Chapter 2 of Suppes' develops the notion of sentential inference and how to construct rules of inference. However, there is one part in his development that seems incorrect no matter how I look at it.
The chapter starts with asserting that every rule of inference must satisfy two criteria:
1) Given a set of premises, the rules of logical derivation must permit us to infer ONLY those conclusions which logically follow from the premises.
2) Given a set of premises, the rules of logical derivation must permit us to infer ALL conclusions which logically follow from the premises.
He then gives a motivation for what it means for a conclusion to logically follow from premises. He starts with defining the notion of a sentential interpretation of another sentence, and the definition is:
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 atomic sentences of Q by other (not necessarily distinct) sentences.
From that he gives a sufficient condition for one sentence to logically follow another:
(i) Q logically follows from P if every sentential interpretation of the implication "if P then Q" is true
Further in the section, he then says that Criterion 1 for a rule of inference says that the rules of inference must be sound. Then he says that we can combine Criterion 1 and (i) to obtain the following:
(*) First we see that Q should be derivable from P by use of the rules only if every interpretation of "if P then Q" is true.
This seems incorrect, because from Criterion 1, I interpret that to mean:
- "If we can infer a conclusion from a set of premises, then the conclusion logically follows from the premises."
Then from (i):
- "If every sentential interpretation of the implication 'if P then Q' is true then Q logically follows from P"
To me this does not guarantee that if Q is derivable from P, then every interpretation of "if P then Q" is true (as what (*) seems to state). This is so because (i) seems to imply that there could be a conclusion that logically follows from some premises where not every sentential interpretation of the implication 'if P then Q' is true, but by criterion 1, we must be able to infer the conclusion from the premises since the conclusion logically follows from the premises.
This seems very incorrect. Does he use only if in (*) in the idiomatic way (i.e. he means only if as if)? Is that fact that Criterion 1 uses "premises" that make it so there are some sentences P that can never be premises? This seems like an unusual mistake to make in a chapter where he is trying to be precise in his language, which makes me sure that I am missing something.