# Quantified Logic and Unquantified Modal Logic

Is there a need to study unquantified modal logic if one knows the quantified PC logic very well? There seems to be an obvious connection between Possibility and the Existential Quantifier, and Necessity and the Universal Quantifier. Yes, modal logic textbooks have an alphabet soup of various axiomatic systems K, S4, S5, etc.... but at the end of the day, can all of these be reduced to quantified PC with the right choice of axioms appended to your language.

It would seem nice to have a minimal theory without unnecessary jargon like "possible worlds" for example. ... unless, of course, you just want these terms that's okay,... but it would be nice to say these theories are "isomorphic" in a sense.

Is there really any difference between the two?

One possible response is that modal logic also covers things outside of Possibility and Necessity (P&N), for example "Temporal Modal logic" or "Provability" or even things like "Obligation" etc . So the question is two fold: (1) Are there limits to quantified PC logic (even with axioms added) such that it cannot get the P&N modal logic systems we want, and separately (2a) is there REALLY a difference between P&N modal logics and not-P&N modal logics (i.e. can we treat them like P&N modal logic anyway)? and (2b) even if they are different (i.e. the answer to 2a is "yes"), is there a way to reduce (at least some of) these to PC (a rephrasing of question (1) to other systems)

• In effect, the Kripke fashion of interpreting modal logic is just a second order logic with a set of 'worlds' which gets quantified over and the notion that worlds exhibit or do not exhibit traits described by propositions. So yes, you can just do modal logic as second order logic. Possibly P is just Exists W in Worlds with W exhibits P, Necessarily P is just All W in Worlds imply W exhibits P. You can then explore the lattices of Worlds and Facts which have the same sort of dual relationship as models and conditions. Until you choose a set of Worlds, all modal statements are true. – jobermark Nov 7 '17 at 20:52
• "Possible Worlds" don't cease to exist, because the option of changing reality is the point of moods. But they can exist as models, and not 'realities'. By making the added layer of second-order aspects implicit, you get a simpler representation. but a less interesting overall view of modality, in my opinion. – jobermark Nov 7 '17 at 20:55
• Possible duplicate of Is there modal logic without possible worlds? Unquantified logic, modal or not, is much weaker in its expressive power than quantified one. There is a difference, however, between quantified modal logic and its semantics in terms of possible worlds. Giving alternative semantics that avoids them is an active project. – Conifold Nov 7 '17 at 20:57

## 1 Answer

Philosophy.stackexchange doesn't seem to allow mathjax, so this answer is a bit messy; if anyone knows how to make mathematical notation display properly here, feel free to edit.

You ask:

Is there a need to study unquantified modal logic if one knows the quantified PC logic very well?

This suggests that you view the role of a logic as letting you express more things - so that more expressive logics are better. However, this isn't universally true: there are lots of times when we actually care about weaker logics, which have better properties (e.g. decidability), from both mathematical and philosophical perspectives. Propositional modal logic is in fact a fragment of predicate (= first-order) logic; this fragment is quite small, and much better behaved than full predicate logic, and plays an important role in the general study of nice fragments of predicate logic.

I am told (although I haven't read it) that the text to recommend here is Modal languages and bounded fragments of predicate logic.

How can we view propositional modal logic as a fragment of predicate logic? Well, you had the right basic idea - possibility ("Poss") and necessity ("Nec") should correspond to "exists" and "for all," respectively; let me give the details.

There is a natural way to convert a Kripke frame with a valuation (W, R, V) in propositional language {p_i: i in I} into a structure M in the sense of predicate logic: the language has unary predicates U_i for each p_i, a binary relation S corresponding to R, and the elements of the structure are exactly the worlds in the frame.

In this spirit, there is a natural way to convert a sentence phi in propositional modal logic to a formula phi'(x) in predicate logic, which intuitively has the property that phi holds at world w of (W, R, V) iff M satisfies phi'(w). The details are a bit tedious, but I'll give an example which should make it clear: if phi is the sentence

"Poss(p_0 implies Nec(p_1 implies p_2))",

then phi'(x) is the formula

"There is some y such that xRy and (if U_0(y), then for every z(if yRz, then U_1(z) implies U_2(z)))."

Now this translation is faithful in the following sense: if every world of a frame-with-valuation (W, R, V) satisfies phi, then the associated structure M satisfies "forall x, phi'(x)." There are other senses in which this translation is faithful, as well; propositional modal logic can, in every sense, be faithfully be embedded in predicate logic.

(I've focused on frame-oriented modal logic, but this really extends further.)

The converse is extremely false!

First of all, note that most structures in predicate logic don't come from frames-with-valuations - e.g. in what sense can you think of a field as coming from a frame-with-valuation?

Even once we restrict attention to structures which come from frames-with-valuations in the above sense, propositional modal logic is still very weak: for example, how would you try to express the predicate logic sentence

"There is exactly one world w such that for every world u, either u sees w or w sees u."

in the context of propositional modal logic?

All this is to say that, compared to predicate logic, propositional modal logic is extremely weak. However, in many senses this can be viewed as a positive: philosophically we may doubt the "definiteness" of predicate sentences in general (stemming from a doubt about the meaningfulness of iterated quantification over the entire domain), while having much more faith in sentences expressible in a particularly nice fragment of it; mathematically, modal logic and its variants exhibit a number of nice properties (e.g. decidability) which full predicate logic does not have, making it interesting from both pure and applied standpoints.

Let me contrast two different foundational goals (or directions, since there's no expectation of eventual completion) for formal logic:

• To express as wide a collection of propositions as possible. In this sense, of course propositional modal logic has nothing to offer over predicate logic. (EDIT: actually, this isn't entirely true; if we ask what kinds of frames validate given propositional modal sentences, things get very interesting: in a precise sense, we can in fact get full second-order logic from questions of validation - see this answer of mine for some more details/citations.)

• To study the mathematical commitments/costs involved in formalizing classes of propositions. E.g. if we want to express all statements involving first-order quantification, we have to stomach undecidability; conversely, predicate logic does have several nice properties such as compactness, which other more expressive logics lack. A star theorem in this regard is Lindstrom's theorem, which states roughly that there is no logic strictly more expressive than predicate logic which has both the compactness and Lowenheim-Skolem properties. (If you're interested in this sort of thing, you should look at the collection Model-theoretic logics.)

In my opinion, neither of these is more important than the other (and of course there are many, many others besides); where propositional modal logic really shines is with respect to the second goal.

So, should you study propositional modal logic? It depends: what do you want logic to do for you?