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Chapter Nine

Predicate Logic Proofs

1. Proving Validity

The eighteen valid argument forms plus CP and IP that are the proof machinery of sentential logic are incorporated intact into predicate logic.

However, for proofs in predicate logic we must introduce four new rules of implication.

These rules of implication tell us when taking off quantifiers in justified and when replacing them is justified.

2. The Four Quantifier Rules

- Universal Instantiation (UI)
- Universal Generalization (UG)
- Existential Instantiation (EI)
- Existential Generalization (EG)

3. The Five Main Restrictions

- An EI must be done to a quasivariable
- A quasivariable introduced into a proof by rule EI must not have occurred as a quasivariable previously in the proof.
- UG cannot be performed on a constant.
- If a variable is free in an EI line, we cannot use UG to bind that variable.
- When we make an assumption with a quasivariable, we cannot bind that variable with UG so long as we are relying on the assumption in which it occurs.

4. Precise Formulation of the Four Quantifier Rules

Rule UI: (u) (…u…)/Therefore, (…w…)

Provided: 1. (…w…) results from replacing each occurrence of u free in (…u…) with a w that is either a constant or a variable free in (…w…) (making no other changes).

Precise Formulation of the Four Quantifier Rules, continued

Rule EI:

(∃u) (…u…)/Therefore, (…w…)

Provided: 1. w is not a constant; 2. w does not occur previously in the proof; 3. (…w…) results from replacing each occurrence of u free in (…u…) with a w that is free in (…w…) (making no other changes).

Precise Formulation of the Four Quantifier Rules, continued

Rule UG:

(…u…)/Therefore, (w) (…w…)

Provided: 1. w is not a constant; 2. u does not occur free previously in a line obtained by EI; 3. u does not occur free previously in an assumed premise that has not yet been discharged; 4. (…w…) results from replacing each occurrence of u free in (…u…) with a w that is free in (…w…) (making no other changes) and there are no additional free occurrences of w already contained in (…w…)

Precise Formulation of the Four Quantifier Rules, continued

Rule EG:

(…u…)/Therefore, (∃w) (…w…)

Provided: 1. (…w…) results from replacing at least one occurrence of u, where u is a constant or a variable free in (…u…) with a w that is free in (...w…) (making no other changes), and there are no additional free occurrences of w already contained in (…w…).

5. Mastering the Four Quantifier Rules

- Do not try to do two things at once, such as change bound x’s to free x’s and y’s, bind both an x and a y at once, and so on.
- Do not violate the two restrictions having to do with constants.
- When using EI, check to make sure that the variable we are introducing does not occur free on any earlier line.
- When using UG, check to make sure that the variable we are binding is not free in an EI line or an undischarged assumed premise.

Mastering the Four Quantifier Rules, continued

- If you must use EI, do so as soon as possible.

Remember: These rules are to be applied to whole lines of proofs only!

6. Quantifier Negation

The four other inference rules to be introduced into our predicate logic proof procedure are all referred to by the name Quantifier Negation (QN)

Quantifier Negation, continued

Adding (x) to an expression does the same job as adding ˜(∃x) ˜ to that expression, and, similarly, adding ˜(∃x) ˜ to an expression does the same as adding (x) to it.

The first version of rule QN allows us to make inferences from of these sorts of expressions to the other.

Quantifier Negation, continued

The other three varieties of Rule QN are similar to the first one.

The first tells us that we can move from “Everything has weight,” to “There isn’t anything that does not have weight,” the second from “Something has weight” to “It’s not the case that nothing has weight,” the third from “Everything is such that it doesn’t have weight” to “It’s not the case that something has weight,” and the fourth from “There is something that doesn’t have any weight” to “It’s not the case that everything has weight”.

Quantifier Negation, continued

The four quantifier negation rules require us to do exactly the same thing:

- Change the quantifier in question from an existential to a universal quantifier, or vice versa
- Remove any negation signs there may have been either to the left or to the right of that quantifier
- Put negation signs in whichever of these two places there may not have originally been one.

Key Term

- Quasivariable

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