A definition of truth entails a notion of consequence. Then again, it doesn't.

A definition of truth entails a notion of consequence. A definition of truth will define truth relative to (a language L and) a formal object x--"truth at x"--which could be (i) an index, (ii) a context, (iii) an index-context pair, (iv) a suitably restricted set of index-context pairs, e.g., those pairs where the index and context are related in a certain way (as in "the index of the context", or "the index initialized by the context"). Suppose we have some such formal object x relative to which truth (in L) is defined. Then, given a set of sentences \Gamma and a sentence s, s is a consequence of \Gamma iff, when all the sentences in \Gamma are true relative to x, s is also true relative to x. We are done.

Then again, a definition of truth does not entail a notion of consequence. In our search for a notion of consequence, we are seeking to do justice to plausible-seeming inference patterns, such as Lukasiewicz's Principle, De Morgan's Law, the Choice Inference, or Stalnaker's "Direct Argument".

It could turn out that our inference patterns do not track the notion of consequence defined above. What they could be giving us data about is (i) the constraints the context places on the index--how to define a "index of the context." (ii) in a more dynamic (or "reasonable inference") vein, perhaps we should see the premises in \Gamma as being true at an ordered series of contexts, rather than a single context, where the inference is valid if the conclusion is true "after" all the premises are true (here, and not in the first definition of consequence, is it the case that the order of the premises matters). (iii) in an "informational consequence" vein, we could think of the inferences as preserving something other than truth (or, what comes to the same thing, we could define a new kind of truth--truth relative to only one parameter of the index.*) Following the first convention rather than the second, we will say that an inference pattern may preserve acceptance (aka certainty or Supertruth) rather than truth.

If our formal semantics can account for the inference pattern in question in any of these ways, we may consider our semantics to have dealt satisfactorily with the inference pattern, even if it is not "valid" in the sense first defined above (valid with regard to truth at L and x).

We achieve synthesis of the two points of view on truth and consequence by noting that it's true that a definition of truth entails A notion of consequence, but there may be many notions of consequence which are of interest to us, and which are useful in explaining inference patterns.

*truth relative only to the i-parameter, and not the w-parameter. For e.g. supervaluationist Supertruth, this is indeed truth relative to only part of the index. Sentences (the full battery of sentences--both modalized and unmodalized) are true at a world and a set of worlds accessible from that world, but given S5 (which defines an i for any w) and an interest in preserving supertruth, we actually don't need the world parameter. We have effectively cut down on the battery of sentences by considering only the preservation of supertruth.