# Proving "if Q is derivable from P, then all sentential interpretations of P->Q are true" from Suppes criterion I

Question. I have trouble understanding how Patrick Suppes in his "Introduction to logic" derives the important sentence (A2) from another two, (A1) and (CrI).

Introduction.

He uses criterion I:

(Cr1): "Given a set of premises, the rules of logical derivation must permit to infer ONLY those conclusions which logically follow from the premises."

He also uses sentence:

(A1): "Q logically follows from P if every sentential interpretation of the implication P→Q is true."

He derives:

(A2):"Q should be derivable from P by use of the rules only if every interpretation of P→Q is true".

My attempt.

It is hard for me to comprehend proofs in natural language. So, I am using symbolizations.

I use "⇒" to symbolize if-then of the metalanguage and "→" for if-then of the object language. All quantifiers belongs to metalanguage.

I understand criterion I as $$\forall P,Q [L(P,Q)\Rightarrow\exists R[R(P,Q)]],$$ where R(P,Q) is "Q can be derived from P via rule R", L(P,Q) is "Q logically follows from P".

I understand (A1) as $$\forall P,Q [L(P,Q)\Leftarrow\forall x[x \in \operatorname{sInt}(P→Q)\land x=1]],$$ where sInt(P→Q) is "set of all sententional interpretations of P→Q".

I understand (A2) as $$\forall P,Q [\exists R[R(P,Q)]\Rightarrow\forall x[x \in \operatorname{sInt}(P→Q) \land x=1]].$$ In some other place of the books he writes that "P only if Q" is same "Q→P".

In my understanding, from (CrI) and (A2), we can infer only: $$\forall P,Q [\exists R[R(P,Q)]⇐\forall x[x \in \operatorname{sInt}(P→Q) \land x=1]].$$

So, for me, the derivation A1+Cr1→A2 seems counter-intuitive.

P.S. Please keep explanations simple. I am a chemist who has logic as a hobby. There are many things that are very obvious for you and are far from obvious for me. That is because you work a lot with these things and I see them rarely. I have gone through six textbooks on logic with exercises.

Scans.

Criterion I

(A1)

(A2)

• $A\implies B$ is true whenever $B$ is true , even if $A$ is false. Commented Jul 9 at 11:27
• Could you explain in more detail? I know all the rules for sentential logic, and it didn't help me to solve this issue. I have been struggling for 2 days with no success. Commented Jul 9 at 11:34
• (A2) is a restatement of (Cr1). The idea is: assume that we have a purported rule FR ("fake" rule) that violates (Cr1), i.e. such that Q is derivable from P with FR but $P \to Q$ is NOT True for every interpretation. This means that there is an intepretation such that $P \to Q$ is False; but by (A1) this means that Q does NOT logically follows from P, contrary to (Cr1). Commented Jul 9 at 12:14
• Could you explain how to derive from sentences "there is a false sentential interpretation of P->Q" and (A1) "Q logically follows from P if every sentential interpretation of the P->Q is true" the sentence that "Q does not follow from P". Commented Jul 9 at 13:13
• See Logical consequence and Tautological consequence for the particular case of propositional logic. Commented Jul 9 at 13:31

(I) IF every sentential interpretation of $$P \to Q$$ is true, THEN $$Q$$ logically follows from $$P$$.

And:

(IV) IF $$Q$$ is tautologically implied by $$P$$, THEN $$Q$$ logically follows from $$P$$.

By (CI): Given a set of premises, the rules of logical derivation must permit to infer ONLY those conclusions which logically follow from the premises,

we have that a sound set of rules must permit to infer a conclusion $$Q$$ from premise $$P$$ only if $$Q$$ logically follows from $$P$$, and this in turn means that:

a sound set of rules should allow the derivation of $$Q$$ from $$P$$ only if every interpretation of $$P \to Q$$ is true.

Then apply the note of 7th line page 22: "we obtain a complete [with "IFF"] characterization of logical consequnece by omitting the restriction to sentential interpretations."