# what's the difference between Syntactic consequence ⊢ and Semantic consequence ⊨ [duplicate]

Can you help me to differentiate the Syntactic consequence $\vdash$ and Semantic consequence $\vDash$ ?

I think $A \vdash B$ means, "$A$ proves $B$" and $A \vDash B$ means , if $A$ is true, then $B$ is true.

If so, what is difference $A \to B$ and $A\vDash B$ ?

## 1 Answer

$$A \vdash_S B$$ means there is a derivation, in the proof-system $$S$$, from the premise $$A$$ to the conclusion $$B$$. [If context fixes the relevant system $$S$$ we suppress the subscript.]

$$A \vDash_L B$$ means that on every possible interpretation of the non-logical vocabulary of language $$L$$, if $$A$$ comes out true, so does $$B$$. [If context fixes the relevant language $$L$$ we suppress the subscript.]

Both those are metalinguistic claims, the symbolism abbreviating mathematical English (or mathematical Spanish, or whatever), making claims about the relation between the well-formed formulas (wffs) $$A$$ and $$B$$, looking from the outside of their formal language, so to speak.

$$A \to B$$, by contrast, is a wff that belongs to the object language, to the formal language of which $$A$$ and $$B$$ are wffs (typically, but far from always, interpreted as expressing the truth-functional conditional).

On the truth-functional interpretation, if the atomic wff $$p$$ happens to be false and the atomic wff $$q$$ happens to be false too, then $$p \to q$$ evaluates as true. But of course we don't have $$p \vDash q$$ ($$q$$ isn't true on every valuation which makes $$p$$ true).

And so it goes ......

[Oh, I seem to have said much the same before, in slightly different words, at https://math.stackexchange.com/questions/286077/implies-vs-entails-vs-provable, so check that out too to see whether that helps as well.]

• Peter, as a footnote to your answer, I already pointed to the linked thread in my comment and have suggested it as a duplicate. Apr 18, 2013 at 17:17
• Ah: I just went onto auto-pilot when I saw the qn: that's what forty years of giving intro logic lectures does for you ... Sad, eh? Apr 18, 2013 at 17:21
• As a footnote to your comment, I can relate! :-) Apr 18, 2013 at 17:25