Modal Logic: Axioms and Systems for Alethic Modal Logic

Author: Thomas Metcalf
Category: Logic and Reasoning, Metaphysics
Word count: 992

Philosophical arguments are usually about what’s actually the case: whether it’s actually true that God exists or whether it’s actually true that we have free will.

But commonly, we also want to talk about whether something is possible or impossible, or whether it might be necessarily true: could not have been false.[1]

We use the term “modal logic” (where “modal” refers to there being different modes or ways in which something can be true or false) to refer to a special extension of standard logic.[2] Alethic modal logic deals with necessary and possible truth and falsity.[3]

This essay is an introduction to alethic modal logic.

An image of multiple instances of the planet Earth.
An image of multiple instances of the planet Earth.

1. Modal Concepts

Modal concepts, in this context,[4] are concepts such as possibility and necessity:[5]

  • if a proposition is necessarily true, then it has to be true: it could not possibly have been false. Maybe the proposition that all squares have four sides is necessarily true;
  • if a proposition is impossibly true, then it could not have been true. Maybe the proposition that 3 > 5 is impossibly true;
  • if a proposition is possibly true, then it could have been true. The proposition that there are twelve planets in this solar system could have been true (although it actually isn’t).

2. Adding Modality to Symbolic Logic

In standard (non-modal) sentential logic, we create formulas by sticking sentences together (abbreviated by letters such as “A” and “B”) with constants such as “not” and “if … then ….”[6] To name possibility, impossibility, and necessity explicitly, we need some special constants. The most common are the box (“□”) and the diamond (“◊”). The box means “necessarily” and the diamond means “possibly.”[7]

So if we want to symbolize the sentence “1+2=3,” we can just write “M.” But if we want to say that it’s necessarily true that 1+2=3, we can write, “□M.” If I want to say that a unicorn exists, I can write “U,” but if I want to write that it’s possible for a unicorn to exist, I can write “◊U.”

3. Basic Modal-Logic Axioms

Even if we have some new symbols, they won’t do us any good if we don’t know how to use them. We have to choose a set of axioms:[8] basic propositions that tell us which formulas can be derived from which.

Given the meanings of “necessarily” and “possibly,” we can define one in terms of the other. If “¬” means “not” and “↔” means “if and only if,” then for any formula φ (which itself may contain boxes or diamonds):[9]

(Dual)[10]

□φ ↔ ¬◊¬φ

◊φ ↔ ¬□¬φ

¬□φ ↔ ◊¬φ

¬◊φ ↔ □¬φ

For example, “not necessarily not” and “possibly” seem to entail the same claim. We can call all four of these axioms “(Dual)” because they take advantage of dualities between “◊” and “□”: how formulas with one can be translated into equivalent formulas with the other.

Another axiom that seems true is that if a formula follows from no assumptions (i.e. no matter what), then it’s necessarily true:

(N) If φ is a theorem (i.e., it’s unconditionally true, or true based on no assumptions, in the system), then □φ is a theorem.

For example, it seems that no matter what else is true about the world, if φ then φ, for any φ. So we can conclude, □(φ → φ).

This axiom also seems true:[11]

(K) □(φ → ψ) → (□φ → □ψ)

For example, if it’s necessarily true that if 5 > 3 then 3 < 5, then: if it’s necessarily true that 5 > 3, then it’s necessarily true that 3 < 5. But if φ isn’t necessarily true, then ψ might not be necessarily true. For example, if it’s necessarily true that all squares have four sides, but it’s not necessarily true that this building is square, then it won’t be necessarily true that the building has four sides; someone could destroy one of the sides.

Another plausible axiom is:[12]

(M) □φ → φ

Intuitively, everything that must be true is true.

4. More Modal-Logic Axioms and Systems

We could have a system of modal logic that only contained (Dual), (N), (K), and (M).[13] But it might be incomplete: there might be true conclusions that we can’t prove within the system, even given true premises.

So we might want to include other axioms in our system:[14]

(B) φ → □◊φ

(4) □φ → □□φ

(5) ◊φ → □◊φ

These are more controversial. For example, the (B) axiom entails that if something could have been necessarily true, then it is true.[15] But it’s not initially clear why that would be the case. Similarly, just because something is possible, does that entail that it couldn’t have been impossible? Axiom (5) says “yes.”

5. Systems and Axioms

The axioms we accept will determine which system we end up using. Here’s a summary of which axioms are true according to which systems (axioms in parentheses, systems named in boldface):[16]

(Dual), (N), and (K) (M) (B) (4) (5)
K Yes No No No No
M or T Yes Yes No No No
B Yes Yes Yes No No
S4 Yes Yes No Yes No
S5 Yes Yes Yes Yes Yes

The more axioms your system contains, the more arguments the system will regard as valid: the more it can prove. But if you use an axiom that’s not true, then the resulting system may be unsound: it may lead you from true premises to false conclusions. That’s not ideal; part of the point of having a system of logic is to ensure that that doesn’t happen.[17]

Which of these axioms (if any) are actually true depends on how possibilities affect other possibilities. This is a complicated but interesting topic.[18]

6. Conclusion

There are several other useful systems of alethic modal logic,[19] and many non-alethic modal-logic systems.[20] However, when philosophers say “modal logic” by itself, they’re usually referring to a system of alethic modal logic that contains some or all of the axioms described herein.

Notes

[1] For more about modality in general, see Possibility and Necessity: An Introduction to Modality by Andre Leo Rusavuk. For a discussion of how we might know or reasonably believe modal claims, see Modal Epistemology: Knowledge of Possibility and Necessity by Bob Fischer.

As for philosophical arguments, one might hold that many arguments in ethics, for example, contain a premise that purports to be a necessary truth. This would be the case if most moral claims are necessarily true if they’re true at all. One might say something similar for arguments in epistemology, if facts about whether some evidence E justified some hypothesis H given some background-knowledge K are necessary truths. And one might think that many claims in metaphysics are necessarily true if they’re true at all. For more along these lines, see Philosophy and its Contrast with Science and What is Philosophy? by Thomas Metcalf.

[2] For more about standard logic, see Formal Logic: Symbolizing Arguments in Sentential Logic by Thomas Metcalf and Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic by Timothy Eshing, as well as Open Logic Project (n.d.).

[3] This essay is about alethic modal logic, the logic concerning the different ways things can be true or false. Instead of using that term in the title, which could put off lay readers, I’ve elected to explicitly mention possibilities and necessities in the title. There are many other modal logics, some of which aren’t about possibility and necessity; for example, there are modal logics about obligation and permission, about knowledge, and about time (Garson, 2022). Note also that modal logic is normally considered an extension of first-order logic, but one can interpret matters in reverse (van Benthem, 2010, ch. 27).

[4] The context is alethic modality; see n. 3.

[5] It’s standard to talk about possible worlds, but that can be confusing to lay readers. Some might mistake possible worlds (pace David Lewis (2001)) for something like parallel universes, and there is of course debate about their nature anyway. So I’ve elected to try to avoid such talk except for in notes.

[6] See also Formal Logic: Symbolizing Arguments in Sentential Logic by Thomas Metcalf and Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic by Timothy Eshing.

[7] See Garson (2022). Other influential authors have used “L” and “M” (Hughes and Cresswell 1996, 14-15), based on notation from Jan Łukasiewicz (cf. Simons, 2022, sect. 5).

[8] An axiom is a proposition that we take as given, although axioms can be argued for or justified (in the sense that one can argue that adding a certain axiom to a system is necessary to make it sufficiently strong). I’m stating these as material conditionals and biconditionals (that could be entered on a new line, in a natural-deduction system, with no assumptions), but we could also have a natural-deduction system in which the axiom tells us that if we have the antecedent on one line, then we can infer the consequent on a new line, and so on. After all, if Δ ∪ { φ } ⊢ ψ (i.e. the set of assumptions containing the assumptions in set Δ of formulas plus the formula φ) entails (in the system) the formula ψ, then in that system, φ → ψ follows from the set Δ. This fact follows from the Deduction Theorem (cf. Franks, 2021).

[9] Note that strictly speaking, these formulas won’t be “true” because they contain unbound metavariables (with the “φ” standing in for sentence variables, they’re variables of variables, so we call them “metavariables”). Thus, you can imagine them all with a quantifier as in, “for all formulas φ, ….” (See Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic by Timothy Eshing.) Also, these are not always noted as axioms (sometimes they’re simply stated as definitions), but I don’t think there’s any harm in explicitly describing them as axioms.

[10] These equivalencies are sometimes called “duals” or “dualities” (van Benthem, 2010, sect. 2.1).

[11] The system was named after Saul Kripke, one of the most important contributors to the development of modal logic. See e.g. Kripke (1959 and 1963) and Lemmon and Scott (1977, 29) for the name. I’ve stated these as conditionals rather than sequents, because we’re talking about axioms, and so I want to suggest entering these (conditionals) as new lines (based on no assumptions) in a natural-deduction proof. However, they arguably work as sequents too if you replace the arrow with a turnstile (“⊢” symbol), which means (roughly) “entails, in our logical system, that,” and given the left-hand-side on some line, entering the right-hand-side as a new line. One might argue that sequents are about the meta-logic question of what can be proven in the system given what, while conditionals are about the object-language question of what’s true in the system. There is some dispute here (Church, 1996, 165), which we don’t need to get into; see Klement (n.d., §§ 5-6) for more. I add that we can imagine systems of modal logic that don’t even contain (K); some of these are called “non-normal” modal logics (cf. Priest, 2012, ch. 3).

[12] This axiom will be true whenever accessibility is reflexive: that is, whenever every world is possible relative to itself. No matter which world turns out to be actual, that very world will also be possible. If so, then another useful, true formula will be “φ → ◊φ.” This is intuitive: surely everything that’s actually true is possibly true. As before, here and throughout, φ can contain its own boxes and diamonds, so given this axiom, “□□φ → □φ” represents true formulas, as does “□φ → ◊□φ,” and so on.

[13] The resulting system is sometimes called “M” or “T” (Garson, 2022, § 2).

[14] The (B) axiom will be true whenever accessibility is symmetric: that is, whenever if world w2 is possible relative to w1, w1 will be possible relative to w2, for any worlds w1 and w2. If so, another useful, true formula will be “◊□φ → φ.” The (4) axiom will be true whenever accessibility is transitive: that is, whenever if world w3 is possible relative to w2 and w2 is possible relative to w1, w3 will be possible relative to w1, for any worlds w1-w3. If so, another useful, true formula will be “◊◊φ → ◊φ.” In fact, any string of ◊s (including only one ◊) can be replaced by any string of ◊s, and any string of □s (including only one □) can be replaced by any string of □s. Recall that given M, any string of □s can be replaced by one □, and any ◊ can be replaced by any string of ◊s  (Garson, 2022, § 2). So with M and (4), you can switch out a string of any length of either symbol with a string of any length of that symbol. For example, “◊φ → ◊◊◊φ” and “□◊φ → □□□□□◊φ” will be true for any formula φ. The (5) axiom will be true whenever accessibility is Euclidean: that is, whenever: if worlds w2 and w3 are possible relative to w1, w2 and w3 will also be possible relative to each other, for any worlds w1-w3. Note that as with the axioms (Dual), there are dual theorems for each of these: (B Dual) ◊□φ → φ; (4 Dual) ◊◊φ → ◊φ; and (5 Dual) ◊□φ → □φ. I’m calling these duals “theorems” but one might just as well introduce them as their own axioms. Indeed, some of them might be more intuitive than the “original” axioms (B), (4), and (5).

[15] That is, the (B) axiom plus the (Dual) axioms entails the following theorem: (B Dual) ◊□φ → φ. Given (B), if ¬φ then □◊¬φ. Given (Dual), □◊¬φ → ¬◊□φ.

[16] Garson (2022, § 2). Here I follow Garson in using boldface to refer to the system and plain font in parentheses to refer to the axioms, although some authors (e.g. Hughes and Cresswell 1996) use the opposite convention. We get the names “S4” and “S5” from Lewis and Lankford (1932, 501). Note that I described creating S5 by adding either (B) and (4), or (5), to M. But (B), (4), and (5) will be true in S5. So you could get S5 by adding (B), (4), and (5). The soundness of S5 is sufficient for the truth of (B), (4), and (5).

[17] A system is sound whenever everything provable in the system is true. A system is complete whenever every true statement is provable in the system. See e.g. Garson (2022, § 6) for more.

[18] See notes 12, 14, and 16.

[19] These other systems might be useful for different varieties of modality, for example, because we think that those varieties of modality are governed by different metaphysical or physical laws. If so, then we might think that one of these systems is sound and complete for what’s metaphysically possible, while a different one is sound and complete for what’s physically possible. See Kment (2022) for more about the varieties of modality.

[20] See Garson (2022) for an extensive discussion. These include logics for obligation and permission, logics for time and temporal facts, and more.

References

Church, A. (1996). Introduction to Mathematical Logic. Princeton University Press.

Franks, C. (2021). The deduction theorem (before and after Herbrand). History and Philosophy of Logic, 42(2), 129–159.

Garson, J. (2022). Modal logic. In E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Summer 2022 Edition

Hughes, G. E. and Creswell, M. J. (1996). A new introduction to modal logic. Routledge.

Klement, K. C. (N.d.). Propositional logic. In J. Fieser and B. Dowden (Eds.), Internet Encyclopedia of Philosophy 

Kment, B. (2022). Varieties of modality. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2022 ed.). 

Kripke, S. A. (1959). A completeness theorem in modal logic. Journal of Symbolic Logic, 24(1), 1-14.

Kripke, S. A. (1963). Semantical analysis of modal logic, I. Mathematical Logic Quarterly, 9(5-6), 67-96.

Lemmon, E. J. & Scott, D. (1977). The “Lemmon Notes”: An Introduction to Modal Logic. Basil Blackwell.

Lewis, D. (2001). On the Plurality of Worlds. Wiley.

Lewis, C. I. & Langford, C. H. (1932). Symbolic Logic. Century Company.

Open Logic Project. (N.d.). Open Logic Project. Open Logic Project.

Priest, G. (2012). An Introduction to Non-Classical Logic. Cambridge University Press.

Simons, P. (2022). “Jan Łukasiewicz.” In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2022 ed.). 

Van Benthem, J. (2010). Modal Logic for Open Minds. CSLI Publications.

Related Essays

Possibility and Necessity: An Introduction to Modality by Andre Leo Rusavuk

Modal Epistemology: Knowledge of Possibility and Necessity by Bob Fischer

Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic by Timothy Eshing

Formal Logic: Symbolizing Arguments in Sentential Logic by Thomas Metcalf

Modal Ontological Arguments for the Existence of God by Thomas Metcalf

Philosophy and its Contrast with Science by Thomas Metcalf

What is Philosophy? by Thomas Metcalf.

Translation

Turkish

Acknowledgments

The author is grateful to the editors of 1000-Word Philosophy, to Timothy Eshing, and to Andre Leo Rusavuk for helpful comments on earlier drafts of this essay.

About the Author

Tom Metcalf is an associate professor at Spring Hill College in Mobile, AL. He received his PhD in philosophy from the University of Colorado, Boulder. He specializes in ethics, metaethics, epistemology, and the philosophy of religion. shc.academia.edu/ThomasMetcalf

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