logic teaching question

I'm teaching Symbolic Logic this term, and we just introduced the notion that two syntactically distinct sentences can mean exactly the same thing (e.g. DeMorgan's Laws). In class yesterday, I asked the students to come up with a criterion for when two sentences are identical in meaning. We eventually reached the "official" answer: S_1 and S_2 are synonymous just in case they have the same truth-value in all models (ok, we're not using the notion of models; for us, it's "... in all possible arrangements of the board" in Tarski's World -- we're using Barwise and Etchemendy's Language, Proof, and Logic.)

Along the way to the official answer, though, a student gave the following characterization:
Sentences S_1 and S_2 are synonymous just in case the set of all sentences that follow from* S_1 is identical to the set of all sentences that follow from S_2.
After class, I jotted down a proof-sketch showing that the student's characterization is equivalent to the "official" one. But I'm not 100% confident in it, so I'm just curious whether anyone can see a counter-example (i.e. are there any 2 sentences that meet one characterization but not the other?).
* EDIT: As Bryan made me realize in the comments, I should specify that 'follows from' here is semantic, not syntactic/ proof-theoretic; i.e. 'Conclusion C follows from premise P' means that every model (or world, or construction, or whatever it is that makes sentences true in your formal semantics) in which P is true is also a model in which C is true.


Bryan said...

Hi Greg,

I think this equivalence is guaranteed by (and only by) the soundness and completeness of a proof system w.r.t. a semantics.

If two sentences have the same truth conditions, then the same sentences follow validly from each in an argument. Given completeness, this implies that the same sentences can be derived from each as a theorem -- which is the definition your student suggests.

The converse can be proved in the same way, assuming soundness.

Greg Frost-Arnold said...

Hi Bryan --

Thanks for stopping by. The original post should have been clearer. I did not mean, in the student's definition, that 'C follows from P' means that there exists a proof of C from P (in proof system S). But I think you are right that if that is what 'follows from' means, then the equivalence is immediate, if S is sound and complete.

But I meant 'follows from' semantically, i.e., 'C follows from P' means 'In every model (or whatever you're using for making sentences true) where P is true, C is also true.'

P.D. Magnus said...

If A entails B and B entails A, then there is no model in which one is true and the other false. Therefore, they have the same truth value in every model.

So you can define equivalence as mutual entailment or as having the same truth value in every sentence, and either way you pick out the same sentences.

Anonymous said...

Greg - if the set of sentences A and B entail is S in both cases, then both A and B belong to S (everything entails itself.) Since they both belong to the set of things the other entails, they both entail each other. (And of course, the converse is obvious.)

Greg Frost-Arnold said...

Andrew --
That was the little argument I used for that direction (student def'n --> standard def'n); glad to hear it occurred to you too.