1st-order Predicate Logic (FOL) Now a “real logic”; Think and concentrate ! First-Order Predicate Logic

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

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: Every individual, if it is a Monkey, then it likes Bananas Judy is an individual with the property of being a Monkey  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

Formal language of FOL (first-order predicate logic) Alphabet Logical symbols individual variables: x, y, z, ... Symbols for truth-connectives: , , , ,  Symbols for quantifiers: ,  Special symbols Predicate: Pn, Qn, ... n – arity = number of arguments Functional: fn, gn, hn, ... -- „ -- Auxiliary symbols: (, ), [, ], {, }, ... First-Order Predicate Logic

Formal language of FOL Grammar terms: each symbol for a variable x, y, ... is a term 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, …) only expressions due to i. and ii. are terms First-Order Predicate Logic

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

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

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

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

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

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

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

First-Order Predicate Logic 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

Substitution of terms for variables Ax/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

First-Order Predicate Logic 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

First-Order Predicate Logic 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

First-Order Predicate Logic Semantics of FOL !!! x P(x, f(x)) we have to specify first, x P(x , f(x)) how to understand these formulas: What do they talk about; we have to choose the universe of discourse: any non-empty set U   What does the symbol P denote; it is binary, with two arguments; it has to denote a binary relation R  U  U What does the symbol f denote; it is an unary, one-argument symbol; it has to denote a function F  U  U, denoted F: U  U First-Order Predicate Logic

First-Order Predicate Logic Semantics of FOL !!! A: x P(x, f(x)) we have to specify B: x P(x , f(x)) how to understand these formulas: Let U = N (the set of natural numbers) 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, …}) 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

First-Order Predicate Logic 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

First-Order Predicate Logic 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

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

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 choose a Universe U = N to the symbol P assign a relation:  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

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