PREDICATE LOGIC. 1 Basic Concepts. Jorma K. Mattila LUT, Department of Mathematics and Physics

Transcription

1 PREDICATE LOGIC Jorma K. Mattila LUT, Department of Mathematics and Physics 1 Basic Concepts In predicate logic the formalism of propositional logic is extended and is made it more finely build than propositional logic. Thus it is possible to present more complicated expressions of natural language and use them in formal inference. Example 1.1. Consider the inference All ravens fly. Peter is a raven. So, Peter flies. In the view of propositional logic, the sentences are totally different atoms, and thus they have no such common parts needed in the analysis of the inference. However, there are clearly similar parts in the sentences. In the first sentence there is spoken about ravens and flying, in the second sentence about ravens and Peter, and in the last sentence there is spoken about Peter and flying. So, points of contacts can be found. Predicate language, as propositional language, too, consists of the following parts: Syntax: Alphabet and rules for formation wff s Semantics: Questions connecting models: interpretation, determining truth values, i.e. how to translate wff s for example into natural language, and what are the exact conditions for their truth. Proof theory: The principle of calculus, axioms and inference rules. Theorems are deduced from axioms by means of inference rules. (Axioms are 1

2 premises that are taken the basic truths of the system. Thus they must be logically true.) Let us denote the pure predicate language by the symbol P. 1.1 Syntactic Components of Predicate Logic Predicate logic contains all the components of propositional logic, including propositional variables and constants. In addition, predicate logic contains terms, predicates, and quantifiers. Terms are typically used in place of nouns and pronouns. They are combined into sentences by means of predicates. For example, in the sentence "John loves Mary", the nouns are "John" and "Mary", and the predicate is "loves". The same is true if this sentence is translated into predicate logic, except that "John" and "Mary" are now called terms. Predicate logic uses quantifiers to indicate if a statement is always true, if it is sometimes true, or it is never true. In this sense, the quantifiers are used to correspond words such as "all", "some", "never", and related expressions. Definition 1.1. Alphabet of P consists of connectives (familiar from propositional logic),,, and and parentheses (, ) and dot,. Other symbols belonging to alphabet are individual constants: a, b, c,... variables: x, y, z,... predicate symbols P, Q, R,... function symbols f, g,... identity symbol = quantifiers and. Constants are thought to be names of creatures of (real or ideal) world (more precisely: interpretation maps constants to some elements of a set A of creatures). Variables refer to creatures, too, but not to any certain ones. They correspond nearest to pronouns, like he, she, it,.... Expressions including so-called 2

3 free variables can have a truth value not until variables have got their constant values The Universe of Discourse To explain the main concepts of this section, we use the following logical argument: 1. Jane is Paul s mother. 2. Jane is Mary s mother. 3. Any two persons having the same mother are siblings. 4. Paul and Mary are siblings. The truth of the statement "Jane is Paul s mother" can only be assessed within a certain context. There are many people named Jane and Paul, and without further information the statement in question can defer to many different people, which makes it ambiguous. To remove such ambiguities, we introduce the concept of a universe or a domain. Definition 1.2. The universe of discourse or domain is the collection of all persons, ideas, symbols, data structures, and so on, that affect the logical argument under consideration. The elements of the universe of discourse are called individuals In the argument concerning Mary and Paul, the universe of discourse may, for example, consist of the people living in a particular house or on a particular block. Many arguments involve numbers and, in this case, one must stipulate whether the domain is the set of natural numbers, the set of integers, the set of real numbers, or even the set of complex numbers. In fact, the truth of a statement may depend on the domain selected. The statement "There is a smallest number" is true in the domain of natural numbers, but false in the domain of integers. 3

4 To avoid trivial cases, one stipulates that every universe of discourse must contain at least one individual. Hence, the set of all natural numbers less that 0 does not constitute a universe because there is no negative natural numbers. Instead of the word individual, one sometimes uses the word object, such as in "the domain must contain at least one object". To refer to a particular individual or object, identifiers are used.these identifiers are called individual constants. Each individual constant must uniquely identify a particular individual and no other one. For example, if the universe of discourse consists of persons, there must not be two persons with the same name Predicates Generally, predicates make statements about individuals. To illustrate this notion, consider the following statements: (a) Mary and Paul are siblings. (b) Jane is the mother of Mary. (c) Tom is a cat. (d) The sum of 2 and 3 is 5. In each of these statements, there is a list of individuals, which is given by the argument list, together with phrases that describe certain relations among or properties of the individuals mentioned in the argument list. These properties are referred to as predicates. For example, in the statement (a), the argument list is given by "Mary" and "Paul", in that order, whereas the predicate is described by the phrase "are siblings". The entries of the argument list are called arguments. In this sense, arguments are terms, i.e. variables or individual constants. In predicate logic, each predicate is given a name, which followed by the list of arguments. For example, to express "Jane is the mother of Mary" one would choose an identifier, say, "mother, to express the predicate "is mother of", and one would write mother(jane, Mary). Very often only single letters are used for predicate names and terms. We usually write, for example, M(j, m) instead 4

5 of mother(jane, Mary). The order of the arguments is important. Clearly, the statement M(j, m) and M(m, j) have completely different meaning. Each predicate is associated with arity number. It indicates the number of elements in the argument list of a predicate.unary predicates describe properties of of objects, for example, "P (x)" indicates that x has the property P. The interpretation of a predicate P in a set of objects, A, is the set of those elements of A that have the property P, i.e., {α A P (α)}. A predicate with arity n is often called an n-place predicate. These predicates indicate relations between objects. For example, if Q is a two-place predicate then we interpret Q as a binary relation on a universe of discourse A, i.e. as the pairs of elements (α, β) A A to denote that α is in the relation Q to β. Hence we can write {(α, β) A A Q(α, β)}. Example 1.2. The predicate "is a cat" is one-place predicate. The predicate "is the mother of" is two-place predicate, i.e., its arity is 2. The predicate in the statement "The sum of 2 and 3 is 6" (which is false) contains the three-place predicate "is the sum of". The identity = is a constant two-place predicate always having the same interpretation: identity on a set of terms A is {(α, β) α, β A}. The identity symbol can be inserted only between two terms. Definition 1.3. In predicate language P, an atomic formula (or an atom) is (a) a predicate name followed by an argument list, (a) an identity t 1 = t 2, where t 1 and t 2 are terms (i.e. individual constants or variables). 5

6 Atomic formulas are statements, and they can be combined by logical connectives in the same way as atoms in propositional logic. For example, using above mentioned natural language counterparts, we can write M(j, m) M(m, j) to mean that "If Jane is Mary s mother then Mary is not Jane s mother." If all arguments of a predicate are individual constants, then the resulting atomic formula must either be true or false. For example, if the universe of discourse consists of Jane, Doug, Mary, and Paul, we have to know for each ordered pair of individuals whether or not the predicate "is the mother of" (or "mother" for short) is true. This can be done in the form of a table. Any method that assigns truth-values to all possible combinations of individuals of a predicate is called an assignment of the predicate.for example, the following table is an assignment of the predicate "mother". Doug Jane Mary Paul Doug F F F F Jane F F T T Mary F F F F Paul F F F F In general, if a predicate has two arguments, its assignment can be given by a table in which the lows correspond to the first argument and columns to the second. In a finite universe of discourse, one can represent the assignments of predicates with arity n by n-dimensional arrays. For example, properties are assigned by one-dimensional arrays, predicates of arity 2 by two-dimensional arrays, and so on. Note that mathematical symbols and are predicates. However, these predicates are normally used in infix notations. By this, we mean that they are placed between the arguments. For example, we usually write 2 > 1 instead of > (2, 1). 6

7 1.1.3 Variables and Instantiations Often, one does not want to associate the arguments of an atomic formula with a particular individual. To avoid this, variables are used. Variable names are frequently chosen from the end of the alphabet: x, y, z etc., with or without subscripts. Examples of expressions containing variables include cat(x) hastail(x), dog(y) brown(y), grade(x) (x 0) (x 100). As in propositional logic, expressions can be given names, i.e. meta-variables are used. For example, one can give the name A to the expression A := B(x) C(x) which means that when we write A we really mean "B(x) C(x)". Example 1.3. Consider a statement "If x is a cat then x has tails." We formalize it as follows: C(x) := "x is a cat", T (x) := "x has tails", then the formalization of the whole sentence is A := C(x) T (x). This is an open formula, because there exists a variable x in the formula. We cannot give a truth-value to it before we know the value of of x. If S is a domain of objects, where x can have values, we can choose some object and replace it to all occurrences of x in the formula. After doing this, we can state the truth-value of the corresponding closed formula. Suppose that a S, and a refers to Tom. When we replace a to the instances of x of the formula, we have C(a) T (a), and its interpretation in natural language is "If Tom is a cat then Tom has tails". Generally, if A is an expression, the expression obtained by replacing all occurrences of a variable x in A by term t is denoted by S x t A. According to Example 1.3, S x aa stands for C(a) T (a). 7

8 Definition 1.4. Let A represent an expression, x represent a variable, and t represent a term. Then S x t A represents the expression obtained by replacing all occurrences of x in A by t. S x t A is called an instantiation of A, and t is said to be an instance of x. Example 1.4. Let a, b, and c be individual constants, P and Q be predicate symbols, and x and y be variables. Find S x a(p (a) Q(x)), S y b (P (y) Q(y)), SaQ(a), y S y a(p (x) Q(x)). Solution: Sa(P x (a) Q(x)) is P (a) Q(a), and S y b (P (y) Q(y)) is P (b) Q(b). Since (Q(a)) does not contain any y, replacing all occurrences of y by a leaves Q(a) unchanged, which means that S y aq(a) = Q(a). Similarly, S y a(p (x) Q(x)) = P (x) Q(x) Quantifiers Consider the following statements: 1. All cats have tails. 2. Some people like their meat raw. 3. Everyone gets a break once in a while. All these statements indicate how frequently certain things are true. In predicate logic, one uses quantifiers in this context. Specially, we will discuss two quantifiers: the universal quantifier, which indicates that something is true for all individuals, and the existential quantifier, which indicates that a statement is true for some individuals. 8

9 Definition 1.5. Let A represent an expression, and let x represent a variable. If we want to indicate that A is true for all possible values of x, we write x A. Here x is called the universal quantifier, and A is called the scope of the quantifier. The variable x is said to be bound by the quantifier. The symbol is pronounced "for all". The quantifier and the bounded variable that follows have to be treated as a unit, and this unit acts somewhat like a unary connective. Statements containing words like "every", "each", and "everyone" usually indicate universal quantification. Such statements must typically be reworded such that they start with "for every x", which is then translated to x. Example 1.5. Express "Everyone gets a break once in a while" in predicate logic. Solution: We define B to mean "gets a break once in a while". Hence, B(x) means that x gets a break once in a while. The word "everyone" indicates that this is true for all x. This leads to the following translation x B(x). Example 1.6. Express "All cats have tails" in predicate logic. Solution: We first have to find the scope of the universal quantifier, which is "If x is a cat, then x has tails". After choosing descriptive predicate symbols, we express this by the following compound formula: cat(x) hastails(x). This expression must be universally quantified to yield the required solution: x (cat(x) hastails(x)). At last, when we let C stand for "cat" and T stand for "hastails", we have the result x (C(x) T (x)). Under quantification, we are able to use what ever variable we like, for example y (C(y) T (y)) tells the same thing. 9

10 Definition 1.6. Let A represent an expression, and let x represent a variable. If we want to indicate that A is true for at least one value of x, we write x A. This statement is pronounced "There exists an x such that A". Here x is called the existential quantifier, and A is called the scope of the existential quantifier. The variable x is said to be bound by the quantifier. Statements containing such phrases as "some" and "at least one" suggest existential quantification. They should be rephrased as "there is an x such that", which is translated by x. Example 1.7. Let P be the property "like their meat raw". Then x P (x) can be translated as "There exists people who like their meat raw" or "Some people like their meat raw". Example 1.8. If the universe of discourse is a collection of things, x blue(x) should be understood as " There exist objects that are blue" or "Some objects are blue". Quantifiers x and x have to be treated like unary connectives. The quantifiers are given a higher precedence than all binary connectives. For example, if P (x) and Q(x) means that x is living and that x is dead, respectively, then one has to write x(p (x) Q(x)) to indicate that everything is either living or dead. x P (x) Q(x) means that everything is living or x is dead. The variable x in a quantifier is just a placeholder, and it can be replaced by any other variable name not appearing elsewhere in the expression. For example, x P (x) and y P (y) mean the same thing: they are logically equivalent. The expression y P (y) is called a variant of x P (x). Definition 1.7. An expression is called a variant of x A if it is of the form y S x y A, where y is any variable name, and S x y A is the expression obtained from A by replacing all instances of x by y. Similarly, x A and y S x y A are variants of one another. Quantifiers may be nested, as demonstrated by the following examples. 10

11 Example 1.9. Translate the sentence "There is somebody who knows everyone" into the language of predicate logic. To do this, use K(x, y) to express the fact that x knows y. Solution: The best way to solve this problem is to go in steps. We write informally x(x knows everybody). Here "x knows everybody" is still in English and means that for all y it is true that x knows y. Hence, x knows everybody = y K(x, y). We now add the existential quantifier and obtain x y K(x, y). Example Translate "Everybody has somebody who is his or her mother" into predicate logic. Solution: We define M to be the predicate "mother"; that is, M(x, y) stands for "x is the mother of y". The statement "Someone is the mother of y" becomes x M(x, y). To express that this must be true for all y, we add the universal quantifier, which yields y x M(x, y). The statement "Nobody is perfect" also includes a quantifier, "nobody", which is the absence of an individual with a certain property. In predicate logic, the fact that nobody has property P cannot be expressed directly. To express the fact that there is no x for which an expression A is true, one can either use x A or x A. For example, if P represents the property of perfection, both x P and x P indicate that nobody is perfect. In the first case, we say, in verbal translation, "It is not the case that there is somebody who is perfect", whereas in the second case we say "For everyone, it is not the case that he or she is perfect". The two methods to express that "nobody is A" must of course be logically equivalent, i.e., x A x A. (1.1) 11

12 According to Definitions 1.5 and 1.6, the variable appearing in the quantifier is said to be bound. For example, in the expression x (P (x) Q(x)), x appears three times, and each time x is bound variable. Any variable that is not bound is said to be free. Later, we will see that the same variable can occur both bound and free in an expression. For this reason, it is important to also indicate the position of the variable in question. Example Find the bound and free variables in z (P (z) Q(x)) y Q(y). Solution: Only the variable x is free. All occurrences of z are bound, and so are all occurrences of the variable y. Note that the status of a variable changes as expressions are divided into subexpressions. For example, in x P (x) x occurs twice, and it is bound both times. This statement contains P (x) as a subexpression. Nevertheless, in P (x), the variable x is free. Instantiations only affect free variables. Specifically, if A is an expression, S x t A only affects the free occurrence of the variable x. For example, S x y x P (x) is still x P (x), that is, the variable x is not free. However S x y (Q(x) x P (x)) yields Q(y) x P (x). Hence, instantiation treats the variable x differently, depending on whether it is free or bound, even if this variable appears twice in the same expression. Obviously, two things are only identical if they are treated identically. This implies that, if a variable appears both free and bound within the same expression, we have in fact two different variables that happen to have the same name. From this it follows that if several quantifiers use the same bound variable for quantification, then all these variables are local to their scope, and they are therefore distinct. To illustrate this, consider the statement "y has a mother". If M is the predicate name for "is mother of", then this statement translates into x M(x, y). One obviously must not form the variant y M(y, y), which means that y is her own mother. For similar reasons, there are restrictions on instantiations. For example, the instantiation S y x( x M(x, y)) is illegal. because its result is x M(x, x). In such cases, one tampers with the way in which a variable is defined, and this undesired side effects. 12

13 If all occurrences of x in an expression A are bound, we say "A does not contain x free". If A does not contain x free, then the truth-value of A does not change if x is instantiated to an individual constant. A is independent of x in this sense Restrictions of Quantifiers to Certain Groups Sometimes, quantification is over a subset of the universe of discourse. Suppose, for instance, that animals form the universe of discourse. How can one express sentences such as "All dogs are mammals" and "Some dogs are brown"? Consider first the statement "All dogs are mammals". Since the quantifier should be restricted to dogs, one rephrases the statement as "If x is a dog, then x is a mammal". This immediately leads to Generally, the sentence x(dog(x) mammal(x)). x(p (x) Q(x)) can be translated as "All individuals with property P also have property Q". Consider now the statement "Some dogs are brown". This statement means that there are some animals that are dogs and that are brown. Of course, the statement "x is a dog and x is brown" can be translated as dog(x) brown(x). "There are some brown dogs" can now be translated as The statement x(dog(x) brown(x)) x(p (x) Q(x)) can in general be interpreted as "Some individuals with property P also have property Q". Note that if the universal quantifier is to apply to individuals with a given property we use the conditional to restrict the universe of discourse. On the other hand, if we similarly restrict application of the existential quantifier, we use conjunction. 13

14 Finally, consider statements containing the word "only", such as "only dogs bark". To convert this into predicate logic, this must be reworded as "It barks only if it is a dog", or, equivalently, "If it barks, then it is a dog". One has therefore x (barks(x) dog(x)). 2 Interpretations and Validity 2.1 Introduction This section gives the semantical approach to predicate logic, and it deals with interpretations of logical statements and with the soundness of logical arguments. Interpretations are obviously fundamental to predicate logic, and they are therefore important in their own right. Moreover, interpretations allow one to distinguish between arguments that are sound and arguments that are not. Soundness is closely related to validity. Generally, an expression A is valid if A is true for all interpretations. Valid expressions in predicate logic play the same role as tautologies in propositional logic. In particular, logical implications and logical equivalences are defined as valid implications and valid equivalences, respectively. For consideration of truth of sentences of predicate language, we divide the alphabet of L into two parts, such that predicate symbols, function symbols, and individual constants form non-logical alphabet and other symbols logical alphabet of L. Non-logical alphabet refer to those things illustrated by the language, i.e., to objects and their mutual relationships outside the language. Hence, in each case, we can restrict L by giving a list of the non-logical alphabet under consideration. For example, L = {P, Q, R; a, b, c, d} is a predicate language having the predicate symbols P, Q, and R and the individual constants a, b, c, and d as its non-logical alphabet. 2.2 Interpretations An interpretation of a logical expression contains the following components: 14

15 1. There must be a universe of discourse. 2. For each individual, there must be an individual constant that exclusively refers to this particular individual, and to no other. 3. Every free variable must be assigned a unique individual constant. 4. There must be an assignment for each predicate used in the expression, including predicates of arity 0, which represents propositions. The truth status of a sentence can be determined after an interpretation is given to the sentence or to the whole L, i.e. we can speak about truth in some model. Let L = {P, Q,... ; a, b,...} be a predicate language and A a non-empty set. Definition 2.1. An interpretation of L in a set A is a valuation V that associates an element of A to every constant symbol and an n-ary relation to every n-ary predicate symbol in L. Example 2.1. Let L = {P, Q; a, b} be a predicate language where the both predicate symbols are binary predicates. Let A = Z + (the set of positive integers). We can choose an interpretation to L, for example, such that V (a) := 3, V (b) := 2, V (P (x, y)) := x < y, andv (Q(x, y)) := x y. A model of L consists of a non-empty set and an interpretation according to the following definition. Definition 2.2. A model of L is an ordered pair M = (A, V ) where A is a non-empty set and V is an interpretation of L in the set A. A is the universe of discourse of M. A term of predicate logic has an interpretation according to the following definition. 15

16 Definition 2.3. (a) If a term t is a constant symbol then its interpretation in the a A is a certain element V (t) A. (b) If a term t is a variable symbol then it can have any element of A as its value in a given interpretation V, i.e., V (t) A arbitrarily. Example 2.2. Consider the predicate language in Example 2.1 in the model M = (Z +, V ) where V is the valuation of Example 2.1. Determine the truthvalue of a formula P (c, b) Q(c, a) in the model M. We have the interpretation 3 < for the formula, and hence it is true in M. We consider more exactly the truth of a formula in a model. Definition 2.4. Let M = (A, V ) be a model, and α a formula of L. (a) If P is a k-ary predicate symbol and a 1,..., a k A are constants then P (a 1,..., a k ) is true in M, denote M = P (a 1,..., a k ), iff the k-tuple (a 1,..., a k ) belongs to the relation V (P ). (b) A formula α is true in a model M iff α is false in M, i.e., M = α iff M α. (c) M = α β iff M α or M = β. (d) M = x α iff M = S x aα(x) for any a A. Metatheorem 2.1. Let M = (A, V ) be a model, α and β formulas of L, and γ an open formula of L where x is free. Hence we have (a) M = α β iff M = α M = β. (b) M = α β iff M = α or M = β. (c) M = α β iff M = α and M = β. (d) M = α β iff either M = α and M = β or M α and M β. (e) M = x γ(x) iff M = S x aγ(x) for any a A. 16

17 (f) M = x γ(x) iff M = S x aγ(x) for some a A. Proof. The assertions of Metatheorem follow from Definition 2.4 by the mutual dependency of connectives and that of quantifiers. Note that only closed formulas can have a truth-value. It is nonsense to speak about the truth status of open formulas. On the other hand, we can speak about satisfaction of open formulas in a sense that in an interpretation variables of an open formula can have such values belonging to the universe of discourse of the correspondent model, and according to the interpretation, it is possible to get a true expression from the formula. It may also happen that a given expression is not true and not false in a model. For this reason, we define for every statement α in L a set of relevant models as follows: M α = {M M = (A, V ), such that either M = α or M α}. Definition 2.5. A statement α is valid iff M = α for all M M α. A statement α is satisfiable iff M = α for some M M α. A statement α is refutable iff M α for some M M α. A statement α is contradiction iff M α for all M M α. The closure of an open formula α is the closed formula obtained from α by bounding universally all occurrences of the free variables in α. Hence, according to Definition 2.5, the validity of an open formula means the consideration of truth of the closure of the formula in all relevant models. Definition 2.6. Let be a set of statements of L and M = (A, V ) a model. M is the model of the set of statements iff M = α for all statements α of the set, denoted by M =. Now the concept of semantic entailment (or logical consequence) in L can be defined. Definition 2.7. A statement α is a semantic entailment of a set of statements iff α is true in every model of. 17

18 Example 2.3. Show that the statement y x Q(x, y) x y Q(x, y) is valid. Let M = (A, V ) be any relevant model of the statement. If M y x Q(x, y) then M = y x Q(x, y) x y Q(x, y) by Definition 2.4 (c). Thus, we consider the case where M = y x Q(x, y). In this case, M = x Q(x, b) for some element b A by Metatheorem 2.1 (f). Hence, for some element b A it holds that for all elements a A, M = Q(a, b) by Definition 2.4 (d). Hence, for all elements a A, there exists an element b A, such that M = Q(a, b). Hence, M = y Q(a, y) holds foa all a A by Metatheorem 2.1 (f), and thus M = x y Q(x, y) holds by Definition 2.4 (d). Hence, in this case, too, M = y x Q(x, y) x y Q(x, y) holds by Definition 2.4 (c). Because M was an arbitrarily chosen relevant model of α, the formula y x Q(x, y) x y Q(x, y) is valid. Example 2.4. Show that the statement x y Q(x, y) y x Q(x, y) is not valid. Consider the formula in the model M = (N, V ) (N = {1, 2,...} is the set of natural numbers) where V (Q) = {(a, b) N N a < b}. Hence, M = x y Q(x, y), because for all natural number, there exists a natural number b, such that M = Q(a, b), i.e. a < b. On the other hand, M y x Q(x, y) because there does not exist a natural number b, such that M = Q(a, b) for all natural numbers a, because the set N does not have the greatest element. Hence, M is such a model where our statement is not true. From this it follows that the statement is not valid. Consider the satisfaction of an arbitrary formula more closely. Let M = (A, V ) be a model of L and t 1,..., t n are terms in L. 18

19 Definition 2.8. Let P be a unary predicate symbol. Then a formula P (t) is satisfied in a model M = (A, V ) iff in the interpretation V, t has a value in A, such that it has the property V (P ), i.e. V (t) V (P ). Example 2.5. Consider a formula P (x), and let V (P (x)) := x is a father, i.e. V (P (x)) = {x x is a father. Let A be the set of students. Hence the formula P (x) is satisfied in the model M = (A, V ) iff V gives x a value a A, such that a is a father, i.e. x is a student who is also a father. Definition 2.9. Let P be a n-ary predicate symbol, n 2. A formula P (t 1,..., t n ) is satisfied in a model M = (A, V ) iff in the interpretation V, the terms t 1,..., t n have such values V (t i ) A (i = 1,... n) which are in the relation V (P ), i.e. (V (t 1 ),..., V (t n )) V (P ). Example 2.6. Consider a formula P (xc), and let M = (A, V ) be a model, such that A is the set of soldiers of Finnish army, V (P (x, y)) means that x is the superior of y, and V (c) is a soldier named Karhunen (we use the individual constant k to refer him/her), i.e. k A. Hence, the formula P (x, c) is satisfiable in M iff a A, such that (a, k) V (P (x, y)), i.e. the formula P (xc) is satisfiable in M iff a is a superior of k, for example, a is a captain and Karhunen is a sergeant. If we want, we can use the satisfiability in a model of a formula when determining the truth status of a statement in a given model. However, we have to distinguish the case "a statement to be satisfiable" mentioned in Definition 2.5 from the case "a formula to be satisfiable in a model" mentioned in Definitions 2.5 and 2.9. Example 2.7. Show that the formula x (P (x) P (x)) is valid. Let M = (A, V ) be an arbitrary relevant model. Hence M = x (P (x) P (x)) iff M = P (a) P (a) for all a A. 19

20 This is the case iff M = P (a) or M = P (a) for all a A, i.e. M = P (a) or M P (a) for all a A. It is obviously clear that P (a) is either true or false in any model where a is any element of the domain of the model. From this it follows that x (P (x) P (x)) is true in any relevant model and hence valid. Example 2.8. Show that the formula x (P (x) P (x)) is a contradiction. Let M = (A, V ) be an arbitrary relevant model. Hence M x (P (x) P (x)) iff M = x (P (x) P (x)) iff M = x (P (x) P (x)) iff M = x ( P (x) P (x)), which holds by Example 2.7. Hence the formula x (P (x) P (x)) is a contradiction. Example 2.9. Show that a formula x (P (x) Q(x)) is satisfiable. Consider the model M = (Z, V ), such that V (P (x)) = {x x > 2} and V (Q(x)) = {x x < 10}. M = x (P (x) Q(x)) iff M = P (a) Q(a) for some a Z. This is the case iff M = P (a) and M = Q(a), i.e. M Q(a). For example, the value a = 15 satisfies the last condition, because 15 V (P ) (i.e. 15 > 2) and 15 / V (Q) (i.e ). Hence, the formula is satisfiable. Example Examine, whether the formula x (P (x) Q(x)) is refutable. Consider the model M = (Z, V ) where V (P (x)) = {x x = 2} and V (Q(x)) = {x x < 10}. 20

21 The interpretation in the model M is as follows: x (x = 2 x 10) which is false for any integers x Z. Hence, M x (P (x) Q(x)) and thus the formula is refutable. Example Examine, whether the set of formulas = { x(p (x) Q(x)), x( P (x) Q(x))} is satisfiable. We refer to the first formula by (1) and to the second formula by (2). Let M = (A, V ) be a model, such that A = {1, 2, 3, 4}, V (P (x)) = {x x > 2}, and V (Q(x)) = {x x > 1}. M = x(p (x) Q(x)) iff M = P (a) Q(a) for all a A. Clearly, x > 2 x > 1 holds for all elements of A. Hence, the formula (1) is true in M. M = x( P (x) Q(x)) iff M = P (a) Q(a) for some a A. Clearly, x 2 x > 1 holds for example for 2 A. Hence, (2) is true in M. Because there exists at least one model, such that the formulas (1) and (2) are true in it, the set is satisfiable. Example Examine, whether in the following set of formulas the last formula follows logically from the two first formulas: = { x(p (x) Q(x)), x(p (x) R(x)), x(q(x) R(x))} We refer to these formulas from the first to the third by the respective numbers (1), (2), and (3). We consider the formulas in the model M = (A, V ) where A = {1, 2, 3, 4}, V (P (x)) = {x x is odd}, V (Q(x)) = {x x < 5}, and V (R(x)) = {x x is positive}. (1) M = x(p (x) Q(x)) iff M = P (a) Q(a) for all a A. All the odd numbers are less that 5 in A. Hence the formula (1) is true in M. 21

22 (2) M = x(p (x) R(x)) iff M = P (a) R(a) for all a A. All the odd numbers are positive in A. Hence the formula (2) is true in M. (3) M = x(q(x) R(x)) iff M = Q(a) R(a) for all a A. Because the elements of A which are less than 5, are not positive, the formula (3) is not true in M. Because there exists at least one model of formulas (1) and (2), where the formula (3) is not true, the formula (3) is not a logical consequence of the formulas (1) and (2). 3 Axiomatization and Proof Theory In predicate logic, the concepts deduction, proof, and theorem are defined in the similar way as in propositional logic. 3.1 Axiom Schemes Axiom schemes of predicate logic are created by extending those of propositional logic. Hence, we add to the axioms of propositional logic additional axioms concerning quantification. Also, we add to the set of inference rules a rule concerning the use of universal quantifier. The meta-variables appearing in the axioms coming from propositional logic refer to formulas of predicate logic. Definition 3.1 (Axioms). Let α, β, and γ be formulas of predicate logic, then the following formulas are the axiom schemes of predicate logic: A1 α (β α), A2 (α (β γ)) ((α β) (α γ)), A3 ( β α) (α β), A4 x α S x aα, A5 x(α β) (α x β) if x is not free in α. 22

23 Definition 3.2 (Inference Rules). The inference rules of predicate logic are as follows: R1 Modus ponens (MP): R2 Universal Generalization (UG): α α β β α x α, if x does not appear as a free variable in any premises. According to the rule Universal Generalization, since x becomes bound in the process, we say that the universal generalization is over x, or that one generalizes over x. We will justify universal generalization later. At the moment, it must be pointed out that universal generalization is subject to restrictions, If one generalizes over x, then x must not appear in any premise as free variable, If x does appear free in any premise, then x always refers to the same individual, and it is fixed in this sense. For example, if P (x) appears in a premise, then P (x) is only true for x and not necessarily true for any other individual. If x is fixed, one cannot generalize over x. Generalizations from one particular individual toward the entire population are unsound. If, on the other hand, x does not appear in any premise or if x is bound in all premises, then x is assumed to stand for everyone, and universal generalization may be applied without restriction. Example 3.1. To demonstrate universal generalization, consider the following problem whose domain consists of a group of computer science students. Of course, all computer science students like programming. The derivation must prove that everyone in the domain likes programming. If P (x) and Q(x) stand for "x is a computer science student" and "x likes programming", respectively, the premises become x P (x), x (P (x) Q(x)). 23

24 The desired conclusion is x Q(x). Hence, we have to prove that x P (x), x (P (x) Q(x)) x Q(x). The deduction is as follows: Formula Rule Comment 1. x P (x) premise Everyone is a CS major. 2. x (P (x) Q(x)) premise CS majors like programming. 3. P (x) Sx, x 1 x is a CS major. 4. P (x) Q(x) Sx, x 2 If x is a CS major, x likes programming. 5. Q(x) MP, 3,4 x likes programming. 6. x Q(x) UG, 5 Everyone likes programming. In the proof, Q(x) is derived in line 5, which means that x likes programming. This statement is then generalized to x Q(x). This generalization is only possible because all instances of x in the premises are bound. If the premise x P (x) is replaced by P (x), then universal instantiation over x is no longer sound. This is the case because x is fixed and universal generalization over fixed variables is unsound. Example 3.2. As a second example, we derive y x P (x, y) from x y P (x, y). We have the deduction: Formula Rule Comment 1. x y P (x, y) premise 2. y P (x, y) Sx x Drop the first quantifier. 3. P (x, y) Sy y Drop the second quantifier. 4. x P (x, y) UG, 3 This is a sound generalization. 5. y x P (x, y) UG, 4 Generalize again to obtain the desired conclusion. On the line 3, we drop the second quantifier to obtain an expression without quantifiers. Then we use UG to add the quantifiers back in reverse order. On the line 4, it is sound to generalize, because the premise does not contain x as a free variable. All occurrences of x in the premises are bound. The situation is similar on the line 5. 24

25 Example 3.3. In the third example of UG, we show that the variable x in a universal quantifier may be changed to the variable y i.e. we prove x P (x) y P (y): Formula Rule Comment 1. x P (x) premise 2. P (y) Sy x, 1 Instantiate the premise for y. 3. y P (y) UG, 2 Generalize P (y) to obtain the conclusion. In fact, when we use the argument of instantiation St x the axiom A4 in the following way: in deductions, we use Formula Rule Comment 1. x P (x) premise Premise, (or deduced from earlier steps). 2. x P (x) P (t) A4 A4 is used in its original meaning. 3. P (t) MP, 1,2 This is the result of St x applied to 1. Hence, we have proved that x P (x) P (t). This justifies the use of instantiation in that form used in the previous inferences. Thus, we have a derived the rule of universal specification (US): x α(x) S x t α(t) Example 3.4. Consider an additional example about the rule US. We prove the following inference to be correct: Healthy people live long. Socrates was healthy. Socrates lived long. To do the derivation, let H(x) indicate that x is a healthy people, L(x) x lives (or lived) long, and s stands for Socrates. We have the deduction: 1. x (H(x) L(x)) premise 2. H(s) premise 3. H(s) L(s) US, 1 4. L(s) MP, 2,3 25

26 3.2 Deduction Theorem and Universal Generalization In the deduction theorem, one assumes B, proves C, using the assumption B like a premise, and concludes, that B C. Once this is done, B is discharged. The question now is how to treat free variables occurring in B. First, while B is used as an assumption, that is, as long as B is not discharged, B has to be treated like any other premise. In particular, should B contain x as a free variable, then one must not generalize over x. However, as soon as B is discharged, this is no longer true. Once B is discharged, it has no effect whatsoever on the status of any variable. Hence, if x is not free in any other premise, one can universally generalize over x even if x appears free in B. The deduction theorem is now demonstrated by an example. Example 3.5. Let S(x) stand for "x studied" and P (x) stands for "x passed". The premise is that everyone who studied passed. Prove that everyone who did not pass did not study. Solution: The premise "everyone who studied passed" can be translated as x(s(x) P (x)), and the statement "everyone who did not pass did not study" becomes x( P (x) S(x)). Formula Rule Comment 1. x(s(x) P (x)) premise Everyone who studied passed. 2. S(x) P (x) US, 1 If x studied, x passed. 3. P (x) assumption Assume that x did not pass. 4. S(x) MT, 2,3 x cannot have studied. 5. P (x) S(x) DT, 3,4 Apply DT and discharge P (x). 6. x( P (x) S(x)) UG, 5 Anyone who did not pass cannot have studied. To obtain the result, the assumption P (x) is introduced in line 3. As long as this assumption is not discharged, no generalization over x is allowed. To indicate that an assumption is in effect, lines 3 and 4 are indented. However, once the deduction theorem is applied, the indentation is removed and the assumption P (x) is discharged, and one can generalize over x. This is done in line 6. In all other aspects, the proof is self-documenting. x(s(x) P (x)) is a premise, and x( P (x) S(x)) follows. To arrive 26

27 at the desired conclusion, one uses universal generalization. This can be done because x is not free in any premise. 3.3 Dropping the Universal Quantifiers In mathematics, universal quantifiers are frequently omitted. For example, in the statement x + y = y + x, both x and y are implicitly universally quantified. This causes problems when such statements are used as premises because, according to our rules, any variable appearing free in a premise is foxed in the sense that throughout the proof it is bound to one and the same individual. To get around this difficulty, we single out certain variables in the premises and explicitly state that these variables are not fixed. All variables that are not fixed will be called true variables. A variable may be universally generalized if and only if it is a true variable. If a variable appears in a premise, then it is assumed to be fixed, unless it is explicitly stated that the variable is a true variable. By using true variables, one can omit many universal quantifiers and this, in turn, simplifies proofs. Moreover, we allow from now on that any true variable can be instantiated to any term. The same effect can, of course, be achieved by using universal generalization first, following by universal instantiation. However, direct instantiation is shorter and often clearer. Until now, instantiations were always represented by the symbols S, such as S x y. From now on, we will frequently make use of the notation x := y to indicate that x is replaced by y. In some programming languages, the notation := means "assign to", which is the same as "instantiate to". Example 3.6. Let P (x, y, z) : x + y = z. Given the premises P (x, 0, x) and P (x, y, z) P (y, x, z), where x, y, and z are true variables, prove that 0 + x = x, i.e., prove P (0, x, x). Solution: The following derivation is used to prove P (0, x, x). Note that the first two lines are premises and that x, y, and z are explicitly declared as true variables. We have the deduction: 27

28 1. P (x, y, z) P (y, x, z) Premise: x + y = z y + x = z, x, y, and z are true. 2. P (x, 0, x) Premise: x + 0 = x, x is a true variable. 3. P (x, 0, x) P (0, x, x) Line 1 with x := x, y := 0, z := x. 4. P (0, x, x) MP, 2,3; 0 + x = x. All true variables are strictly local to the line on which they appear. Hence, if the true variable x appears on two different lines, then these two instances of x are really two different variables. For example, in the proof of Example 3.6, x in line 1 and x in line 2 are two different variables. When doing the proof, one obviously has to establish some type of connection between the variables, and this connection is through instantiation. Of course, instantiations must not be made blindly. Instead, one has to do the instantiations in such a way that progress is made toward the desired conclusion. How this is done in detail depends on the general strategy, and in each proof, some type of strategy should be followed. However, there are some general principles that are helpful, and one of them is unification. Definition 3.3. Two expressions are said to unify if there are legal instantiations that make the expressions in question identical. The act of unifying is called unification. The instantiation that unifies the expressions in question is called a unifier. Example 3.7. Suppose the expressions Q(a, y, z) and Q(y, b, c) are expressions appearing on different lines. Show that the two expressions unify, and give a unifier. Here, a, b, and c are fixed, and y and z are true variables. Solution: Since y in Q(a, y, z) is a different variable than y in Q(y, b, c), rename y in the second expression to become y 1. This means that one must unify Q(a, y, z) with Q(y 1, b, c). An instance of Q(a, y, z) is Q(a, b, c), and an instance of Q(y 1, b, c) is Q(a, b, c). Since these two instances are identical, Q(a, y, z) and Q(y, b, c) unify. The unifier is a = y 1, b = y, c = z. There may be several unifiers. For example, if a and b are constants, then R(a, x) and R(y, z) have the unifier y = a, z = x, which yields the common instance R(a, x). However, there is also the unifier y = a, x = b, z = b, which 28

29 yields the common instance R(a, b). However, R(a, b) is an instance of R(a, x), and the unifier y = a, x = b, z = b is in this sense less general than the unifier y = a, z = x. Of course, we always want to find the most general unifier, if one exists. The solution of Example 3.6 involved unification. Specifically, to make use of the modus ponens, line 2 was unified with the antecedent of line 1. Generally, unification is performed in such a way that some rule of inference can be applied after unification. Example 3.8. Clearly, if x is the mother of y and if z is the sister of x, then z is the aunt of y. Suppose now that Brent s mother is Jane and Liza is Jane s sister. Prove that Liza is Brent s aunt. Solution: If "mother(x, y)" is the predicate that is true if x is the mother of y, and if "sister(x, y)" and "aunt(x, y)" are defined in a similar fashion, one can state the premises as follows: 1. mother(x, y) sister(x, y) aunt(x, y) 2. mother(jane, Brent) 3. sister(liza, Jane) The problem is now to create an expression that unifies with the antecedent of line 1. To do this, one combines lines 2 and 3 to obtain 4. mother(jane, Brent) sister(liza, Jane) This expression can be unified with mother(x, y) sister(z, x) by setting x := Jane, y := Brent, and z := Liza. This yields 5. mother(jane, Brent) sister(liza, Jane) aunt(liza, Brent) The conclusion that Liza is Brent s aunt now follows from 4 and 5 by modus ponens. 29

30 3.4 Rules for Existential quantifiers Existential Generalization If there is any term t for which P (t) holds, then one can conclude that some x satisfies P (x). Hence, P (t) logically implies that x P (x). More generally, x α can be derived from S x t α, where t is any term. This leads to the following rule of inference, existential generalization (EG): S x t α x α Example 3.9. Let c be Aunt Cornelia, and let P (x) stand for "x is over 100 years old". Then one has P (c) x P (x) The reason is that if one replaces x by c in P (x) then one finds P (c). The following example demonstrates demonstrates how to use existential generalization within a formal proof. The premises of our derivation are 1. Everybody who has won a million is rich. 2. Mary has won a million. We want to show that these two statements logically imply that 3. There is somebody who is rich. If somebody were asked to demonstrate that the conclusion follows from the premises, he or she would probably argue as follows. If everybody who wins a million euros is rich, then Mary is rich if she wins a million. Since we know that Mary has won a million, we apply modus ponens and conclude that Mary is rich. There is thus somebody, Mary, who is rich. This argument is now formalized. W (x) means that x has won a million, R(x) means that x is rich, and m stands for Mary. Hence, we prove that x (W (x) R(x)), W (m) x R(x). The deduction is as follows: 30

31 1. x (W (x) R(x)) premise 2. W (m) R(m) US, 1, x := m 3. W (m) premise 4. R(m) MP, 2,3 5. x R(x) EG, 4 It was stated earlier that x P (x) is logically equivalent to x P (x). We now prove the first half of this statement by showing that x P (x) x P (x). The deduction is as follows: 1. x P (x) premise 2. P (x) assumption 3. x P (x) EG, 2 4. P (x) x P (x) DT, 2,3 5. P (x) MT, 1,4 6. x P (x) UG, 3 We will later prove x P (x) x P (x) The two proofs together establish the logical equivalence of x P (x) and x P (x). Existential Spesification If x α is true, then there must be some term t that satisfies α; that is, S x t α must be true for some t. For example, if P (x) stands for "x does somersaults", then x P (x) means that S x t P (x) = P (t) must be true for some t. The problem is that we do not know whether it is Aunt Eulalia, Uncle Petronius or even somebody else who makes somersaults. In a proof, the question must therefore be kept as to who the individual is who makes somersaults. To do this, a new variable, say b, is selected to denote this unknown individual. This leads to the following rule of inference, existential specification (ES): x α S x t α The variable introduced by existential instantiation must not have appeared earlier as a free variable. For example, when applying ES to the two statements 31

32 "There exists someone who is over 100 years old" and "There exists someone who makes somersaults", one must not use the same variable b for existential instantiation in both cases. Otherwise, one could conclude that b is both over 100 and makes somersaults, which certainly does not follow logically. Similarly, one cannot use any variable that appears free in any of the premises. Hence, ES must not introduce any variable that has appeared already as a free variable in the derivation. Moreover, the variable introduced is fixed in the sense that one cannot use universal generalization over this variable. For example, if b makes somersaults, then one cannot use UG to conclude that everyone makes somersaults. Moreover, a variable with an unknown value must not appear in the conclusion, and since any variable introduced ny ES is unknown, it must not appear in the conclusion either. For the purpose of demonstration, suppose that there is someone who won a million euros, and we want to prove that there is someone who is rich. Hence, the premises are 1. Someone has won a million euros. 2. Everybody who has won a million is rich. We want to show that these two statements logically imply 3. There is somebody who is rich. Hence, we have to prove x (W (x) R(x)), x W (x) x R(x). The deduction is as follows: 1. x W (x) premise 2. W (b) ES, 1, x := b 3. x (W (x) R(x)) premise 4. W (b) R(b) US, 3, x := b 5. R(b) MP, 2,4 6. x R(x) EG, 5 In this proof, existential specification is used on line 2, where the winner is called b. Once this is obtained, the second premise is given on line 3, and this premise is instantiated with x := b on line 4. Note that one must not derive lines 3 and 4 before lines 1 and 2; that is, one must not apply the universal 32