Blog posts about modal logic

• Typesetting modal logic
• Modal and temporal logic for security
• Word problems, modal logic, and regular expressions
• Axioms for S1 through S5
• Naming and numbering modal logic systems
• Modal logic and science fiction
• Dual axioms
• Temporal and polymodal logics

I created the image above by asking DALL-E 2 for an image of diamonds and boxes, a sort of pun on the ◇ (“diamond”) and □ (“box”) operators in modal logic.

My posts on modal logic have mostly been about monomodal logic, logic with one modal operator. This may not seem accurate because I’ve talked about □ (“box”) and ◇ (“diamond”). But these are really just one mode: you can define either in terms of the other.

◇p = ¬ □ ¬p
□p = ¬ ◇ ¬p

Temporal logic

Temporal logic is an example of a polymodal operator. There are many varieties of temporal logic, but one version defined by Arthur Prior in the 1950s has two basic modes: P (past) and F (future). Prior called his system tense logic but the term temporal logic is more common now.

We can interpret Fp to mean that at some point in the future, p will be the case.

We can interpret Pp to mean at some point in the past p was the case.

In temporal logic we have two analogs of the box operator, and each has its dual counterpart analogous to diamond.

Gp = ¬ F ¬p
Hp = ¬ P ¬p

So Gp can be read as “from now on p will be the case.”

And Hp can be read as “always up until now p has been the case.”

The combination GFp can be interpreted as “eventually p.”

The combination PHp can be interpreted as “up until some point in the past, p.”

I’ve mentioned before how Kripke semantics—models of “accessible worlds”—sounds like science fiction. Temporal logic can have that flavor too when you get into whether you view time as one linear stream or a branching tree, whether time is discrete or continuous, etc.

I mentioned briefly at the bottom of my post on duality that the algorithm given there for monomodal logics extends to polynomial logics.

In that post I said that the shell command

    tr '□◇→' '◇□←'

can turn a modal proposition into its dual. We could find the dual of a temporal proposition with the shell command

    tr 'FGPH→' 'GFHP←'

In other words, we replace Fs with Gs, Gs with Fs, Ps with Hs, Hs with Ps, and reversing the direction of implication.

More modes

Polymodal logics can have a lot more than two modes. For example, in security applications of modal logic, you might have a modal operator for every user of a system. Here we might model what a user knows rather than what is true: something might be true without a particular agent knowing that it’s true.

Simulation

Obviously monomodal logics are contained inside polymodal logics; if you have several modalities, box could be one of them. But here’s the amazing converse: polymodal logics can be simulated by monomodal logics.

In the abstract to their paper “Normal monomodal logics can simulate all others.” Marcus Kracht6 and Frank Wolter announce

This paper shows … that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations … sheding new light on the power of normal monomodal logic.

Related posts

• Modal axiom cube
• Simplifying Boolean expressions
• Intuitionistic logic in The Gondoliers

Axioms in modal logic often say that one sequence of boxes and diamonds in front of a proposition p implies another sequence of boxes and diamonds in front of p. For example, Axiom 4 says

□ p → □□ p

and Axiom 5 says

◇p → □◇p.

Every axiom has a dual form. The dual form of Axiom 4 is

◇◇p → ◇p

and the dual of Axiom 5 is

◇□ p → □ p.

Computing duals

There’s a simple way to compute the dual of such axioms:

Rotate all the squares 45° and rotate the arrow 180°.

This turns boxes into diamonds, diamonds into boxes, and flips the direction of implication.

Shell and Perl

We could do this using the tr utility at the command line

    $ echo '□□◇□◇p → □◇p' | tr '□◇→' '◇□←'
    ◇◇□◇□p ← ◇□p

We could also do the same thing in Perl, using its tr operator

    $prop = "□□◇□◇p → □◇p";
    ($dual = $prop) =~ tr/□◇→/◇□←/;
    print "$prop\n$dual\n";

This prints

    □□◇□◇p → □◇p
    ◇◇□◇□p ← ◇□p

It’s important to note that tr in both its incarnations does simultaneous replacement. It did what we expected, so it might be hard to notice.

tr takes two strings of the same length as arguments. Call the first one from and the second to. The easiest way to implement tr would have been to replace the first character of from with the first character of to, then replace the second character of from with the second character of to, etc.

This would have turned all our boxes into diamonds, then turned all diamonds into boxes, and so we’d be left with nothing but boxes! Our sequence □□◇□◇ would have turned into □□□□□.

Proof

Why is the rule above valid?

Let ○ stand for either a box or a diamond and suppose we start with

○₁ ○₂ … ○_m p → ○_m+1 ○_m+2 … ○_n p

where p is an arbitrary proposition.

Now let ○’ stand for the dual of ○. So if ○ is a box, ○’ is a diamond, and vice versa. Then

○ p = ¬○’ ¬p

by definition. (If you take □ as primary, then the equation above is the definition of ◇. If you take ◇ as primary, it’s the definition of □.) Apply this rule everywhere.

¬○’₁ ¬¬○’₂ ¬… ¬○’_m ¬p → ¬○’_m+1 ¬¬○’_m+2 … ¬○’_n ¬p

Now cancel out all the pairs of consecutive negations.

¬○’₁ ○’₂ … ○’_m ¬p → ¬○’_m+1 ○’_m+2 … ○’_n ¬p

Now take the contrapositive: (¬P → ¬Q) → (Q → P).

○’_m+1 ○’_m+2 … ○’_n ¬p → ○’₁ ○’₂ … ○’_m ¬p

Since p was an arbitrary proposition, we can replace p with ¬p.

○’_m+1 ○’_m+2 … ○’_n p → ○’₁ ○’₂ … ○’_m p

What we have above is the proposition we started with, with all the boxes replaced with diamonds, all the diamonds replaced with boxes, and the direction of the implication reversed.

More modalities

Notice that the theorem and proof still holds if there are multiple modalities. Suppose we have a set of modalities K_i. You could interpret

K_i p

as saying the ith agent knows proposition p is true. Then the dual is defined by

K’_i p = ¬ K_i ¬ p,

which could be interpreted as saying the ith agent does not know p to be false.

You could form the dual of a proposition involving K and K‘ expressions by adding primes to terms that don’t have them, and removing primes from terms that do, and turning the implication around. The proof would be the same as above, only we don’t restrict ○ to being □ or ◇.

Related posts

• Linear logic arithmetic
• Modal logic and science fiction
• Big logic

The previous post pointed out the analogy between models for modal logic (i.e. Kripke semantics) and science fiction. Rules for relationships between points in a Kripke model are analogous to rules for interplanetary travel in a fictional universe. This post will expand on this last point.

The following cube shows the relationships between eight different axiom systems for modal logic. Unfortunately, as I’ve written about before, the names of these systems are more historical than mnemonic.

An arrow from A to B means the set of propositions provable in A contain the propositions provable in B.

For example, all the systems represent by the cube are “normal,” meaning that they all require Axiom K. So any proposition provable in K is provable in the other systems as well.

The axes of the cube correspond to rules for how “worlds” are connected in Kripke models. Imagine that K in the diagram is the origin. The x axis (from K to KB) corresponds to symmetry, the y axis (from K to K4) corresponds to transitivity, and the z axis (from K to T) corresponds to reflexivity.

S4 example

S4 in the diagram has coordinates (0, 1, 1), meaning that Kripke models for S4 must be transitive and reflexive. In SF terms, this says you can travel from any planet back to itself, and if it is possible to travel from planet x to planet y, and possible to travel from planet y to planet z, then it must be possible to travel from x to y.

Why weaker systems than S5?

Models of S5 are symmetric: if you can travel from x to y, you can travel from y to x. S4 does not have have this requirement.

The axioms for S5 seem so sensible that you may wonder what the point is in having weaker systems. It all depends on how you interpret the modal qualifier □. In some applications of modal logic, □ has properties that correspond to models with unusual restrictions on accessibility.

We’re not trying to use logic to describe networks, though we could, but rather creating networks to model systems of logic. The resulting networks may have strange geometry if they weren’t motivated by geometry to begin with.

Axiom K

Axiom K (named for Saul Kripke) requires that

□(p → q) →  (□p → □q)

This says that if the proposition p → q is true in all accessible worlds, and if p is true in all accessible worlds, then q must be true in all accessible worlds.

Axiom T

Axiom T requires that

□p →p.

If something is true in all worlds accessible from a world w, and w is accessible from itself, then it must be true on w.

Axiom 4

Axiom 4 requires

□p → □□p.

Standing on a world w, the proposition □p says that p is true on all worlds accessible from w. So if x is a world accessible from w, p is true there. And if y is a world accessible from x, then by transitivity that world must be accessible from x, so p is true there. So □p is true on x. And since □p is true on any world accessible from w, then we can say □□p.

Axiom B

Axiom B requires

p → □◇p.

One way to state this is that if a proposition is true, then it is necessarily possibly true.

Imagine you’re on a world w there p is true. Symmetry says that if a world x accessible from w, then w is accessible from x. Now □◇p holds because in any world x accessible from w, we have ◇p because w is accessible from x.

Axiom 5

Axiom 5 requires

◇p →□◇p.

In words, whatever is possible is necessarily possible. A Kripke model for axiom 5 must be Euclidean: If from world w you can access worlds x and y, then from x you can access y. A relation that is symmetric and transitive is Euclidean. If x is accessible from w, then symmetry says w is accessible from x, and transitivity says you can go on from w to y.

Systems versus Axioms

One of the confusing things about modal logic is the complicated relation between the names of axioms and systems of axioms. Sometimes it’s simple. For example, K4 means the system with axioms K and 4.

The vertices of the cube above are systems, not axioms. And system B, confusingly, requires more than axiom B, and is in fact stronger than system KB because it adds axiom T. Chellas [1] calls this system KTB, a more verbose but more transparent name.

Provenance of the cube

The cube at the top of this post is a simplification of similar diagrams that appear in The Standford Encyclopedia of Philosophy and Wikipedia. Both may trace their origin to figure 4.1 in [1]. I don’t know that either diagram came from there, or whether they were developed independently, but [1] predates both. However, the graph in Chellas is less clear. The edge directions are implicit, as is the connection to properties of the models.

Related posts

• Linear logic
• Logic in moral terminology
• When are formal proofs worth the effort?

[1] Brian F. Chellas. Modal Logic: An Introduction. 1980,

Modal logic extends classical logic by adding one or more modes. If there’s only one mode, it’s usually denoted □. Curiously,  □ can have a wide variety of interpretations, and different interpretations motivate different axioms for how □ behaves. Modal logic is not one system but an infinite number of systems, depending on your choice of axioms, though a small number of axiom systems come up in application far more than others.

For a proposition p, □p is often interpreted as “necessarily p” but it could also be read, for example, as “it is provable that p“. Thanks to Gödel, we know some theorems are true but not provable, so p might be true while □p is false.

Kripke semantics interprets □p to be true at a “world” w if p is true in all worlds accessible from w. The rules of a logic system transfer to and from the set of models for that system, where a model is a directed graph of worlds, and an oracle (a “valuation function”) that tells you what’s true on each world. Axioms for a logic system correspond to requirements regarding the connectivity of all graph models for the system.

All this talk of what worlds are accessible from other worlds sounds a lot like science fiction. For example, if the planet Vulcan is accessible from Earth, and p is the statement “The blood of sentient beings is red,” then p is true on Earth, but not necessarily true on Earth since it’s not true on Vulcan, a world accessible from Earth.

Modal logic defines ◇ by

◇p = ¬ □ ¬ p.

For a proposition p, ◇p can be read as “possibly p.” A proposition ◇p is true on a world w if there is some world accessible from w where p is true.

So in the Star Trek universe, if p is the statement “Blood is green” then p is false on Earth, and so is □p, but ◇p is true because there is a world accessible from earth, namely Vulcan, where p is true.

You could have all kinds of fun making up rules about which worlds are accessible from each other. If someone from planet x can reach planet y, and someone from planet y can reach planet z, can someone from x reach z? Sounds reasonable, and if all worlds have this property then your Kripke frame is said to be transitive. But you could create a fictional universe in which, for whatever reason, this doesn’t hold.

Is a world accessible from itself? Depends on how you define accessibility. You might decide that a non-space faring world is not accessible from itself. But if every world is accessible from itself, your Kripke model is reflexive. If a Kripke frame is reflexive and transitive, the corresponding logic satisfies the S4 axioms. (More on this in the next post.)

Johan van Benthem gives an example in his book Modal Logic for Open Minds that’s scientific but not fictional. If you define a “world” to be a point in Minkowski space-time, then the worlds accessible from a given world are in that world’s future light cone. Propositions in this logic satisfy

◇□ p →□◇ p

and in fact the logic satisfies a system of axioms known as S4.2.

More modal logic posts

• Naming modal logic systems
• Typesetting modal logic
• Modal logic and security

Linear logic has connectives not used in classical logic. The connectives ⊗and & are conjunctions, ⊕ and ⅋ are disjunctions, and ! and ? are analogous to the modal operators ◻ and ◇ (necessity and possibility).

Another way to classify the connectives is to say ⊕ and & are called additive, ⊗ and ⅋ are multiplicative, and ! and ? are called exponential.And still another classification says that ⊕, ⊗, and ! have positive polarity, while &, ⅋, and ? have negative polarity.

This post will show that these arithmetical names are justified by analogy.

For one thing, multiplication-like connectives distribute over addition-like connectives.
2

Also, the exponential-like operators behave analogously to the equation

with respect to the addition-like and multiplication-like connectives.

In both equations, if you apply an exponential-like operator to the result of applying an addition-like operator, you get a multiplication-like operator applied to the exponential-like operator applied to two addition-like arguments separately.

The term “polarity” is justified by the fact that linear negation flips the polarity of connectives.

In the following analogs of De Morgan’s laws, negation turns conjunctions into disjunctions and vice versa, and it reverses the polarity of connectives.

Negating exponential connectives also flips polarity.

These rules are analogous to the rules for negating quantifiers in classical logic.

More logic posts

• Minimizing Boolean expressions
• Rotating symbols in LaTeX
• Modal and temporal logic for security

Stanislaw Ulam once said

Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.

There is only one way to be linear, but there are many ways to not be linear.

A similar observation applies to non-classical logic. There are many ways to not be classical.

Modal logic axioms

Modal logic extends classical logic by adding modal operators □ and ◇. The interpretation of these operators depends on the application, but a common interpretation is “necessarily” and “possibly.”

There are a bewildering number of possible axioms to choose from in order to create a system of modal logic, and these often have cryptic names like “K”, “G”, and “5”. Seldom is any reason given for the names, and after digging into this a little, I think I know why: There isn’t much reason. Or rather, the reasons are historical rather than mnemonic.

For example, Axiom T is so named because Kurt Gödel published a paper that called some set of axioms “System T.” Of course that raises the question of why Gödel called his system “T.”

Axiom 5 gets its name from C. I. Lewis’ classification of logic systems. But Lewis’ numbering of systems S1 through S5 was essentially arbitrary, other than larger numbers corresponding to more axioms.

Unless you’re interested in the history for its own sake, it’s probably not worth questioning why rules are called what they are.

Lemmon numbering

Although there’s not much logic behind the names of model logic axioms, there is a convenient way to catalog many of them. E. J. Lemmon [1] observed that many axioms for modal logic can be written in the form

Here’s a list of some common modal logic axioms, along with their names and Lemmon numbers.

[1] E. J. Lemmon, An Introduction to Modal Logic, American Philosophical Quarterly, Monograph 11. Oxford, 1977.