Metalanguage for actual, modal, and intentional contexts
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John F Sowa
Jun 21, 2026, 12:26:43 AMJun 21
to ontolog-forum, CG
There has been some discussion about modal and intentional logics. But it's possible to replace all such logics with ordinary first order logic just by using metalanguage. The following diagram shows the kinds of relations that can be used to replace those logics with FOL. The relation named 'possible' relates a diagram that describes the actual state to a diagram of the modal state. The relation named 'intended' relates a diagram of an actual world and a person to a diagram of a state that person intended.
With ordinary FOL plus these two new relations, It's possible to define and use an open ended variety of ways of talking about and relating contexts. New options may include contexts that are desired, feared, wanted, rejected, preferred, despised, foolish, or brilliant. There is no limit to the number of ways of thinking, talking,, and representing relations among contexts.These options provide a much richer and more flexible range of options than a modal logic that only supports a two-way distinction of necessary or possible. They show that metalanguage with ordinary FOL provides a much greater range of options than any kind of modal or intentional logic.
Furthermore, it does not require any special kind of theorem prover. Any system that can reason about physical things can be used to reason about the way people think and talk about anything --- including themselves. It's simpler and more general than inventing a large number of special purpose logics.
John
João Oliveira Lima
Jul 13, 2026, 7:04:28 AM (10 days ago) Jul 13
to ontolo...@googlegroups.com
John,
Your post reminded me of a proposal Carlos Alchourrón made in 1969, in "Logic of Norms and Logic of Normative Propositions." For the deontic case he does almost exactly what you describe for modal and intentional logic.
Alchourrón separates two things that are usually run together: the logic of norms themselves (his "deontic logic," with object-language operators O, P, Ph) and the logic of propositions about norms (his "normative logic"). For the latter he introduces a relational operator Nx, where "Nx α" reads "agent x has issued a norm to the effect that α." That is precisely a metalinguistic relation between an agent and a norm — the same shape as your "intended" relation among an actual world, an agent, and an intended state. He indexes it by author and iterates it freely: "Ny O Nx O p" = "y has ruled that it is obligatory for x to rule that p," which is your open-ended nesting of contexts, and he argues it survives self-reference.
The payoff is exactly the richer, more flexible range you point to. Making the operator metalinguistic lets him draw distinctions that collapse in ordinary modal logic: strong permission (Ps = NxP, an actual act of permitting) versus weak permission (the mere absence of a prohibition). The strong operators assert that a normative act occurred, the weak ones deny it. From that single move he can represent gaps (nothing has been normed about p) and inconsistencies (p normed both permitted and prohibited), define the consistency and completeness of a normative system, and disambiguate slogans like "whatever is not forbidden is permitted." And none of it needs a special calculus: his decision procedure is ordinary truth tables layered over propositional logic: your point that a system able to reason about ordinary things can reason about how people think and talk.
The one caveat I would add is Alchourrón's own: he insists the metalinguistic system presupposes, but is not identical with, the logic of norms. He metalinguizes only the "x issued / normed …" layer; the deontic operator itself never disappears, it sits inside the N relation, and his decision method for the normative formulas explicitly presupposes the deontic one, which in turn presupposes propositional logic. So at least in his hands the metalanguage does not replace the modal logic; it relocates the speech-act layer into ordinary truth-apt propositions and presupposes the modal core rather than dissolving it. (He also shows the two calculi become isomorphic only under completeness and consistency; drop those assumptions and the metalinguistic system is strictly the richer of the two, which supports your case, but also shows the two layers are genuinely distinct.)
So I would state the limit this way: what the FOL-plus-metalanguage layer captures cleanly is the truth of propositions about which norms or possibilities, intentions obtain; those propositions really are just true or false. What it does not by itself supply is the logic of the norms or modalities as such, which stays presupposed inside the relation. The reduction is real and, as you say, more general; but its ordinary-truth part is confined to propositions about contexts, not to the modal or prescriptive force of the contexts themselves.
Best,
Joao Lima
Your post reminded me of a proposal Carlos Alchourrón made in 1969, in "Logic of Norms and Logic of Normative Propositions." For the deontic case he does almost exactly what you describe for modal and intentional logic.
Alchourrón separates two things that are usually run together: the logic of norms themselves (his "deontic logic," with object-language operators O, P, Ph) and the logic of propositions about norms (his "normative logic"). For the latter he introduces a relational operator Nx, where "Nx α" reads "agent x has issued a norm to the effect that α." That is precisely a metalinguistic relation between an agent and a norm — the same shape as your "intended" relation among an actual world, an agent, and an intended state. He indexes it by author and iterates it freely: "Ny O Nx O p" = "y has ruled that it is obligatory for x to rule that p," which is your open-ended nesting of contexts, and he argues it survives self-reference.
The payoff is exactly the richer, more flexible range you point to. Making the operator metalinguistic lets him draw distinctions that collapse in ordinary modal logic: strong permission (Ps = NxP, an actual act of permitting) versus weak permission (the mere absence of a prohibition). The strong operators assert that a normative act occurred, the weak ones deny it. From that single move he can represent gaps (nothing has been normed about p) and inconsistencies (p normed both permitted and prohibited), define the consistency and completeness of a normative system, and disambiguate slogans like "whatever is not forbidden is permitted." And none of it needs a special calculus: his decision procedure is ordinary truth tables layered over propositional logic: your point that a system able to reason about ordinary things can reason about how people think and talk.
The one caveat I would add is Alchourrón's own: he insists the metalinguistic system presupposes, but is not identical with, the logic of norms. He metalinguizes only the "x issued / normed …" layer; the deontic operator itself never disappears, it sits inside the N relation, and his decision method for the normative formulas explicitly presupposes the deontic one, which in turn presupposes propositional logic. So at least in his hands the metalanguage does not replace the modal logic; it relocates the speech-act layer into ordinary truth-apt propositions and presupposes the modal core rather than dissolving it. (He also shows the two calculi become isomorphic only under completeness and consistency; drop those assumptions and the metalinguistic system is strictly the richer of the two, which supports your case, but also shows the two layers are genuinely distinct.)
So I would state the limit this way: what the FOL-plus-metalanguage layer captures cleanly is the truth of propositions about which norms or possibilities, intentions obtain; those propositions really are just true or false. What it does not by itself supply is the logic of the norms or modalities as such, which stays presupposed inside the relation. The reduction is real and, as you say, more general; but its ordinary-truth part is confined to propositions about contexts, not to the modal or prescriptive force of the contexts themselves.
Best,
Joao Lima
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