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Suppes gives a natural deduction system in his book Introduction to Logic (Van Nostrand Reinhold Company 1957). First he gives some rules for making sentential inferences, and then later for inferences involving quantifiers. One of the rules for sentential inferences is "Rule T" on page 28 which says:

We may introduce a sentence S in a derivation if there are preceding sentences in the derivation such that their conjunction tautologically implies S.

Then later, when he is introducing the rules for inference with quantifiers (example 1, page 59, reproduced below), he gives a derivation where he uses rule T on a nonsentence (he uses rule T on a formula with free variables). So apparently S does not have to be a sentence in the wording of rule T?

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  • Welcome to Philosophy.SE. You'd be surprised how many academics share copyrighted material publicly. Although perhaps in this case the copyright is expired, I don't know. In any case, I added the example as a picture in case the link goes dead. – user2953 Jan 12 '18 at 20:18
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Rule T here allows one to write any tautological consequence of input sentences. Every sentential logic inference rule describes a particular tautological consequence. You can see that if you write out the inference rules as sentences. For example, a rule sometimes called “reiteration” can be written as “If P then P.” That sentence is logically true, and “tautology” is another way of saying “logically true.”

So, in this case, since the law of hypothetical syllogism is logically true, it can be invoked with the rule T. Any other sentence logic rule can also be invoked this way, as can a whole series of them. To put it in different terms, any sentence logic proof can be done with rule T in one step, using this rule, once you know what’s provable in sentence logic.

(As for what he’s doing with the unbound variables, I’m not sure. I’d have to see how he defines well-formed formulas, or what the context of this example is.)

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  • Often free variables are used as “parameters” that function in the same way as “dummy constants” in other approaches (e.g., existential instantiation requiring a “new/fresh” constant). I believe Mendelson does it this way. I’d guess there’s a renaming (variable substitution) rule as well. I’d be curious how he formulates the restriction on Universal Generalization to maintain soundness. – Dennis Jan 13 '18 at 0:00
  • @Dennis sounds likely! – ChristopherE Jan 13 '18 at 0:03
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See Patrick Suppes's textbook: Introduction to Logic (1957).

Rule T (page 28) is formulated in the context of sentential logic.

In the context of quantified logic (page 54), Suppes define (as usual) sentence as a "closed" formula, i.e. a formula without free occurences of variables.

In the summary of the rules (page 99), Rule T is summarized with:

use of tautologies.

Thus, you are formally right: the avoid misunderstanding, in quantificational logic the Rule T must be re-phrased as follows:

we may introduce a formula S in a derivation if there are preceeding formuals ...

Note : of course, the use of the rule made by Suppes is correct.

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  • Actually this is very confusing. Suppes talks about his metavariable use on pages 123 and the bottom of 126. He says P,Q,R,S (in bold) are sentential variables (and defines what a sentential variable is on 123) BUT "sentence" in this context means the more general one of formula (126). He defines formulas on page 52. – isthisreallife Jan 13 '18 at 13:07

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