INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW PREDICATE LOGIC

Size: px

Start display at page:

Download "INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW PREDICATE LOGIC"

Donald Bruce
5 years ago
Views:

1 1 CHAPTER 7. PREDICATE LOGIC 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW PREDICATE LOGIC 1 Predicate Logic 1.1 Introduction There are various arguments which cannot be dealt with within the confines of propositional logic. For example All human beings are rational. Some animals are human beings. Hence, some animals are rational. Any friend of Martin is a friend of John. Peter is not John s friend. Hence, Peter is not Martins friend. The successor of an even integer is odd. 2 is an even integer. Hence, the successor of 2 is odd. We introduce notation to study the internal structure of statements - to deal with expressions like any, all and some. Let P (x) denote x has the predicate or property P. Then x.p (x) means that all x has the property P.

2 2 CHAPTER 7. PREDICATE LOGIC 2 Also x.p (x) means that some x has the property P. The symbols and are termed quantifiers - is the universal quantifier and is the existential quantifier. Definition x.p (x) is true if and only if the truth set of P is all of U. Definition x.p (x) is true if and only if the truth set of P is not φ. Now consider some simple examples of translations: i John is brilliant B(j) ii If Brian practices he will win P (b) W (b) iii All grass is green y.(g(y) R(y)) iv Every dog has its day v Some person done this x.(g(x) D(x)) x.(p (x) D(x)) vi There is a winning combination y.(c(y) W (y)) Note We use capital letters usually for predicates and small letters to indicate terms or objects. Furthermore, situations involving the universal quantifier are represented by the implication connective. Situations involving the existential quantifier are represented by conjunction. Finally, so far the predicates used are one place predicates. We extend our notation to two place, three place predicates in general n ary predicates. i Everyone loves logic x.l(x, l) ii Every even number is divisible by 2 x.(e(x) D(x, 2)) iii There is no prime number between 23 and 29 x.(p (x) B(x, 23, 29)) iv Any two numbers can be added together x. y. z.(x + y = z) v Everyone is loved by someone x. y.l(y, x) vi Someone is loved by everyone x. y.l(y, x)

3 3 CHAPTER 7. PREDICATE LOGIC 3 Note The order of the universal and existential quantifiers is significant. Also with each statement there is a suitable domain of interpretation U. Consider the statement Everyone is loved by someone. If U = all people we have x. y.l(y, x) If we were to assume that the domain of interpretation is not restricted to people we might use x.(p (x) y.(p (y) L(y, x))) where P (x) is the predicate of being a person. Example Using predicates, quantifiers and logical connectives we can now convert each of the following arguments to predicate logic. All human beings are rational. Some animals are human beings. Hence, some animals are rational. x.(h(x) R(x)), x.(a(x) H(x)) x.(a(x) R(x)) Any friend of Martin is a friend of John. Peter is not John s friend. Hence, Peter is not Martins friend. x.(f (x, m) F (x, j)), F (p, j) F (p, m) The successor of an even integer is odd. 2 is an even integer. Hence, the successor of 2 is odd. x.((i(x) E(x)) O(s(x))), I(2) E(2) O(s(2))

4 4 CHAPTER 7. PREDICATE LOGIC Functions In the last argument we used a function. Function notation easily fits into our predicate logic symbolism. In general f(a 1, a 2,..., a n ) is a function of n arguments (more generally represented as fi n ). For example, The square of an odd number is odd may be represented as x.(o(x) O(s(x)) where s(x) is a single variable function that returns the square of x. Exercise Let U = all people. Translate the following, using functions where suitable. i Anyone who is persistent can learn logic ii No politician is honest iii Not all birds can fly iv John hates everyone who does not hate himself v John likes Kate s sister vi John is afraid of Kate s mother s sister s husband [Solutions:] i x.(p (x) L(x)) ii x.(p (x) H(x)) or x.(p (x) H(x)) iii x.(b(x) F (x)) iv x.( H(x, x) H(j, x)) v L(j, s(k)) vi A(j, h(s(m(k))))

5 5 CHAPTER 7. PREDICATE LOGIC Terms and Well-formed Formulae Definition We define terms recursively as follows: i Every constant is a term. ii Every variable is a term iii If t 1, t 2,..., t n are terms then fi n(t 1, t 2,..., t n ) is a term. iv Nothing else is a term. String of constants, variables and functions must be properly formed for the purposes of logic. We define the correct formation of formulae. Definition Well-formed formulae are defined recursively as follows: i Pi n(t 1, t 2,..., t n ) is a well-formed formulae, where the t i are terms. ii If A and B are well-formed formulae so are A, A B, A B, A B and A B. iii If A is a well-formed formulae so is x.a. iv If A is a well-formed formula so is x.a. v Nothing else is a well-formed formulae. Remark For the wff x.a(x) the scope of the quantifier is A(x). x.(a(x, y) A(x)) x.a(x, y) A(x) x.(a(x, y) x.a(x)) the scope of x is A(x, y) A(x). the scope of x is A(x, y). the scope of x is A(x, y) x.a(x) and the scope of x is A(x). Consider the following wff x.p (x, y) The occurrences of the variable x are said to be bound. The occurrences of y is said to be free. When a variable x occurs within the scope of x of x it is said to be bound. An occurrence that is not bound is free.

6 6 CHAPTER 7. PREDICATE LOGIC 6 Example For the wff x.( y.a(x, y, z) z.a(z)) we can say that the scope of x is y.a(x, y, z) z.a(z). The scope of y is A(x, y, z). The scope of z is A(z). Every occurrence of x and y is bound. The first occurrence of z is free, the other occurrence is bound. Exercise For each quantifier in thee following examples state its scope. For each variable state whether each occurrence is bound or free. i x.b(y, z) ii x. y.a(x, y, z) iii x. y.a(x, y, z) z.a(x, z, z) 1.2 Resolution in Predicate Logic Recall that the resolution principle was used to investigate if a sequence of compound statements are inconsistent. Again, in the context of an argument if we negate the conclusion of the argument and show, using resolution, that it is inconsistent with the premises of the argument, then the argument will be valid. This approach is very mechanical but is relying on an inconsistency (contradiction) being established. If an inconsistency cannot be established the argument may be invalid. Recall the resolution principle of propositional logic. Theorem 1 The Resolution Principle A resolvent of two clauses C 1 and C 2 is a logical consequence of C 1 C 2. C 1 C 2 res(c 1, C 2 ) Resolution in predicate logic is bases on the same principle in fact the wff are, more or less, reduced to propositions in order to apply the method. Hence the terms normal form, clauses, literals, resolvents and resolution all must be brought to mind again. However, as you would expect, complications arise with the richer notation of predicate logic and there can be alot of preparation before the resolution principle can be applied.

7 7 CHAPTER 7. PREDICATE LOGIC Prenex Normal Form Remember to apply the resolution technique in propositional logic our wff must be in conjunctive normal form and are represented as clauses. Clauses in predicate logic are similar but there is extra preparation due to quantifiers being present. The procedure is to move all quantifiers to the front of the wff, thereby getting what is known as prenex normal form. For example, the first of the two following wff is in prenex normal form, the second is not. x. y. z.(a(x, y) (B(x) C(x, z))) x. y.a(x, y) z.(b(x) C(x, z)) In order to convert an arbitrary wff to prenex normal form we first eliminate and in favour of and, and make use of the following equivalences: 1. x.a(x) x.b(x) x.(a(x) B(x)) 2. x.a(x) x.b(x) x.(a(x) B(x)) 3. x.a(x) y.b(y) x. y(a(x) B(y)) 4. x.a(x) y.b(y) x. y(a(x) B(y)) 5. x.a(x) y.b(y) x. y(a(x) B(y)) 6. x.a(x) y.b(y) x. y(a(x) B(y)) 7. x.a(x) y.b(y) x. y(a(x) B(y)) 8. x.a(x) y.b(y) x. y(a(x) B(y)) 9. x.a(x) y.b(y) x. y(a(x) B(y)) 10. x.a(x) y.b(y) x. y(a(x) B(y)) 11. x.a(x) x.b(x) x.(a(x) B(x)) 12. x.a(x) x.b(x) x.(a(x) B(x)) Note that identities 11 and 12 require an appropriate change of variables. Example We can convert a wff to prenex normal form as follows: x.a(x) x.b(x) x.a(x) x.b(x) x. A(x) x.b(x) x.( A(x) B(x))

8 8 CHAPTER 7. PREDICATE LOGIC 8 Example In a similar way we have x.a(x) x.b(x) x.a(x) x.b(x) x. A(x) x.b(x) x. A(x) y.b(y) x. y.( A(x) B(y)) Example In a similar way we have x. y.[(a(x, y) B(y, z)) x.c(x, z)] x. y.[ (A(x, y) B(y, z)) x.c(x, z)] x. y.[ A(x, y) B(y, z) x.c(x, z)] x. y.[ A(x, y) B(y, z) w.c(w, z)] x. y. w.[ A(x, y) B(y, z) C(w, z)] Exercise Convert the following wff to prenex normal form i x.a(x) x.b(x) ii x. y.[ z.((a(x, y, z) B(y)) x.c(x, z)] Finally, a further transformation must be performed before we have clauses to which the resolution principle can applied. We must attempt to remove the existential quantifier from our wff in prenex normal form (if it exists). This technique is called Skolemisation Skolemisation When a wff is in prenex normal form we can use the now familiar existential rule x will allow us to remove an existential quantifier. Recall x (Existential Instantiation) x.a(x) A(c) where c is a term which has not been used in the derivation so far. However, it would be incorrect to do the following: x. y.m(x, y) x.m(x, c)

9 9 CHAPTER 7. PREDICATE LOGIC 9 Say, we let M(x, y) denote that the mother of x is y. Our premise is for all x, there is a y such that the mother of x is y. Our deduction is for all x there is a term (person) c such that the mother of x is c i.e., c is the mother of all x (c is everyones mother). This is not a correct deduction. The following is the correct deduction: x. y.m(x, y) x.m(x, m(x)) where the single variable function m(x) returns the mother of x. It should be that y varies with the value of x, hence we use the function m(x). This function returns for each x that persons mother.. The function used may be single-variable, bi-variable etc, depending on the number of s preceding x. y.a(x, y) x.a(x, f(x)) x. y. z.a(x, y, z) x. y.a(x, y, f(x, y)) x. y. z. u.[a(x, y, z) B(u, z)] x. y. z.[a(x, y, f(x, y)) B(u, f(x, y))] Notice that in the last wff that the function that replaces z does not involve u since u occurs to the right of z. Now the statements are universally quantified so we can drop all universal quantifiers and write the wff in clause form. Definition The process of eliminating the existential quantifier and replacing the corresponding variable by either a constant (Skolem constant) or a function (Skolem function) is called Skolemisation. Example We have already seen how to bring the following wff to prenex normal form as follows x. y.[(a(x, y) B(y, z)) x.c(x, z)] x. y.[ (A(x, y) B(y, z)) x.c(x, z)] x. y.[ A(x, y) B(y, z) x.c(x, z)] x. y.[ A(x, y) B(y, z) w.c(w, z)] x. y. w.[ A(x, y) B(y, z) C(w, z)] Now to bring to Skolem standard form. Since there are two s preceeding we need only replace the variable w with a function of two variables, say f(x, y).

10 10 CHAPTER 7. PREDICATE LOGIC 10 x. y.[ A(x, y) B(y, z) C(f(x, y), z)] The wff is now universally quantified so we can represent it in clause form as { A(x, y), B(y, z), C(f(x, y), z)} Exercise Write each of the following in Skolem standard form, and write the result as a set of clauses i x.a(x) x. y.(b(x) C(x, y)) ii x.a(x) x. y.(b(x) C(x, y)) iii x. y.[(a(x, y) B(x, y)) x.c(x, z)] The Resolution Principle Consider the following argument from earlier All human beings are rational. Some animals are human beings. Hence, some animals are rational. x.(h(x) R(x)), x.(a(x) H(x)) x.(a(x) R(x)) where H(x), R(x) and A(x) denote the properties of being human, rational and an animal. Taking each wff of the argument in turn (remembering to negate the conclusion) we bring to clause form as follows: x.(h(x) R(x)) x.( H(x) R(x)) x.(a(x) H(x)) A(a) H(a) x.(a(x) R(x)) x. (A(x) R(x)) x.( A(x) R(x)) We can use the resolution principle to show that these statements cannot be simultaneously be true i.e., they are inconsistent. Remembering that the clauses involving x are universally quantified, our clauses are

11 11 CHAPTER 7. PREDICATE LOGIC 11 { H(x), R(x)}, {A(a)}, {H(a)}, { A(x), R(x)} Constructing a resolution tree we get { H(x), R(x)} {H(a)} { A(x), R(x)} {A(a)} {R(a)} { A(a)} We have applied the resolution principle to the above clauses. It has yielded a contradiction so we conclude the the statements are inconsistent. This further implies that the argument from which these statements have come is a valid argument. Definition A resolution deduction of from a set of clauses S is called a refutation of S. Example Consider the following argument: All students are determined. Anyone who is determined and hardworking will succeed. Michael is a hardworking student. Therefore, Michael will succeed. S(x) x is a student D(x) x is a determined H(x) x is a hardworking I(x) x is will succeed x.(s(x) D(x)), x.((d(x) H(x)) I(x)), S(m) H(m) I(m) Taking each wff of the argument in turn (remembering to negate the conclusion) we bring to clause form as follows:

12 12 CHAPTER 7. PREDICATE LOGIC 12 x.(s(x) D(x)) x.( S(x) D(x)) x.((d(x) H(x)) I(x)) x.( (D(x) H(x)) I(x)) x.( D(x) H(x) I(x)) S(m) H(m) S(m) H(m) I(m) I(m) We can use the resolution principle to show that these statements cannot be simultaneously be true i.e., they are inconsistent. Remembering that the clauses involving x are universally quantified, our clauses are { S(x), D(x)}, { D(x), H(x), I(x)}, {S(m)}, {H(m)}, { I(m)} Constructing a resolution tree we get { S(x), D(x)} { D(x), H(x), I(x)} {S(m)} {H(m)} { I(m)} { S(m), H(m), I(m)} { H(m), I(m)} {I(m)} We have applied the resolution principle to the above clauses. It has yielded a contradiction so we conclude the the statements are inconsistent. This further implies that the argument from which these statements have come is a valid argument.

13 13 CHAPTER 7. PREDICATE LOGIC 13 Exercise Consider the following argument: Some students attend logic lectures diligently. No student attends boring logic lectures diligently. Sean s lectures on logic are attended diligently by all students. Therefore, none of Sean s lectures are boring. S(x) x is a student L(x) x is a logic lecture A(x, y) x attends y diligently B(x) x is boring G(x, y) x is given by y s Sean. x.(s(x) y.(l(y) A(x, y))), x.(s(x) y.((l(y) B(y)) A(x, y))), x.((l(x) G(x, s)) z.(s(z) A(z, x)) x.((l(x) G(x, s)) B(x)) Convert each wff to clause form and use the resolution principle to establish its validity the argument is valid.

14 14 CHAPTER 7. PREDICATE LOGIC Formal Proofs in Predicate Logic Review of Formal Proofs in Propositional Logic We consider a more formal or syntactic treatment of proofs and deductions. Our objective is to prove logical forms from given logical forms using formal rules of deduction. The rules of deduction are like the rules of a game, such as chess. They permit certain logical moves. They must be adhered to in order to play the formal game of logic. The rules of deduction will ensure true conclusions will be deduced from true premises. Remark We will use the logical constant to replace the symbol 0. This constant has the false value. It is known by the latin name falsum. The rules of deduction are intended to represent in a formal way the intuitive or natural methods of reasoning used by humans. I ( introduction) A B A B E ( elimination) A B A E ( elimination) A B B I ( introduction) A A B I ( introduction) B A B E ( elimination) A B.. A B C C C

15 15 CHAPTER 7. PREDICATE LOGIC 15 If C is derived from A, and C is derived from B, then C may be derived from A B. The reason A and B are crossed out is that they are no longer needed - A B takes their place. If A and B are assumptions then they are said to be discharged. I ( introduction) A. C A C If C can be derived from the assumption A, then we can discharge the assumption and conclude that A C. For example, say we conclude that The cat drank the cream from the assumption The saucer is empty. Then it is reasonable to make the assertion If the saucer is empty, then the cat drank the cream. We do not hold on to the assumption The saucer is empty, since it is contained in the assertion. E ( elimination) A A C C If A holds, and A implies C, then C holds. This is the modus ponens rule of reasoning. C From a contradiction, any conclusion C can be drawn. RAA (Reductio Ad Absurdum) A. A

16 16 CHAPTER 7. PREDICATE LOGIC 16 This rule formalises the famous proof by contradiction method of arguing. If assuming that A is not the case leads to a contradiction, then we conclude that A is the case. Using the I rule and noting that A A we get a complementary form of this rule: A. A We will refer to this form of the rule as RAA also. Finally we have a final rule. Id (Identity) A A Any formula can be derived from itself. Remark The method of natural deduction was introduced by the German/Polish mathematician Gerhard Gentzen ( ) in a paper published in He intended to set up a formal system which comes as close as possible to actual reasoning. Let us consider how we use he above rules of deduction to establish the validity of an argument in a more efficient way. Recall the form of an argument A 1, A 2,..., A n C Note If the set of assumptions are empty we simply write C Further Rules of Deduction The rules of deduction established for propositional logic are used in a similar way along with four new rules to deal with quantifiers. x (U niversal Instantiation) x.a(x) A(c)

17 17 CHAPTER 7. PREDICATE LOGIC 17 where c is some arbitrary element of the universe. x (Existential Instantiation) x.a(x) A(c) where c is a term which has not been used in the derivation so far. I (U niversal Generalisation) A(c) x.a(x) where A(c) holds for every element c of the universe and x must not appear as a free variable in A(c). I (Existential Generalisation) A(c) x.a(x) where c is an element of the universe and x must not appear as a free variable in A(c). The following laws of logic establish a relationship between and. They can be easily confirmed by way of example. x.a(x) x. A(x) x.a(x) x. A(x) Natural Deduction We can illustrate each of the above rules with the following examples. The first example illustrates universal instantiation x.

18 18 CHAPTER 7. PREDICATE LOGIC 18 Example x.(a(x) B(x)), A(c) B(c) Proof 1. x.(a(x) B(x)) Hypothesis 2. A(c) Hypothesis 3. A(c) B(c) by x 4. B(c) by E with 2, 3. This next example illustrates the use of existential instantiation x. Example x.(a(x) B(x)), x.a(x) B(c) Proof 1. x.(a(x) B(x)) Hypothesis 2. x.a(x) Hypothesis 3. A(c) by x 4. A(c) B(c) by x 5. B(c) by E with 3, 4. This next example illustrates the use of universal generalisation. Example x.(a(x) B(x)), x.a(x) x.b(x) Proof 1. x.(a(x) B(x)) Hypothesis 2. x.a(x) Hypothesis 3. A(c) B(c) by x 4. A(c) by x 5. B(c) by E

19 19 CHAPTER 7. PREDICATE LOGIC x.b(x) by I (universal generalization) on 5. Remark Consider the previous argument x.(a(x) B(x)), x.a(x) x.b(x) An example of a corresponding hypothetical argument could be as follows: For every x if x > 1, then x 1 > 0. Also for every number x, x > 1. Hence we conclude that for every x, x 1 > 0. Finally, a short example to illustrate the use of existential generalisation. Example x.a(x) x.a(x) Proof 1. x.a(x) Hypothesis 2. A(c) x 3. x.a(x) by I (existential generalization) on 2. Example x.a(x) B(x) x.a(x) y.b(y) Proof 1. x.a(x) B(x) Hypothesis 2. A(c) B(c) by x 3. A(c) by E 4. x.a(x) by I 5. B(c) by E 6. y.b(y) by x 7. x.a(x) y.b(y) by I on 4, 6.

20 20 CHAPTER 7. PREDICATE LOGIC 20 Example x.(a(x) B(x)), x.a(x) x.b(x) Proof 1. x.(a(x) B(x)) Hypothesis 2. x.a(x) Hypothesis 3. A(c) B(c) by x (c has not been used so far) 4. A(c) by x 5. B(c) by E 6. x.b(x) by I (existential generalization) on 8. Example x.(a(x) B(x)), x.(b(x) C(x)), y.a(y) z.c(z) Proof 1. x.(a(x) B(x)) Hypothesis 2. x.(b(x) C(x)) Hypothesis 3. y.a(y) Hypothesis 4. A(c) by x (c has not been used so far) 5. A(c) B(c) by x 6. B(c) by E 7. B(c) C(c) by x 8. C(c) by E 9. z.c(z) by I (existential generalization) on 8.

21 21 CHAPTER 7. PREDICATE LOGIC 21 Example x.a(x) B(x), B(c) A(c) Proof 1. x.a(x) B(x) Hypothesis 2. B(c) Hypothesis 3. A(c) B(c) by x 4. A(c) Assumption 5. B(c) by E 6. by 2, A(c) by RRA, discharging assumption. Remark We can use the deduction theorem in our proofs which we discussed in section 1.6 axiomatic propositional logic. When a wff of the form C D is to be establish, it may be useful to include C among the hypotheses, derive D from the expanded set of hypotheses, and then conclude C D from the original set using the deduction theorem. Recall, again, the deduction theorem. Theorem 1 (The Deduction Theorem) Let H be a set (possibly empty) of wff of AL, and let A and B be any wff of AL. If H {A} B then H A B Example x.(a(x) B(x)) ( x.a(x) x.b(x)) Proof 1. x.(a(x) B(x)) Hypothesis 2. x.a(x) Hypothesis 3. A(c) by x (c has not been used so far)

22 22 CHAPTER 7. PREDICATE LOGIC A(c) B(c) by x 5. B(c) by E 6. x.b(x) by I (existential generalization) on x.(a(x) B(x)), x.a(x) x.b(x) 8. x.(a(x) B(x)) x.a(x) x.b(x) by deduction theorem 9. x.(a(x) B(x)) ( x.a(x) x.b(x)) by deduction theorem Example x.(a(x) B(x)), x.a(x) x.b(x) Proof 1. x.a(x) Hypothesis 2. x. B(x) Hypothesis 3. A(c) x 4. B(c) x 5. (A(c) B(c)) (A B) (A B) 6. x. (A(x) B(x)) I 7. x.a(x), x. B(x) x. (A(x) B(x)) 8. x.a(x) x. B(x) x. (A(x) B(x)) by Deduction T heorem 9. x.a(x) ( x. (A(x) B(x)) x. B(x) A B B A 10. x.a(x) x.(a(x) B(x)) x.b(x) x. A(x) x.a(x) 11. x.a(x), x.(a(x) B(x)) x.b(x) by Deduction T heorem This final proof above involves the use of many formal rules of logic - including the converse of the deduction theorem.

23 23 CHAPTER 7. PREDICATE LOGIC 23 Exercise Derive each of the following theorems using the deduction theorem i ii x.a(x) x.a(x) x.(a(x) B(x)) ( x.a(x) x.b(x)) iii x. y.a(x, y) x.a(x, x) Exercise Use predicates, quantifiers and logical connectives to convert this argument to predicate logic. All computer science graduates are people. Some computer science graduates are logical thinkers. Therefore, some people are logical thinkers. Write a formal proof for this argument. Exercise Use predicates, quantifiers and logical connectives to convert this argument to predicate logic. All students are determined. Anyone who is determined and hardworking will succeed. Michael is a hardworking student. Therefore, Michael will succeed. Write a formal proof for this argument. Exercise Use predicates, quantifiers and logical connectives to convert this argument to predicate logic. Some students are anxious. Some students do not study. If a student is anxious he will not pass his examination unless he studies. Therefore, some students will not pass their examination. Write a formal proof for this argument.

24 24 CHAPTER 7. PREDICATE LOGIC 24 Exercise Use predicates, quantifiers and logical connectives to convert this argument to predicate logic. The horse that is registered for today s race is not a thoroughbred. Every horse registered for today s race has won a race this year. Therefore, a horse that has won a race this year is not a thoroughbred. Write a formal proof for this argument.

Denote John by j and Smith by s, is a bachelor by predicate letter B. The statements (1) and (2) may be written as B(j) and B(s).

Denote John by j and Smith by s, is a bachelor by predicate letter B. The statements (1) and (2) may be written as B(j) and B(s). PREDICATE CALCULUS Predicates Statement function Variables Free and bound variables Quantifiers Universe of discourse Logical equivalences and implications for quantified statements Theory of inference