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Between the two horns of Jørgensen's dilemma, the authors opt for that according to which logic deals not only with truth and falsity but also with those concepts not possessing this semantic reference. Notwithstanding the “descriptive” prejudice, deontic logic has gained validity among modal logics. The technical foundation proposed consists in an abstract characterization of logical consequence. By identifying in the abstract notion of consequence the primitive from which to begin, it is possible to define the connectives - even those of obligation - by means of the rules of introduction or elimination in a context of derivation.

https://www.researchgate.net/publication/229701861_Logic_Without_Truth

What I don't understand is deontic logic should then be able to say that X is permissible or impermissible, and if X is permissible then the statement X is permissible or statement Y should be true, since if they don't recognize as Y being true, they cannot say Y, so then how come some say that deontic logic doesn't deal with truth?

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    Jørgensen's dilemma does seem to move us towards rejecting talk of deontic logic (c.f. issues with "imperative logic" (Conifold showed me this paper some time ago, just to note)). Minimalism about truth generally might push us in the other direction again, though. Or we might talk about deontic (and/or imperative) programming, which can be carried out in a kind of mathematically formalistic way. Jul 9, 2022 at 23:01
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    They mean two things. First, "the notion of inference cannot be characterized by the notions of truth or falsity nor does the meaning of logical connectives depend on the truth value of their compounds". In other words, the logic is not truth-functional, truth values of inferences and connectives are not determined by truth values of their terms. This is not groundbreaking, intuitionistic logic is not truth-functional either. And second, they do not wish to reformulate prescriptions (don't do X) into descriptions (X is impermissible), as you did, and prescriptions do not have truth values.
    – Conifold
    Jul 10, 2022 at 0:32

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Your question is concerned with two important fundamental issues. One is whether we should think of logic as being fundamentally the science of truths or else as the science of consequence. The other issue is whether for a particular domain of discourse, the sentences in that domain are truth-apt, which is to say, they are capable of being true or false. The latter issue is related to the question of realism versus antirealism.

Let's start with the first of these issues. One of the things on which logicians disagree is quite simply what logic is about. For some, the concept of truth is fundamental, and logical consequence is to be understood derivatively as that relation between the premises and conclusion of an argument that is necessarily truth-preserving. This is why you sometimes see books with titles like, "Logic: The Laws of Truth". Others take the view that the concept of consequence is fundamental and that logic need not be concerned with truth. On this view, what matters fundamentally is to be able to say: this sentence follows logically from those sentences, without it being a requirement that the sentences have truth values.

In the paper you reference, Alchourròn and Martino take the latter view. Their argument for this is simply that there are many sentences, such as those that express norms, which can participate in relationships of logical consequence, but which may not have truth values. For example, if A is obligatory, and A entails B, then B is obligatory. And this may hold even if we don't think of obligations as statements of fact that are capable of being true or false. One could easily add other examples. If A! is a command to do A, and it is impossible to do A without doing B, then we might say that A! has the logical consequence B!. If A? is a query for the value of A, and it is impossible to ascertain the value of A without ascertaining the value of B, then we might say that A? has the logical consequence B?. Speaking for myself, I agree with this latter position. One of the grand things about logic is that it is not limited in its application to truths and falsehoods.

This brings us to the second issue. In the case of moral judgments, there is plenty of disagreement among those who theorise about meta-ethics as to whether such judgments are truth-apt or not. Moral realists will say that some moral judgments are true, others are false, but there is always a truth value, even if we don't know it. Antirealists hold that moral judgments are not comparable to statements of fact in this way. There are many varieties of antirealism, such as the emotivism or expressivism espoused by Ayer, or the prescriptivism defended by Hare.

Alchourròn and Martino are saying that provided we allow that logic is not limited to truths, there is no theoretical impossibility to the task of devising a logic of obligation, i.e. a deontic logic. We might even be agnostic as to moral realism. We could imagine a moral realist and an antirealist sitting at a desk together and agreeing on the logic of obligation, without agreeing as to whether moral judgments are truth-apt or not.

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  • Boy, it would be good if that could happen, because changing the laws back and forth won't get us very far.
    – Scott Rowe
    Jul 10, 2022 at 15:07
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In alethic modal logic you can interpret the box and diamond operators as "it is necessarily true that" and "it is possibly true that". However, in deontic logic, you wouldn't interpret the box and diamond as "it is obligatory true that" or "it is permissibly true that", because whether something is true is independent of any norm. This is the sense in which deontic logic doesn't deal with truth (or with facts) in the same way alethic modalities do: it deals with norms.

This difference is reflected in the fact that deontic logic isn't factive: "if p is obligatory, then p is the case" isn't an axiom, and neither is the equivalent "if p is the case, then p is permissible". There's no strong connection between what is the case and what is obligatory or permissible, because deontic modal logic deals with norms, not facts, and norms can be broken. On the other hand, "if it's necessary that p, then p is the case" and "if p is the case, then it's possible that p" are theorems of alethic modal logic. There's a connection between what is the case and what is possible or necessary, because alethic modal logic deals with facts.

You can still say, of course, that it is true that something is permissible or obligatory, or that it is a fact that there is a norm, but this is kind of a "second order" remark, and I believe that this is not what is meant by "dealing with truth" in the passage you cited.

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  • Obligatory quote: "What is necessary is never unwise."
    – Scott Rowe
    Jul 10, 2022 at 15:09

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