Incorrect statement in Suppes' Introduction to Logic

Question

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.

Answer 1

You are misreading the "sufficient (but not necessary)" comment of page 22.

We have to consider also the following parenthetical comment :

(As we shall see in Ch.4, we obtain a complete characterization of logical consequence by omitting the restriction to sentential interpretations).

Consider now Ch.4 : General Theory of Inference, page 68 :

Using this [the general] notion of interpretation we may now define universal valid, logical consequence, [...]

A formula Q logically follows from a formula P if and only if [...].

Thus, it is a definition and as every sound definition is expressed with an iff statement, i.e. with a necessary and sufficient condition.

The comment above (page 22) must be read in the context of the provisional "characterization" of logical consequence restricted to the sentential case.

A simple way out can be (as Enderton does) to use two specific concepts : that of logical consequence (the "general" one) and that of tautological consequence, restricted to the sentential case.

Now the interplay of necessary and sufficient must be clear : to be a tautological consequence is a sufficient (but not necessary) condition for being a logical consequence.

Answer 2

The following description of the two criteria in the OP's post may not be accurate:

The chapter starts with asserting that every rule of inference must satisfy two criteria:

Note that it is not that every rule of inference must satisfy the two criteria, but the set of rules of inference as a whole must satisfy those two criteria. Some sets of rules may not allow us to derive all of the conclusions which logically follow from the premises. Other sets of rules may allow us to derive more conclusions than we should since some of them do not logically follow. The two criteria make sure that the set of rules as a whole derives all and only the conclusions which logically follow.

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Assume the set of rules follow Criteria 1, then every conclusion we can derive from the premises using the set of rules of inference will logically follow from the premises. That is because Criteria 1 guarantees that only conclusions that logically follow can be derived using the set of rules.

Criteria 1 with the sufficient condition of "logically follows", (i), lead to this "test criterion" in case we want to add a new rule to the set: (page 23):

If a new proposed rule of inference permits the derivation of a false conclusion from true premises, reject it.

The OP quotes 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.

Using the interpretation of "only if" in Wikipedia, this is equivalent to:

If first we see that Q should be derivable from P by use of the rules, then every interpretation of "if P then Q" is true.

Using the contrapositive, if it is not the case that every interpretation of "if P then Q" is true, then I should not be able to derive Q from P. So if a new rule allows me to derive such a conclusion, reject that rule.

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However, if those rules do not also follow Criteria 2, then there may be conclusions that logically follow from the premises but this particular set of rules will not be able to derive them. Criteria 2 guarantees that all conclusions that logically follow can be derived using the set of rules.

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Suppes, P. (1957). Introduction to logic. Van Nostrand Reinhold.