The Language of First-Order Predicate Logic

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Adele Simon
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1 The Language of First-Order Predicate Logic (Note: First-Order Predicate Logic differs from ordinary Predicate Logic in that it contains individual variables and quantifiers. The designation first-order reflects the fact that our variables only range over individuals (i.e., the possible denotations for individual constants). A second-order logic is one that also contains variables ranging over sets of individuals, sets of ordered pairs of individuals, sets of ordered triplets of individuals, etc. (i.e., the possible denotations for predicate constants).) Vocabulary (list of basic expressions): (i) predicate constants: MALE, PHILOSOPHER, COMPOSER (1-place), ADMIRE (2-place), BETWEEN (3-place),... (ii) individual constants: a, b, c, d, e, f,... (iii) individual variables: w, x, y, z,... (iv) connectives: ~ (negation) &, v, (conjunction, disjunction, material implication) (v) quantifiers: (universal, read as for all/every individual... ) (existential, read as there is/exists an individual... (vi) parentheses: (, ) Syntax (rules that define well-formed formulas): (i) If P is an n-place predicate constant, and t 1, t 2,..., t n are n individual constants or variables, then P(t 1, t 2,..., t n ) is a formula of PredL. (ii) If A is a formula of PredL, then so is ~A. (iii) If A and B are formulas of PredL, then so are (A & B), (A v B), and (A B). (iv) If A is a formula of PredL, then so are xa and xa, for any individual variable x. (v) Nothing else is a formula. Semantics (rules that assign denotations to expressions): Two-step procedure for assigning denotations to PredL expressions: (A) Provide denotations for individual/predicate constants by defining a model. A model M consists of (i) a set D of individuals, and (ii) Val, which assigns a denotation (= semantic value) to each individual/predicate constant in PredL. Denotations assigned by Val: individual constants denote invidivuals one-place predicate constants denote sets of individuals two-place predicate constants denote sets of ordered pairs of individuals three-place predicate constants denote sets of ordered triplets of individuals 1

2 Semantics (rules that assign denotations to expressions, cont d): (B) Provide rules for assigning denotations (= truth values) to PredL formulas. 1 (i) Let P be a n-place predicate constant, and let c 1, c 2,..., c n be n individual constants. P(c 1, c 2,..., c n ) denotes T if <Val(c 1 ), Val(c 2 ),..., Val(c n )> is a member of Val(P). Otherwise, P(c 1, c 2,..., c n ) denotes F. (Rule (i) simply consolidates the three separate rules that we previously had for basic formulas formed with one-, two- and three-place predicate constants.) (ii) Let A be a formula of PredL, and let x be an individual variable. xa denotes T if A denotes T for at least one individual in D when x assumes that individual as its value. Otherwise, xa denotes F. (iii) Let A be a formula of PredL, and let x be an individual variable. xa denotes T if A denotes T for each individual in D when x assumes that individual as its value. Otherwise, xa denotes F. (iv) The truth values for complex formulas constructed with ~, &, v, and are given by our familiar truth tables: A ~A A B A & B A v B A B T F T T T T T F T T F F T F F T F T T F F F F T 1 Our semantics for First-Order Predicate Logic is actually incomplete. To see this, observe that our syntax rules allow for formulas like PHILOSOPHER(x), where x is an individual variable that is not associated with any quantifier. However, rule (i) does not provide any denotation (T or F) to this formula, since it only applies to formulas consisting of a predicate constant P followed by an appropriate number of individual constants. Moreover, even if we modified rule (i) to allow for individual variables, we still couldn t assign a denotation to PHILOSOPHER(x), since we don t have any way to assign denotations to individual variables. Individual variables resemble individual constants, since they can appear in the list of arguments immediately following a predicate constant. But they are also very different from individual constants, since they do not denote individuals. Rather, they serve as placeholders that indicate relationships between quantifiers and the argument positions of predicate constants. In rules (ii) and (iii), which provide denotations to quantified formulas, individual variables are then allowed to range over the individuals in our domain D. A full semantic treatment of First-Order Predicate Logic would remedy this problem by providing denotations to individual variables, even in the absence of any associated quantifiers. For our purposes, though, the simpler version will suffice. 2

3 Models, Denotations, and Truth Conditions Our semantics for First-Order Predicate Logic constitutes what we have called a denotational theory of meaning. In other words, our semantics takes meaning to be a relationship between expressions and the world, or more precisely, between expressions and a model. This is reflected most clearly in Val, which associates individuals, sets of individuals, sets of ordered pairs of individuals, etc. with the constants (individual or predicate) in our logical language. Formulas constructed out of these basic expressions in turn come to denote truth values. The truth conditions for a formula are then constituted by those conditions that must be satisfied by a model in order for the formula to denote T(rue). Here are some examples showing how the semantics for First-Order Predicate Logic works, and how the truth conditions for formulas can be stated in terms of a model. We will consider the following model: D = {Socrates, Aristotle, Plato, Mozart, Beethoven} Val(s) = Socrates Val(a) = Aristotle Val(p) = Plato Val(m) = Mozart Val(b) = Beethoven Val(MALE) = {Socrates, Aristotle, Plato, Mozart, Beethoven} Val(PHILOSOPHER) = {Socrates, Aristotle, Plato} Val(COMPOSER) = {Mozart, Beethoven} Val(ADMIRE) = {<Socrates, Socrates>, <Socrates, Aristotle>, <Mozart, Beethoven>, <Beethoven, Mozart>, <Plato, Mozart>} Val(BETWEEN) = {<Plato, Socrates, Aristotle>} (1) a. Plato is a philosopher. b. PHILOSOPHER(p) TCs: (1b) denotes T just in case Val(p) is a member of Val(PHILOSOPHER). Note that the truth conditions for (1b) do not make reference to any specific facts about the above model. Rather, they state a general condition ( Val(p) is a member of Val(PHILOSOPHER) ) that must be met in order for (1b) to be true relative to any model M. We can then check whether this general condition is met by the actual model under consideration. Denotation: relative to the above model, (1b) denotes T, since Val(p) = Plato, and Plato is a member of Val(PHILOSOPHER). 3

4 (2) a. Socrates admires himself. b. ADMIRE(s, s) TCs: (2b) denotes T just in case <Val(s), Val(s)> is a member of Val(ADMIRE). Denotation: relative to the above model, (2b) denotes T, since Val(s) = Socrates, and <Socrates, Socrates> is a member of Val(PHILOSOPHER). (3) a. Mozart is a male philosopher. b. MALE(m) & PHILOSOPHER(m) (Note that our syntax rules do not allow us to combine predicates to form a complex predicate like MALE-PHILOSOPHER. Rather, (3a) must be translated as the conjunction of two one-place predicates, each applying to the same individual constant.) TCs: (3b) denotes T just in case both MALE(m) and PHILOSOPHER(m) denote T. This will be so whenever Val(m) is a member of Val(MALE), and Val(m) is also a member of Val(PHILOSOPHER). Denotation: relative to the above model, (3b) denotes F. This is because Val(m) = Mozart, and although Mozart is a member of Val(MALE), Mozart is not a member of Val(PHILOSOPHER). So, the first conjunct denotes T in (3b), but the second conjunct denotes F, rendering the entire conjunction false. Question: How would we translate the following English sentences? What are the truth conditions for their logical translations? And do these translations denote T(rue) or F(alse) relative to the above model? (4) a. Mozart and Beethoven admire each other. b. If Beethoven is a composer, then he is also a philosopher. c. Neither Socrates nor Aristotle admires Mozart. 4

5 (5) a. Everyone is male. b. xmale(x) TCs: (5b) denotes T just in case MALE(x) denotes T for each individual in D when x assumes that individual as its value. Denotation: to determine whether (5b) denotes T relative to the above model, we must examine the truth value of MALE(x) as x successively takes each individual in D as its value. So there are five cases to consider (one per individual in D): x Socrates: MALE(x) denotes T, since Socrates is a member of Val(MALE). x Aristotle: MALE(x) denotes T, since Aristotle is a member of Val(MALE). x Plato: MALE(x) denotes T, since Plato is a member of Val(MALE). x Mozart: MALE(x) denotes T, since Mozart is a member of Val(MALE). x Beethoven: MALE(x) denotes T, since Beethoven is a member of Val(MALE). We see that MALE(x) invariably denotes T as x successively takes each individual in D as its value. So, (5b) denotes T relative to the above model. (6) a. Socrates admires someone. b. zadmire(s, z) TCs: (6b) denotes T just in case ADMIRE(s, z) denotes T for at least one individual in D when z assumes that individual as its value. Denotation: We must ask whether there is an individual in D such that the ordered pair consisting of Socrates (= Val(s)) and that individual is a member of Val(ADMIRE). If there is, then the truth conditions for (6b) will be satisfied, since ADMIRE(s, z) will then denote T for at least one individual in D when z assumes that individual as its value. In fact, when z Aristotle, ADMIRE(s, z) denotes T, since <Socrates, Aristotle> is a member of Val(ADMIRE). So (6b) denotes T relative to the above model. 5

6 (7) a. Plato admires a composer. b. x(composer(x) & ADMIRE(p, x)) TCs: (7b) denotes T just in case COMPOSER(x) & ADMIRE(p, x) denotes T for at least one individual in D when x assumes that individual as its value. This will be so whenever both COMPOSER(x) and ADMIRE(p, x) denote T for at least one individual in D when x assumes that individual as its value. Denotation: We must ask whether there is an individual in D such that the individual is a member of Val(COMPOSER), and the ordered pair consisting of Plato (= Val(p)) and the individual is a member of Val(ADMIRE). If there is, then the truth conditions for (7b) will be satisfied, since both COMPOSER(x) and ADMIRE(p, x) will denote T for at least one individual in D when x assumes that individual as its value. In fact, when x Mozart, COMPOSER(x) denotes T, since Mozart is in Val(COMPOSER), and ADMIRE(p, x) also denotes T, since <Plato, Mozart> is in Val(ADMIRE). So (7b) is true relative to the above model. (8) a. Every philosopher is male. b. x(philosopher(x) MALE(x)) TCs: (8b) denotes T just in case PHILOSOPHER(x) MALE(x) denotes T for each individual in D when x assumes that individual as its value. Denotation: to determine whether (8b) denotes T relative to the above model, we must examine the truth value of PHILOSOPHER(x) MALE(x) as x successively takes each individual in D as its value. Remember that according to the truth table for, if PHILOSOPHER(x) denotes F, then PHILOSOPHER(x) MALE(x) automatically denotes T: PHILOSOPHER(x) MALE(x) P(x) M(x) T T T T F F F T T F F T Thus, for any individual in D that is not a member of Val(PHILOSOPHER), the entire formula PHILOSOPHER(x) MALE(x) will denote T when x takes that individual as its value. There are five cases to consider (one per individual in D): x Socrates: PHILOSOPHER(x) denotes T, since Socrates is a member of Val(PHILOSOPHER), and MALE(x) denotes T, since Socrates is a member of Val(MALE) (ln 1 of truth table; P(x) M(x) denotes T) x Aristotle: PHILOSOPHER(x) denotes T, since Aristotle is a member of Val(PHILOSOPHER), and MALE(x) denotes T, since Aristotle is a member of Val(MALE) (ln 1 of truth table; P(x) M(x) denotes T) 6

7 x Plato: PHILOSOPHER(x) denotes T, since Plato is a member of Val(PHILOSOPHER), and MALE(x) denotes T, since Plato is a member of Val(MALE) (ln 1 of truth table; P(x) M(x) denotes T) x Mozart: PHILOSOPHER(x) denotes F, since Mozart is not a member of Val(PHILOSOPHER) (lns 3/4 of truth table; P(x) M(x) denotes T) x Beethoven: PHILOSOPHER(x) denotes F, since Beethoven is not a member of Val(PHILOSOPHER) (lns 3/4 of truth table; P(x) M(x) denotes T) We see that PHILOSOPHER(x) MALE(x) invariably denotes T as x successively takes each individual in D as its value. So, (8b) denotes T relative to the above model. Question: What s wrong with the translation in (9)? (9) x(philosopher(x) & MALE(x)) (Hint: consider whether the logical formulas in (8b) vs. (9) allow for the existence of non-philosophers.) Is there an English sentence for which (9) would be an appropriate translation? 7

8 (10) a. No composer is a philosopher. b. ~ x(composer(x) & PHILOSOPHER(x)) TCs: (10b) denotes T just in case x(composer(x) & PHILOSOPHER(x)) denotes F. This will be so whenever COMPOSER(x) & PHILOSOPHER(x) denotes F for every individual in D when x assumes that individual as its value. This will be so whenever at least one of the two formulas COMPOSER(x) and PHILOSOPHER(x) denotes F for every individual in D when x assumes that individual as its value. Denotation: to determine whether (10b) is true relative to the above model, we must examine the truth values of COMPOSER(x) and PHILOSOPHER(x) as x successively takes each individual in D as its value. So there are five cases to consider (one per individual in D): x Socrates: COMPOSER(x) denotes F, since Socrates is a not member of Val(COMPOSER). x Aristotle: COMPOSER(x) denotes F, since Aristotle is a not member of Val(COMPOSER). x Plato: COMPOSER(x) denotes F, since Plato is a not member of Val(COMPOSER). x Mozart: PHILOSOPHER(x) denotes F, since Mozart is a not member of Val(PHILOSOPHER). x Beethoven: PHILOSOPHER(x) denotes F, since Beethoven is a not member of Val(PHILOSOPHER). We see that invariably, one of the two formulas COMPOSER(x) and PHILOSOPHER(x) denotes F as x successively takes each individual in D as its value. So, (10b) denotes T relative to the above model. 8

COMP Intro to Logic for Computer Scientists. Lecture 2

COMP Intro to Logic for Computer Scientists. Lecture 2 COMP 1002 Intro to Logic for Computer Scientists Lecture 2 B 5 2 J Twins puzzle There are two identical twin brothers, Dave and Jim. One of them always lies; another always tells the truth. Suppose you