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Patrick Suppes gives a working definition of sentential interpretation, based on a sentence maintaining its form. By working definition, I mean an incomplete definition that is needed for someone to understand something else. Here is his precise working definition:

A sentence P is a sentential interpretation of a sentence Q if and only if P can be obtained from Q by replacing the component sentences of Q by other (not necessarily distinct) sentences.

He then goes on to state a sufficient but not necessary condition for one sentence to be a logical consequence of another as follows:

(I) Q logically follows from P if every sentential interpretation of the implication P → Q is true.

Now, I understand what is meant by the working definition of sentential interpretation, but I do not understand the second part of (I) - how every sentential interpretation of a sentence, such as P → Q, can always be true. May someone please help provide me with examples of sentences P and Q such that every sentential interpretation of the sentence P → Q is true? Would it have to be the case that either (1) every interpretation of P can be true or false, while every interpretation of Q is true (because Q is a tautology); or (2) every interpretation of P is false (because P is a contradiction), while every interpretation of Q can either be true of false?

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  • It means, in more standard terms, tha every interpretation that satisies P will also satisfy Q. Jul 10 at 16:49
  • I assume this is limited to propositional logic, right? Jul 10 at 18:19
  • @MauroALLEGRANZA, I think you skimmed over the definition of "sentential interpretation". It's a syntactical transformation, not an interpretation in the sense of model theory. Jul 10 at 18:22
  • Not exactly :-) The author is trying to use a working definition of logical form for sentences in natural language: thus "Socrates is mortal or Socrates is not mortal" is an example of an "interpretation of formula P or not-P. Thus, replacing into "Socrates is mortal or Socrates is not mortal" the part "Socrates is not mortal" with another sentences, what we get is a complex sentence that is again true. This will be so for every substitution; thus the sentence "Socrates is mortal or Socrates is not mortal" is a tautology. The same for logical consequence. Jul 11 at 7:24

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Suppes is using some rather dated terminology here. It is not so common today to speak of sentential interpretations in the way Suppes defines them. In particular, Suppes is not using the term 'interpretation' in the sense used by Tarski in the context of model theory. Suppes' meaning is more like a substitution instance in which component sentences are substituted consistently by other component sentences.

To say of a sentence that every sentential interpretation of it is true, is another way of saying that the sentence is a tautology of propositional logic. For example, a sentence such as

(A ∧ ¬B) ∨ (B ∨ ¬A)

has this property. Whatever substitutions you make for A and B, this will always come out true.

To say of a sentence of the form "P → Q" that it has this property means the same thing. The symbol → here is material implication. So, for example, the sentence

¬(A ∧ ¬B) → (B ∨ ¬A)

is a tautology. Whatever sentences you substitute for A and B, this comes out true. The connection to logical consequence is that if "P → Q" is a tautology, then Q is the logical consequence of P. So, "(B ∨ ¬A)" is the logical consequence of "¬(A ∧ ¬B)".

More generally, if you have an argument consisting of, say, three premises, viz:

A; B; C; therefore D

then D is the logical consequence of the premises if the corresponding conditional is a logical truth. The corresponding conditional is a sentence consisting of a material implication with the conjunction of the premises in the antecedent, and the conclusion in the consequent. In this case, the corresponding conditional would be the sentence

(A ∧ B ∧ C) → D.

It is perhaps worth noting that the reason Suppes says that his criterion is a sufficient but not necessary condition for logical consequence, is that his criterion is concerned only with sentences. Some authors use the term 'tautological consequence' for this kind of logical consequence. In first order logic, logical consequence can occur in other ways.

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