Formal logic on rightfulness

Question

Is there a kind of logic that could easily formulate this kind of statement: X has the right to do Y?

Or more generally: An object that has the property X (or in a set X) could also choose to have property Y, unless stated otherwise (such as Y combined with something else), while there may or may not be a thing with X already had Y.

It is supposed to work like the law. Part of the question is, what does a right even mean? I think my definition is quite incomplete. But I want a starting point (such as an already existed theory) for formulating things like the law and has this feature, instead of accurately describing the law itself.

Edit: For the purpose of this question, I don't care what rights humans naturally should or should not have, but for a set of rules that contains the idea of rights, whether it is a law or a kind of morality, or something completely unrelated to humans, and whether it is "good" or plain evil, to interpret what it actually means.

Answer 1

Brief answer: See Stig Kanger's 1972 paper "Law and Logic" (Theoria 38, pp. 105-132).

Longer answer: Logical treatment of jurisprudence is, as it happens, at an incipient stage, though the idea is not novel - since someone called Leibniz has passed through this world:

We need a new logic in order to know degrees of probability, since this is necessary in judging the proofs of matters of fact and of morals, where there are unusually good reasons on both sides and we are concerned only to know on which side to tip the scales. But the art of weighing probabilities is not yet even partly explained, though it would be of great importance in legal matters and even in the management of business.

(Philosophical Papers and Letters: A Selection, 2nd ed. by L. E. Loemker, Synthese Historical Library, 1969, vol. 2., p. 260)

Leibniz linked logic and jurisprudence primarily by probability theory. Not surprising, for logic and jurisprudence meet not only at the point of an application of logic to jurisprudence, but also at their common interest in argument text as a self-standing object of study - its normative aspects, its prescriptive form, the importance of fine distinctions in meanings and concepts about what follows from what, and the like. Stephen Toulmin stresses this point to the extent of saying:

Logic (we may say) is generalised jurisprudence. Arguments can be compared with law-suits, and the claims we make and argue for in extra-legal contexts with claims made in the courts, while the cases we present in making good each kind of claim can be compared with each other.

(The Uses of Argument, updated edition, CUP, 2003, p. 7)

Hence, we should be more than cautious against oversimplifying matters, in particular, not be tempted to immediately employ deontic logic (meanwhile, which has its own foundational issues).

Returning to the proposition 'X has the right to do Y' as an example, we need a typology of rights amenable to formal treatment in the first place. Kanger bases his analysis on the jurist W. N. Hohfeld's conceptions proposed in his papers Some Fundamental Legal Conceptions as Applied in Judicial Reasoning 1913 and 1917 (both of them are freely available by Yale University on the website https://digitalcommons.law.yale.edu).

Very roughly, Kanger proposes to expand the standard deontic logic in accordance with his definitions of four simple types of rights, i.e., claim, power, immunity, and freedom, and its correlatives duty, liability, disability, and exposure (no-right), each relation of right, as each jural relation, involves two parties and a state of affairs between them.

An exposition of Kanger's formalisation is, I suppose, beyond the bounds of this question. Overall, setting out a shared framework for logic and law is an issue of ongoing research (see, for example, the JuriLog Project, https://anr.fr/Project-ANR-11-FRAL-0003) and, a text-book style explanation is yet to be available (Deontic Logic and Legal Systems by by P. E. Navarro and J. L. Rodríguez, CUP, 2014, may be helpful in this respect).

Answer 2

Modal Logic

GENERAL

• ♢P is short-hand notation for "P is allowed" or "P is a right"
• ~♢ Q is Q is not allowed
• □ P is "it is necessary that Q" or "it must be that Q"
• ~□ P is "it is not necessary that P" or "P is not mandatory"

EXAMPLES

• ∀ h ∈ HUMANS ♢ h eat ice-cream means something like, "For any human h, h is allowed to eat ice-cream"
• ~♢ Bob not pay income tax is "It is not allowed for Bob to not pay income tax"
• □ Bob pay income tax is "Bob must pay income tax"
• ~□ Bob eat ice-cream is it is not required that Bob eat ice-cream

Note the following:

• ~□ P if and only if ♢(not P)
• Example: "it is not legally required that you wear a blue shirt if and only if you have the legal right to not wear a blue shirt"
• ~♢ P if and only if □(not P)
• Example: I do not have the right to perjure myself if and only if I am required by law to not perjure myself.

Note that "~♢" takes less time to write than "it is not possible." That is why diamond and box are "short-hand" notation. When logicians mathematicians are want to use the same sentence fragment over and over again, the logicians will create a short-name for it. ♢ and □ are nice and short.