Chapter Ten

Chapter Ten

Relational Predicate Logic

1. Relational Predicates

We now broaden our coverage of predicate logic to include relational predicates. This allows us to symbolize

sentences such as “Kareem is taller than Mugsy” as Tkm.

With relational predicates the order in which the letters occurs is significant.

2. Symbolizations Containing Overlapping Quantifiers

Symbolizations may contain quantifiers with overlapping scopes.

When the overlapping quantifiers are of the same types, the order in which they occur is not relevant to the meanings

of the sentences.

But when an existential and a universal quantifier are both involved, order becomes crucial.

3. Expansions and Overlapping Quantifiers

One way better to understand sentences which include overlapping quantifiers is to become familiar with the

expansions of such multiply quantified sentences.

4. Places and Times

The symbolization of statements concerning places or times are especially interesting and can cause trouble.

5. Symbolizing “Someone,” “Somewhere,” “Sometime,” and So On

The word “someone” can be misleading, for it sometimes functions as an existential quantifier and sometimes as a

universal quantifier.

The words “somewhere,” “something,” “sometime,” and so on can also be misleading in this way.

6. Invalidity and Consistency in Relational Predicate Logic

We demonstrate invalidity and consistency in relational predicate logic using the same techniques we employed for

monadic predicate logic.

Invalidity and Consistency in Relational Predicate Logic, continued

For invalidity we produce an interpretation that makes the

premises all true and the conclusion false.

For consistency we need only an interpretation that makes all the sentences true.

Invalidity and Consistency in Relational Predicate Logic, continued

As in monadic predicate logic we can provide a complete interpretation, or we can use the more mechanical method

where we replace the quantified sentences with their expansions.

7. Relational Predicate Logic Proofs

The rules for predicate logic proofs outlined in Chapter Nine were devised to handle relational

predicate logic as well.

However, relational predicate logic is more complex as we can encounter lines with more than one quantifier and more than one type of variable.

8. Strategy for Relational Predicate Logic Proofs

• If a premise contains more than one quantifier, you may have to use EI after you have used UI. But you should usually remove the existential quantifier as soon as possible.

• Sometimes it helps when removing variables to introduce new variables.

9. Theorems and Inconsistency in Predicate Logic

The conclusion of a valid deduction in which there are no given premises is a theorem of logic.

Theorems are sometimes referred to as logical truths, or truths of logic.

Theorems and Inconsistency in

Predicate Logic, continued

A logical contradiction, or a logical falsehood, is a single statement that can be proved false without the aid of

contingent information.

10. Predicate Logic Metatheory

There are two ways a system of proof rules could be deficient: If there are valid arguments that cannot be

proven, the rules would be incomplete; if there are invalid arguments that can be proven, the rules would be unsound.

11. A Simpler Set of Quantifier Rules

The quantifier rules UI, EI, UG, EG and QN, together with the eighteen valid argument forms plus CP and IP, form a complete set of rules for

quantifier logic.

A Simpler Set of Quantifier Rules, continued

But there are simpler sets of quantifier rules.

One very simple set includes only two of the four QN rules, together with Rule UI and Rule EI.

A Simpler Set of Quantifier Rules, continued

Every inference permitted by the simpler rules is also permitted by the standard rules, although the

reverse is not true.