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Source: A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley

[p 415:] Since, however, the first eight of these rules are applicable only to whole lines in an argument, as long as the quantifier is attached to a line these rules of inference cannot be applied—at least not to the kind of arguments we are about to consider. To provide for their application, four additional rules are required to remove quantifiers at the beginning of a proof sequence and to introduce them, when needed, at the end of the sequence. These four rules are called universal instantiation, universal generalization, existential instantiation, and existential generalization.

[p 464:] One further restriction that affects all four of these rules of inference requires that the rules be applied only to whole lines in a proof.

This answer exposes a gap in my understanding: Why can the Eight Rules of Implication apply only to whole lines? Without the above restriction, the argument in the aforesaid post of mine appears to succeed more simply; the proof would start at 11; 3-10 above would never be needed.

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  • @quen_tin Thank you, but can you please explain more? I still do not comprehend. Even if 1 (in that question) is a more complex logical form, if the above is true, then instead of an Indirect Proof, why not use Conditional Proof instead? Then using 1A as the Assumption of the Conditional Proof, you can start the proof at 11; 3-10 above would never be needed.
    – user8572
    Jan 6, 2016 at 21:28
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    With so many questions on that book, why don't you try another one? Maybe there are better ones with better explanations of the rules. Also there seem to be a mistake since a double negation is eliminated in the proof inside the quantifier.
    – Quentin Ruyant
    Jan 6, 2016 at 21:29
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    nothing prevents you from using conditional proof using 1A. You can make any assumption you like in a conditional proof. The point is that you will not be able to prove the same thing.
    – Quentin Ruyant
    Jan 6, 2016 at 21:33
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    Very often the best way to understand a proof is to start from the end (the aim of the proof). Here start at 19: that's what we want to prove. Strategy: make "not 19" as an assumption (3) and derive a contradiction (17). You cannot skip (3) it's the most important part, and that has nothing to do with the restriction you're referring to here.
    – Quentin Ruyant
    Jan 6, 2016 at 21:38
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    line 6: the negation elimination rule is applied inside the quantifiers. If negation elimination is one of the 8 rules in the passage you quote it clearly contradicts the restriction.
    – Quentin Ruyant
    Jan 6, 2016 at 21:48

1 Answer 1

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Logical rules must be sound, i.e. they are devised in such a way that the consequence of a rule is logically implied by the premises.

The "paradigmatic" example is Modus Ponens : if P ⊃ Q and P are true, also Q is.

The issue with your "reformed" quantifier rules is that they are unsound: Qa is not a logical consequence of (∃x)Px ⊃ (∃y)Qy.

To check this, consider a "universe" where there are neither Ps nor Qs; in this interpretation (∃x)Px ⊃ (∃y)Qy is true while Qa is false.

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  • Thanks, but I do not understand your answer. From where did Qa cme? Commence with: 1. (∃x)Px ⊃ (∃y)Qy. If I apply Existential Instantiation, then 1 becomes: 2. Pa ⊃ Qb. Correct? What is wrong with 2?
    – user8572
    Jan 21, 2016 at 20:36
  • Would you please respond in your answer, which is easier to read than comments?
    – user8572
    Jan 21, 2016 at 20:37
  • @Greek-Area51Proposal Incorrect. Existential Generalisation only applies when the existential quantifier has highest operational precedence. In your statement 1., the conditional connective has the highest precedence.
    – Graham Kemp
    Oct 23, 2019 at 0:12

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