Two questions about monotonicity of entailment.

Question

I wonder about two things. First, how do we prove that entailment in some logic is monotonic? The second one - What is the relationship between monotonicity of logic and deduction theorem? It seems rather clear that deduction theorem doesn't hold in nonmonotonic logics. But do we have deduction theorem in every possible monotonic logic?

I will be extremely thankful for answers!

Answer 1

Take your favourite Hilbert-style axiomatization of propositional logic. Mendelson's for example. We can prove that the deduction theorem holds with respect to that theory.

Now kill e.g. all instances of Mendelson's first axiom schemata. The proof of the deduction theorem won't go through any more. Damn!

But the weakened logic is still monotonic. That is to say, if in the weakened logic you have $A \vdash C$, then you'll have $A, B \vdash C$, for any $B$. That's because a proof is still defined to be a sequence of wffs such that each one is either an axiom or follows from early wffs in the sequence by modus ponens -- a definition which ensures monotonicity but says nothing about which axioms are actually available.

Answer 2

The received Tarskian notion of entailment is often explicated using models, through the idea of truth-preservation: $\Gamma\models A$, given a set of formulas $\Gamma\cup\{A\}$, if every valuation satisfying all premises in $\Gamma$ also satisfies the conclusion $A$. An entailment relation is called monotonic if $\Delta\models A$ whenever $\Gamma\models A$ and $\Gamma\subseteq\Delta$. It is easy to check that the Tarskian notion of entailment is monotonic: assume $\Gamma\subseteq\Delta$ and suppose by contraposition that you have a counter-model for $\Delta\models A$, that is, suppose that there is a valuation that satisfies $\Delta$ but does not satisfy $A$; it follows from the assumption that $\Gamma$ is also satisfied, thus you have indeed a counter-model for $\Gamma\models A$. To put it otherwise, if you're dealing with some non-monotonic notion of consequence, this notion will fail to meet the requirements to be characterized by way of the Tarskian notion of entailment.

It should be noted that the deduction theorem is neither sufficient nor necessary for a consequence relation to be monotonic. Furthermore, that there are some Tarskian logics which respect the deduction theorem, and other Tarskian logics that disrespect this theorem.