PART TWO PREDICATE LOGIC

Chapter Seven Predicate Logic Symbolization

Predicate Logic, or Quantifier Logic, is concerned with the interior structure of both atomic and compound sentences.

1. Individuals and Properties In atomic sentences a property is ascribed to some individual. For example, “Art is happy” has the property of “being happy” ascribed to Art. We can symbolize this as Ha, with the uppercase letter denoting the property and the lowercase letter denoting the individual.

Individuals and Properties, continued Properties that hold between two or more entities are called relational properties. Properties such as “is happy” which are ascribed to only one individual are monadic properties.

Individuals and Properties, continued Capital letters used to denote properties are called property constants, and lowercase letters (up to an including the letter t) used to denote things, objects, and individual entities are called individual constants. The lowercase variables u through z are used as individual variables, replaceable by individual constants.

2. Quantifiers and Free Variables In predicate logic two symbols, called quantifiers, are used to state how many. The universal quantifier is used to state that all entities have some property or properties. The existential quantifier is used to assert that some individual or individuals have one or more properties.

Quantifiers and Free Variables, continued In predicate logic, parentheses indicate the scope of a quantifier. For example, the sentence “Everything has mass and is extended” is symbolized as (x)(Mx. Ex) (where Mx = “x has mass” and Ex = “x is extended”). The parentheses around the expression (Mx. Ex) indicate that the scope of the (x) quantifier is the remaining part of the sentence.

Quantifiers and Free Variables, continued The expression (x) (Mx. Ex) is a sentence, but (Mx. Ex) is a sentence form, not a sentence. The expression (x)(Mx). Ex is not a sentence as it contains an individual variable that is not quantified, the Ex. Unquantified variables are called free variables. Quantified variables are called bound variables.

3. Universal Quantifiers When we use the universal quantifier, (x), we are saying something about all the individuals represented by the variable in the quantifier. Since few properties can be ascribed to everything, we might need to restrict our domain of discourse.

4. Existential Quantifiers The existential quantifier ( ∃ x) is used to assert that some entities (at least one) have a given property. For example, “Something is heavy” can be symbolized as ( ∃ x)Hx

5. Basic Predicate Logic Symbolizations Sentences that begin with words such as “all” “every” or “any” can be symbolized using the universal quantifier. (x) (Dx ⊃ Fx) Sentences of the form “Some A’s are B’s” can be symbolized using the existential quantifier. ( ∃ x) (Dx. Fx)

Basic Predicate Logic Symbolizations, continued Sentences of the form “No A’s are B’s” can be symbolized using either the existential quantifier: ˜( ∃ x) (Dx. Fx) Or the universal quantifier: (x) (Dx ⊃ ˜ Fx)

Basic Predicate Logic Symbolizations, continued Sentences of the form “Not all A’s are B’s” can be symbolized using the universal quantifier: ˜(x) (Dx ⊃ Fx) Or the existential quantifier: ( ∃ x) (Dx. ˜ Fx)

6. The Square of Opposition Any sentence that can be symbolized with a universal quantifier can be symbolized with an existential quantifier, and vice versa. A traditional way of illustrating the relationship between the quantifiers is known as the square of opposition.

7. Common Pitfalls in Symbolizing with Quantifiers Some English expressions that look like compound subjects or predicates should not be symbolized that way. Be careful with sentences that contain the word “a” or “any”. These sometimes operate logically like the particle “all”, but often they do not. Be careful when symbolizing conjunctions!

8. Expansions A reliable guide to translation is to compare the truth conditions of the ordinary language sentence we are translating with those of its translation. In a universe with four individual entities (x)Fx would be true if (Fa. Fb). (Fc. Fd), the expansion of (x) Fx with respect to that universe, was true.

Expansions, continued A model universe is a domain of a small number of individuals about which we shall construe our quantified statements.

9. Symbolizing “Only, “None but,” and “Unless”” Statements such as “Only those who study will pass the test” should be symbolized as (x)(Px ⊃ Sx), with the universe of discourse restricted to the students in question. Sentences such as “None but the good die young” should be symbolized as (x)(Yx ⊃ Gx) Sentences such as “No one will pass the test unless he or she studies” should be symbolized as (x)(Px ⊃ Sx).

Key Terms Bound variable Domain of discourse Existential quantifier Expansion Free variable Individual constant Individual variable Model universe Predicate logic

Key Terms, continued Property constants Quantifier Relational property Scope of a quantifier Square of opposition Substitution Universal quantifier Universe of discourse