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Chapter Eleven. Rationale Behind the Precise Formulation of the Four Quantifier Rules. 1. Cases Involving the Five Major Restrictions. Restriction 1 on EI :
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Chapter Eleven Rationale Behind the Precise Formulation of the Four Quantifier Rules
1. Cases Involving the Five Major Restrictions Restriction 1 on EI: We must acknowledge that knowing that somebody is happy does not justify our asserting that a particular named individual is happy. Thus, when EI is used to drop an existential quantifier, the variables thus freed cannot validly be replaced by individual constants.
Cases Involving the Five Major Restrictions, continued Restriction 2 on EI: We cannot derive the claim that there is some object that is (e.g.) both red and square from the fact that something is red and something is square. So, a variable introduced free into a proof by EI must not occur free previously in the proof.
Cases Involving the Five Major Restrictions, continued Restriction 1 on UG: That a particular item had a certain property does not prove the universalizability of this property. So, we cannot use UG on a constant.
Cases Involving the Five Major Restrictions, continued Restriction 2 on UG: That there are certain objects does not by itself justify the conclusion that everything is an object of that type. So, we must forbid the use of UG on a variable introduced free into a proof by EI.
Cases Involving the Five Major Restrictions, continued We cannot use UG on a variable free in a line obtained by EI whether that variable became free by using EI or not. This restriction is nonintuitive!
Cases Involving the Five Major Restrictions, continued Restriction 3 on UG: This restriction rules out the use of UG within the scope of an assumed premise on a variable free in that assumed premise. The point of this is to make sure that the variable bound in a UG step names an arbitrary individual.
2. One-to-One Correspondence Matters We might naively characterize an application of EI or UI as a process in which a quantifier is dropped and all the variables thus freed are replaced by a particular variable. But there are two cases where this one-to-one correspondence cannot be required if our logic is to be complete.
One-to-One Correspondence Matters, continued We cannot require a one-to-one correspondence between x and y variables in the application of UI; all we can require is that for each occurrence of the variable freed by the UI step, there corresponds a variable bound by the quantifier on which we performed UI.
One-to-One Correspondence Matters, continued We cannot require one-to-one correspondence between x and y variables in the application of EG. This is handled by the last clause in restriction 1 on EG.
One-to-One Correspondence Matters, continued In using EG or UG, the replacements for the occurrences of only one variable in the original formula are to be bound in the resulting formula by the newly introduced quantifier. This is eliminated by the last clause in restriction 4 on UG.
One-to-One Correspondence Matters, continued If one occurrence of some variable x is freed by UI and replaced by a free variable, then all x variables freed by this application of UI must be replaced by free y variables.
One-to-One Correspondence Matters, continued In the use of UG, if a free x in the original formula is replaced by a y that becomes bound in the resulting formula, then all free occurrences of x in the original formula must be replaced by bound y variables in the resulting formula. This is taken care of by restriction 4 on UG.
3. Accidentally Bound Variables and Miscellaneous Cases When a quantifier is dropped by UI or EI, all the variables thus freed must be uniformly replaced by free variables (or, in the case of UI, by free variables or constants). The rule UI and the third restriction on rule EI take care of this.
Accidentally Bound Variables and Miscellaneous Cases, continued In using UG or EG, the variables to be quantified by the newly introduced quantifier must not be bound by some other quantifier. (This is prevented by restrictions 4 and 1)
4. Predicate Logic Proofs with Flagged Constants There is an alternative system of predicate logic proof rules which is both sound and complete.
Predicate Logic Proofs with Flagged Constants, continued In this alternative system, the rule QN is the same; there two rules, UI and EI, for taking off quantifiers, and UG and EG are used for putting them back on (although these rules are stated differently).
Predicate Logic Proofs with Flagged Constants, continued The typical sequence is still the same: use UI or EI, use sentential logic, then use UG or EG.
Predicate Logic Proofs with Flagged Constants, continued The difference is that this alternative system uses flagged constants instead of freed variables in the intermediate steps of the proof. Let us call this the flagging system.
Predicate Logic Proofs with Flagged Constants, continued When we flag a constant we “raise a red flag” to note that there is something special about it.
Predicate Logic Proofs with Flagged Constants, continued Flagged constants are subject to three restrictions: • They may not appear in the conclusion of the proof. • They must be new to the proof. • Any constant introduced within a subproof can only be used within that subproof.
Predicate Logic Proofs with Flagged Constants, continued The rule UG is where the flagging method differs most from the standard system. In the flagging system UG involves as subproof. However, the first step consists not of an assumption but of a flagging step.
