)) We have found an interpretation I in which the formula B is true. The Interpretation structure N, <, x2 satisfies the formula B; it is a model of the formula B. How to adjust the interpretation in order it were a model of the formula A? There are infinitely many possibilities, infinitely many models. For instance: N, <, x+1, {N/{0,1}, <, x2, N, , x2, … All the models of the formula A are also models of the formula B (“what holds for all, it holds also for some”) First-Order Predicate Logic 21 Model of a formula, interpretation C: x P(x, f(y)) what are the models of this formula (with a free variable y)? Let us again 1. choose a Universe U = N 2. to the symbol P assign a relation: 3. to the symbol f assign a function: x2 Is the structure IS = N, , power a model of the formula C? In order it were so, the formula C would have to be true in IS for all the valuations of the variable y. Hence the formula P(x, f(y)) would have to be true for all valuations of x and y. But it is not so, for instance, if e(x) = 5, e(y) = 2, then 5 is not 22 First-Order Predicate Logic 22 Model of a formula, interpretation C: x P(x, f(y)) what are the models of this formula (with a free variable y)? The structure N, , x2 is not a model of formula C. A (trivial) model is, e.g., N, N N, x2. The whole Cartesian product N N, i.e. the set of all the pairs of natural numbers, is also a relation over N. Or, the structure N, , F, where F is the function, mapping N N, such that F associates all the natural numbers with the number 0. First-Order Predicate Logic 23