I wish I could give a comprehensive answer to your question, but my knowledge is limited. I'm moved to reply because I think the comments to the question are giving the OP a hard time; I think the question makes excellent sense. Unusually, I disagree with Mauro: logic is not fundamentally about truth, it is about consequence, or deducibility. One of the striking things about the relation of logical consequence is that it is not confined to statements with truth values. Many, perhaps most, speech acts possess their own logic. Obligations can entail other obligations, commands can entail commands, necessities can entail necessities, etc. Validity is not exclusively about preservation of truth. A valid argument in a deontic logic is one that preserves obligation from premises to conclusion. An intuitionistically valid argument is one that preserves constructive provability from premises to conclusion.

Typically, it is a standard move when dealing with such modalities to introduce box/diamond notation and to replace "A is necessary" with "□A is true", for example. We commonly wrap up semantic properties and put them in the box in order to reduce everything down to truth. This tendency towards a semantic monism in which only truth is considered important is fairly pervasive, but I share the OP's desire for it to be justified. After all, many theories of meta-ethics do not assign truth values to moral judgements, so why should we assume that a logic governing moral obligations should be expressed in a truth-theoretic way?

An initial observation might be that since we already have a calculus that expresses the implicational behaviour of propositions that are connected by 'and', 'or', 'not', etc, it might be a duplication of effort to create new rules to show how these connectives behave when they connect things other than propositions. The axiomatization of a modal logic has the effect of demonstrating how to reduce sentences involving connectives between modalities to sentences involving only propositional connectives. For example, the K axiom of modal logic effectively reduces strict implication to material implication. If we can axiomatize a modal logic and perform this reduction correctly, then this suggests that the logics of modalities other than truth are eliminable.

Another point is that we desire our logical operations to be computable. We might argue on the basis of the Curry-Howard correspondence that our best understanding of computability corresponds to the concept of provability afforded by classical logic. And classical logic has the natural semantics of truth and falsehood. This suggests that any logic worthy of the name is ultimately either about truths and falsehoods or about something that looks just like them.

Another consideration is that logic has epistemological consequences. We use logic because we want to come to know things by inference from other things we know. But we usually conceive epistemology as being concerned with knowing things that are true. Think of all those theories of knowledge that are taught in epistemology classes: "X knows that P iff..." and typically one of the conditions is that P is true. Maybe such accounts are too limited and we can have knowledge that is not of things that are true or false, but detractors might claim that we would be describing mere sensibilities and not real knowledge.

But all this still leaves the question that if the logic of modalities is reducible to that of truths, what are modal claims true of, exactly? Are there really necessary truths, moral truths, even aesthetic truths, etc? Modal logic is commonly expressed using Kripkean possible world semantics, but does this mean that if we judge a modal claim to be true, we are committed to the existence of PWs? David Lewis thought so and embraced modal realism, but his position has not proved popular. Others have adopted anti-realist or quasi-realist positions. I suspect that in the last analysis a full answer to your question can only be given on the basis of a comprehensive account of truth and realism and the relationship between them.

Because people that tell the turth are appreciated while people that tell the false are e.g. convicted to prison. – Mauro ALLEGRANZA – 2018-04-22T17:04:44.570

1You might have missed what I'm getting at. My intentions with this question concern the nature of language and logic, rather than the nature of our legal system. Furthermore, can we not conceive of another possible world where one must tell only what is

correctin court? Though, I should reiterate, I'm not concerned with the applied use of semantic values in the world, but with their place within the structure of logic. Many semantic values play important roles within logic which, in some systems, take the place of the designated value. Why and when was this decision made/by who? – BeingOfNothingness – 2018-04-22T17:12:41.757What do you mean by correct if not true? – Canyon – 2018-04-22T17:25:25.780

1A prime example is that in a deductive system with truth as the designated value, it can only express the assertion of the truth of a sentence or the assertion of that sentence's negation. In a deductive system with three-values, the mere assertion of truth is insufficient to express the range of values in the deductive system. Some introduce operators and then use a designated value of correctness to govern the inferential steps in this sort of deductive system. – BeingOfNothingness – 2018-04-22T17:29:11.790

But I only use correctness as an example. Any semantic value can be used as a designated value; i.e. valid inference which preserve a form of reference, valid inference that preserve a certain syntactical structure, and so on.

What process determined the primacy of truth in this above any other semantic value? Or where was it discussed, when and by who? – BeingOfNothingness – 2018-04-22T17:30:51.443

There is a common "thread" in Ancient Greece connecting logic, argument, law, reason, science (in the

polis)... But having said that, logic is fundamentally connected with TRUTH. In a sense, logic is the THEORY of TRUTH: avalidargument is "interesting" exactly because it produces TRUE conclusions from TRUE premsies. – Mauro ALLEGRANZA – 2018-04-22T17:56:51.1101I will agree that logic bears a relation to truth, but it is not the case that logic=the theory of truth. Logic can merely be used to maintain the designated value within the system, which is why correctness-functional entailment (C-Validity) can produce correct conclusions from correct premises. It is simply not the case that logic only deals with truth. Although this is certainly classical logic. But I have seen no arguments to support the notion that the system is fundamentally truth-functional. Certainly no more that a decimal mathematics is more fundamental than a duodecimal one. – BeingOfNothingness – 2018-04-22T20:56:35.087

Ian Rumfitt's "Truth and Meaning" is a prime example of the use of correctness-functional governing inference in a three-valued logical system as a solution to self-referential paradoxes like the liar paradox. I say this to evidence my claims - these are not some idiosyncratic theories of my own, but genuine considerations in non-classical logic. If someone can provide papers/articles/chapters on the (necessary) fundamentality of truth in logic, please share them. My curiosity is burning. – BeingOfNothingness – 2018-04-22T21:00:29.753

1Propositional logic always comes equipped with canonical semantics, where the truth values are

the propositions themselves(modulo whatever relevant form of equivalence). E.g. with classical propositional logic, this algebra of truth values really is a Boolean algebra. – None – 2018-04-23T02:01:53.647@BeingOfNothingness. After this much discussion, perhaps it is time to revise the original question. – Mark Andrews – 2018-04-23T06:14:49.223

Where do you get that the default truth value is true? Assumptions may start as true or false. Taking the truth route is often easier to find what follows from what. Three value logic is not a pure deductive system. Every subject has some deductive reasoning in it. This does not mean every subject is deductive reasoning. Deductive reasoning us worthy because certainty is a result. There is no middle ground between certainty and uncertainty. The three value logic is done for another purpose not the same as propositional logic. – Logikal – 2018-04-23T15:09:32.620

You can see : Yaroslav Shramko & Heinrich Wansing, Truth and Falsehood : An Inquiry into Generalized Logical Values, Springer (2011)

– Mauro ALLEGRANZA – 2018-11-07T13:22:42.850In the end, we must go back to Parmenides : "You must needs learn all things,/ both the unshaken heart of well-rounded reality/ and the notions of mortals, in which there is no genuine trustworthiness. [...] Come now, I shall tell—and convey home the tale once you have heard—/ just which ways of inquiry alone there are for understanding:/ the one, that [it] is and that [it] is not not to be,/ is the path of conviction, for it attends upon true reality"

– Mauro ALLEGRANZA – 2018-11-07T13:24:44.790