10.1 Singular Propositions

This chapter introduces the process of expanding symbolic analysis to permit the evaluation of deductive arguments in which statements that are not truth-functional compounds occur as premisses or conclusions.

The simplest kind of noncompound statement is a singular proposition—a proposition that asserts that a particular individual has (or does not have) some specified attribute. The term "individual" is used to refer not only to individual people, but also to any individual thing of which an attribute can be meaningfully predicated.

To distinguish between individuals and attributes, we introduce two different kinds of symbols. For individual constants—symbols to represent specific individuals—we will use the lowercase letters a through w.

To designate predicated attributes we use uppercase letters.

A singular proposition is represented by an attribute symbol followed by a symbol for the individual of whom or which the attribute is being asserted.

The lowercase letter x is used not as a constant but as an individual variable. The expression Px ("x is president") is a propositional function. In this example, Px is a simple predicate, which is a propositional function having some true and some false substitution instances, each instance of which is an affirmative singular proposition. The singular propositions Pa ("Al Gore is president"), Pg ("George W. Bush is president") are either true or false, but Px is neither true nor false, not being a statement or proposition at all.

10.2 Quantification

The terms with which they begin are quantifiers, and the process of deriving propositions from propositional functions is called quantification.

The symbol (x) is used as a universal quantifier. Placed before a propositional function, it asserts that the predicate following it is true of everything. Thus, (x)Fx means "Given any x, F is true of it."

The symbol (?x) is used as an existential quantifier. Placed before a propositional function it asserts that the predicate following it is true of one or more substitution instances. Thus, (?x)Fx means "There is at least one x such that F is true of it."

The universal quantification of a propositional function is true if and only if all its substitution instances are true.

The existential quantification of a propositional function is true if and only if it has at least one true substitution instance.

To enlarge the concept of propositional functions to encompass negative as well as affirmative propositions, we introduce the concept of negation, symbolized by ~. The proposition "Nothing is mortal" can be paraphrased as "Given any x, it is not mortal" and with the use of the universal quantifier and the negation symbol may be symbolized as (x) ~Mx. The proposition "Some things are not mortal" can by symbolized, by use of the existential quantifier, as (?x) ~Mx.

Using the Greek letter phi (f) to represent any simple predicate whatsoever, the relations between universal and existential quantification are captured by these four logically true biconditionals:

[(x)fx] = [~(?x) ~f x]

[(?x)fx] = [~(x) ~f x]

[(x)~fx] = [~(?x) f x]

[(?x)~fx] = [~(x)f x]

Arranging affirmative and negative universal propositions and affirmative and negative existential propositions on a square of opposition clarifies their relationships.

10.3 Traditional Subject-Predicate Propositions

The concept of a propositional function can be applied to the analysis of the standard-form categorical propositions and the standard-form categorical syllogisms of Aristotelian logic. The four standard-form categorical propositions are:

A: Universal affirmative (for example, "All knights are warriors.")

E: Universal negative (for example, "No knights are warriors.")

I: Particular affirmative (for example, "Some knights are warriors.")

O: Particular negative (for example, "Some knights are not warriors.")

Thus for the A proposition "All knights are warriors" we get:

A: (x)(Kx ? Wx)

For the E proposition "No knights are warriors" we get:

E: (x)(Kx ? ~ Wx)

For the I proposition "Some knights are warriors" we get:

I: (? x) (Kx· Wx)

And for the O proposition "Some knights are not warriors" we get

O: (? x) (Kx· ~Wx)

Using the Greek letters phi (f) and psi (?) to stand for any predicates whatever, the four standard-form categorical propositions may be symbolized as follows:

A: (x)(fx ? ?x)

E: (x)(fx ? ~ ?x)

I : (?x) (fx · ?)

O: (?x) (fx · ~ ?x)

10.4 Proving Validity

The four additional rules thus make it possible to construct formal proofs of validity for arguments whose validity depends on the inner structure of noncompound statements within them. The four additional rules of inference are Universal Instantiation, Universal Generalization, Existential Instantiation, and Existential Generalization.

Using the Greek letter nu (n ) to represent any individual whatever, the rule of Universal Instantiation, abbreviated UI, is stated as

(x)(f x)

? fn (where n is any individual symbol)

The rule of Universal Generalization, abbreviated UG, is stated as

fy

? (x)(fx) (where y denotes "any arbitrarily selected individual")

The rule of Existential Instantiation, abbreviated EI, is stated as

(?x) (fx)

? fn (where n is any individual constant, other than y, having no previous occurrence in the context)

The rule of Existential Generalization, abbreviated EG, is stated as

fn

? (?x) (fx) (where v is any individual constant)

 

 

Summary Table

Rules of Inference: Quantification

10.5 Proving Invalidity

The procedure for proving the invalidity of an argument containing general propositions is the following:

1. Try a one-element model containing the individual a by writing the logically equivalent truth-functional argument for that model with respect to a.
2. If the truth-functional argument can be proved invalid by assigning truth values to its component simple statements, then you are finished; you have proved the original argument invalid. If not, go to step 3.
3. Try a two-element model containing the individuals a and b. If the original argument contains a universally quantified propositional function (x)(fx), use conjunction to join fa and fb. If the original argument contains an existentially quantified propositional function (?x) (fx), use disjunction.
4. If this argument can be proved invalid by assigning truth-values to its component simple statements, then you are finished; you have proved the original argument invalid. If not, go to step 5.
5. Try a three-element model containing the individuals a, b, and c. And so on.

10.6 Asyllogistic Inference

An asyllogistic argument is one in which one or more of the component propositions is of a form more complicated than the A, E, I, and O propositions of the categorical syllogism, and whose analysis therefore requires logical tools more powerful than those provided by Aristotelian logic.

In symbolizing general propositions that result from quantifying more complicated propositional functions, we must take great care to understand the meaning of the English sentence and then symbolize that meaning in terms of propositional functions and quantifiers.

An exceptive proposition asserts that all members of some class, with the exception of the members of one of its subclasses, are members of some other class. Exceptive propositions are in reality compound, because they assert both a relation of class inclusion, and a relation of class exclusion.