Chapter 44 Semantics for ML

The big idea behind the semantics for ML is this. In ML, sentences are not just true or false, full stop. A sentence is true or false at a given possible world, and a single sentence may well be true at some worlds and false at others. We then say that $\mathrm{\square }𝒜$ is true if⁠f $𝒜$ is true at every world, and $\mathrm{◇}𝒜$ is true if⁠f $𝒜$ is true at some world.

That’s the big idea, but we need to refine it and make it more precise. To do this, we need to introduce the idea of an interpretation of ML. The first thing you need to include in an interpretation is a collection of possible worlds. Now, at this point you might well want to ask: What exactly is a possible world? The intuitive idea is that a possible world is another way that this world could have been. But what exactly does that mean? This is an excellent philosophical question, and we will look at it in a lot of detail later. But we do not need to worry too much about it right now. As far as the formal logic goes, possible worlds can be anything you like. All that matters is that you supply each interpretation with a non-empty collection of things labelled possible worlds .

Once you have chosen your collection of possible worlds, you need to find some way of determining which sentences of ML are true at which possible worlds. To do that, we need to introduce the notion of a valuation function. Those of you who have studied some maths will already be familiar with the general idea of a function. But for those of you who haven’t, a function is a mathematical entity which maps arguments to values. That might sound a little bit abstract, but some familiar examples will help. Take the function $x+1$. This is a function which takes in a number as argument, and then spits out the next number as value. So if you feed in the number $1$ as an argument, the function $x+1$ will spit out the number $2$ as a value; if you feed in $2$, it will spit out $3$; if you feed in $3$, it will spit out $4$ …Or here is another example: the function $x+y$. This time, you have to feed two arguments into this function if you want it to return a value: if you feed in $2$ and $3$ as your arguments, it spits out $5$; if you feed in $1003$ and $2005$, it spits out $3008$; and so on.

A valuation function for ML takes in a sentence and a world as its arguments, and then returns a truth-value as its value. So if $\nu$ is a valuation function and $w$ is a possible world, ${\nu }_w(𝒜)$ is whatever truth-value $\nu$ maps $𝒜$ and $w$ to: if ${\nu }_w(𝒜)=F$, then $𝒜$ is false at world $w$ on valuation $\nu$; if ${\nu }_w(𝒜)=T$, then $𝒜$ is true at world $w$ on valuation $\nu$.

These valuation functions are allowed to map any atomic sentence to any truth-value at any world. But there are rules about which truth-values more complex sentences get assigned at a world. Here are the rules for the connectives from TFL:

1. 1.
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   ${\nu }_w(\neg 𝒜)=T$ if⁠f ${\nu }_w(𝒜)=F$

2. 2.
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   ${\nu }_w(𝒜\wedge ℬ)=T$ if⁠f ${\nu }_w(𝒜)=T$ and ${\nu }_w(ℬ)=T$

3. 3.
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   ${\nu }_w(𝒜\vee ℬ)=T$ if⁠f ${\nu }_w(𝒜)=T$ or ${\nu }_w(ℬ)=T$, or both

4. 4.
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   ${\nu }_w(𝒜\to ℬ)=T$ if⁠f ${\nu }_w(𝒜)=F$ or ${\nu }_w(ℬ)=T$, or both

5. 5.
   ​

   ${\nu }_w(𝒜\leftrightarrow ℬ)=T$ if⁠f ${\nu }_w(𝒜)=T$ and ${\nu }_w(ℬ)=T$, or ${\nu }_w(𝒜)=F$ and ${\nu }_w(ℬ)=F$

So far, these rules should all look very familiar. Essentially, they all work exactly like the truth-tables for TFL. The only difference is that these truth-table rules have to be applied over and over again, to one world at a time.

But what are the rules for the new modal operators, $\mathrm{\square }$ and $\mathrm{◇}$? The most obvious idea would be to give rules like these:

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  ${\nu }_w(\mathrm{\square }𝒜)=T$ if⁠f $\forall w^{\prime }({\nu }_{w^{\prime }}(𝒜)=T)$

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  ${\nu }_w(\mathrm{◇}𝒜)=T$ if⁠f $\exists w^{\prime }({\nu }_{w^{\prime }}(𝒜)=T)$

This is just the fancy formal way of writing out the idea that $\mathrm{\square }𝒜$ is true at $w$ just in case $𝒜$ is true at every world, and $\mathrm{◇}𝒜$ is true at $w$ just in case $𝒜$ is true at some world.

However, while these rules are nice and simple, they turn out not to be quite as useful as we would like. As we mentioned, ML is meant to be a very flexible tool. It is meant to be a general framework for dealing with lots of different kinds of necessity. As a result, we want our semantic rules for $\mathrm{\square }$ and $\mathrm{◇}$ to be a bit less rigid. We can do this by introducing another new idea: accessibility relations.

An accessibility relation, $R$, is a relation between possible worlds. Roughly, to say that $R{w}_1{w}_2$ (in English: world $w_1$ accesses world $w_2$) is to say that $w_2$ is possible relative to $w_1$. In other words, by introducing accessibility relations, we open up the idea that a given world might be possible relative to some worlds but not others. This turns out to be a very fruitful idea when studying modal systems. We can now give the following semantic rules for $\mathrm{\square }$ and $\mathrm{◇}$:

1. 6.
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   ${\nu }_{w_1}(\mathrm{\square }𝒜)=T$ if⁠f $\forall w_2(R{w}_1{w}_2\to {\nu }_{w_2}(𝒜)=T)$

2. 7.
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   ${\nu }_{w_1}(\mathrm{◇}𝒜)=T$ if⁠f $\exists w_2(R{w}_1{w}_2\wedge {\nu }_{w_2}(𝒜)=T)$

Or in plain English: $\mathrm{\square }𝒜$ is true in world $w_1$ if⁠f $𝒜$ is true in every world that is possible relative to $w_1$; and $\mathrm{◇}𝒜$ is true in world $w_1$ if⁠f $𝒜$ is true in some world that is possible relative to $w_1$.

So, there we have it. An interpretation for ML consists of three things: a collection of possible worlds, $W$; an accessibility relation, $R$; and a valuation function, $\nu$. The collection of ‘possible worlds’ can really be a collection of anything you like. It really doesn’t matter, so long as $W$ isn’t empty. (For many purposes, it is helpful just to take a collection of numbers to be your collection of worlds.) And for now, at least, $R$ can be any relation between the worlds in $W$ that you like. It could be a relation which every world in $W$ bears to every world in $W$, or one which no world bears to any world, or anything in between. And lastly, $\nu$ can map any atomic sentence of ML to any truth-value at any world. All that matters is that it follows the rules (1)–(7) when it comes to the more complex sentences.

Let’s look at an example. It is often helpful to present interpretations of ML as diagrams, like this:

Here is how to read the interpretation off from this diagram. It contains just two worlds, 1 and 2. The arrows between the worlds indicate the accessibility relation. So 1 and 2 both access 1, but neither 1 nor 2 accesses 2. The boxes at each world let us know which atomic sentences are true at each world: $A$ is true at 1 but false at 2; $B$ is false at 1 but true at 2. You may only write an atomic sentence or the negation of an atomic sentence into one of these boxes. We can figure out what truth-values the more complex sentences get at each world from that. For example, on this interpretation all of the following sentences are true at $w_1$:

If you don’t like thinking diagrammatically, then you can also present an interpretation like this:

$W$:

$1,2$

$R$:

$⟨1,1⟩,⟨2,1⟩$

$\nu$:

${\nu }_1(A)=T,{\nu }_1(B)=F,{\nu }_2(A)=F,{\nu }_2(B)=T$

You will get the chance to cook up some interpretations of your own shortly, when we start looking at counter-interpretations.

We can now extend all of the semantic concepts of TFL to cover ML:

(From now on we will drop the explicit ‘modal’ qualifications, since they can be taken as read.)

We can also extend our use of $\models$. However, we need to add subscripts again, just as we did with $⊢$. So, when we want to say that ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n\therefore 𝒞$ is valid, we will write: ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{\models }_{𝐊}𝒞$.

Let’s get more of a feel for this semantics by presenting some counter-interpretations. Consider the following (false) claim:

In order to present a counter-interpretation to this claim, we need to cook up an interpretation which makes $\neg A$ true at some world $w$, and $\neg \mathrm{◇}A$ false at $w$. Here is one such interpretation, presented diagrammatically:

It is easy to see that this will work as a counter-interpretation for our claim. First, $\neg A$ is true at world $1$. And second, $\neg \mathrm{◇}A$ is false at $1$: $A$ is true at $2$, and $2$ is accessible from $1$. So there is some world in this interpretation where $\neg A$ is true and $\neg \mathrm{◇}A$ is false, so it is not the case that $\neg A{\models }_{𝐊}\neg \mathrm{◇}A$.

Why did we choose the subscript $𝐊$? Well, it turns out that there is an important relationship between system $𝐊$ and the definition of validity we have just given. In particular, we have the following two results:

1. 1.
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   If ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{⊢}_{𝐊}𝒞$, then ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{\models }_{𝐊}𝒞$

2. 2.
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   If ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{\models }_{𝐊}𝒞$, then ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{⊢}_{𝐊}𝒞$

The first result is known as a soundness result, since it tells us that the rules of $𝐊$ are good, sound rules: if you can vindicate an argument by giving a proof for it using system $𝐊$, then that argument really is valid. The second result is known as a completeness result, since it tells us that the rules of $𝐊$ are broad enough to capture all of the valid arguments: if an argument is valid, then it will be possible to offer a proof in $𝐊$ which vindicates it.

Now, it is one thing to state these results, quite another to prove them. However, we will not try to prove them here. But the idea behind the proof of soundness will perhaps make clearer how strict subproofs work.

In a strict subproof, we are not allowed to make use of any information from outside the strict subproof, except what we import into the strict subproof using $\mathrm{\square }$E. If we’ve assumed or proved $\mathrm{\square }𝒜$, by $\mathrm{\square }$E, we can used $𝒜$ inside a strict subproof. And in $𝐊$, that is the only way to import a formula into a strict subproof. So everything that can be proved inside a strict subproof must follow from formulas $𝒜$ where outside the strict subproof we have $\mathrm{\square }𝒜$. Let’s imagine that we are reasoning about what’s true in a possible world in some interpretation. If we know that $\mathrm{\square }𝒜$ is true in that possible world, we know that $𝒜$ is true in all accessible worlds. So, everything proved inside a strict subproof is true in all accessible possible worlds. That is why $\mathrm{\square }$I is a sound rule.

A few moments ago, we said that system $𝐊$ is sound and complete. Where does that leave the other modal systems we looked at, namely $𝐓$, $\mathrm{𝐒𝟒}$ and $\mathrm{𝐒𝟓}$? Well, they are all unsound, relative to the definition of validity we gave above. For example, all of these systems allow us to infer $A$ from $\mathrm{\square }A$, even though $\mathrm{\square }A{⊭}_{𝐊}A$.

Does that mean that these systems are a waste of time? Not at all! These systems are only unsound relative to the definition of validity we gave above. (Or to use symbols, they are unsound relative to ${\models }_{𝐊}$.) So when we are dealing with these stronger modal systems, we just need to modify our definition of validity to fit. This is where accessibility relations come in really handy.

When we introduced the idea of an accessibility relation, we said that it could be any relation between worlds that you like: you could have it relating every world to every world, no world to any world, or anything in between. That is how we were thinking of accessibility relations in our definition of ${\models }_{𝐊}$. But if we wanted, we could start putting some restrictions on the accessibility relation. In particular, we might insist that it has to be reflexive:

In English: every world accesses itself. Or in terms of relative possibility: every world is possible relative to itself. If we imposed this restriction, we could introduce a new consequence relation, ${\models }_{𝐓}$, as follows:

We have attached the $𝐓$ subscript to $\models$ because it turns out that system $𝐓$ is sound and complete relative to this new definition of validity:

1. 1.
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   If ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{⊢}_{𝐓}𝒞$, then ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{\models }_{𝐓}𝒞$

2. 2.
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   If ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{\models }_{𝐓}𝒞$, then ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{⊢}_{𝐓}𝒞$

As before, we will not try to prove these soundness and completeness results. However, it is relatively easy to see how insisting that the accessibility relation must be reflexive will vindicate the R$𝐓$ rule:

To see this, just imagine trying to cook up a counter-interpretation to this claim:

We would need to construct a world, $w$, at which $\mathrm{\square }𝒜$ was true, but $𝒜$ was false. Now, if $\mathrm{\square }𝒜$ is true at $w$, then $𝒜$ must be true at every world $w$ accesses. But since the accessibility relation is reflexive, $w$ accesses $w$. So $𝒜$ must be true at $w$. But now $𝒜$ must be true and false at $w$. Contradiction!

How else might we tweak our definition of validity? Well, we might also stipulate that the accessibility relation has to be transitive:

In English: if $w_1$ accesses $w_2$, and $w_2$ accesses $w_3$, then $w_1$ accesses $w_3$. Or in terms of relative possibility: if $w_3$ is possible relative to $w_2$, and $w_2$ is possible relative to $w_1$, then $w_3$ is possible relative to $w_1$. If we added this restriction on our accessibility relation, we could introduce a new consequence relation, ${\models }_{\mathrm{𝐒𝟒}}$, as follows:

We have attached the $\mathrm{𝐒𝟒}$ subscript to $\models$ because it turns out that system $\mathrm{𝐒𝟒}$ is sound and complete relative to this new definition of validity:

1. 1.
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   If ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{⊢}_{\mathrm{𝐒𝟒}}𝒞$, then ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{\models }_{\mathrm{𝐒𝟒}}𝒞$

2. 2.
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   If ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{\models }_{\mathrm{𝐒𝟒}}𝒞$, then ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{⊢}_{\mathrm{𝐒𝟒}}𝒞$

As before, we will not try to prove these soundness and completeness results. However, it is relatively easy to see how insisting that the accessibility relation must be transitive will vindicate the $\mathrm{𝐒𝟒}$ rule:

The idea behind strict subproofs, remember, is that they are ways to prove things that must be true in all accessible worlds. So the R$\mathrm{𝟒}$ rule means that whenever $\mathrm{\square }𝒜$ is true, $\mathrm{\square }𝒜$ must also be true in every accessible world. In other words, we must have $\mathrm{\square }𝒜{\models }_{\mathrm{𝐒𝟒}}\mathrm{\square }\mathrm{\square }𝒜$.

To see this, just imagine trying to cook up a counter-interpretation to this claim:

We would need to construct a world, $w_1$, at which $\mathrm{\square }𝒜$ was true, but $\mathrm{\square }\mathrm{\square }𝒜$ was false. Now, if $\mathrm{\square }\mathrm{\square }𝒜$ is false at $w_1$, then $w_1$ must access some world, $w_2$, at which $\mathrm{\square }𝒜$ is false. Equally, if $\mathrm{\square }𝒜$ is false at $w_2$, then $w_2$ must access some world, $w_3$, at which $𝒜$ is false. We just said that $w_1$ accesses $w_2$, and $w_2$ accesses $w_3$. So since we are now insisting that the accessibility relation be transitive, $w_1$ must access $w_3$. And as $\mathrm{\square }𝒜$ is true at $w_1$, and $w_3$ is accessible from $w_1$, it follows that $𝒜$ must be true at $w_3$. So $𝒜$ is true and false at $w_3$. Contradiction!

Let’s put one more restriction on the accessibility relation. This time, let’s insist that it must also be symmetric:

In English: if $w_1$ accesses $w_2$, then $w_2$ accesses $w_1$. Or in terms of relative possibility: if $w_2$ is possible relative to $w_1$, then $w_1$ is possible relative to $w_2$. Logicians call a relation that is reflexive, symmetric, and transitive an equivalence relation. We can now define a new consequence relation, ${\models }_{\mathrm{𝐒𝟓}}$, as follows:

We have attached the $\mathrm{𝐒𝟓}$ subscript to $\models$ because it turns out that system $\mathrm{𝐒𝟓}$ is sound and complete relative to this new definition of validity:

1. 1.
   ​

   If ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{⊢}_{\mathrm{𝐒𝟓}}𝒞$, then ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{\models }_{\mathrm{𝐒𝟓}}𝒞$

2. 2.
   ​

   If ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{\models }_{\mathrm{𝐒𝟓}}𝒞$, then ${𝒜}_1,{𝒜}_2,\mathrm{\dots }{𝒜}_n{⊢}_{\mathrm{𝐒𝟓}}𝒞$

As before, we will not try to prove these soundness and completeness results here. However, it is relatively easy to see how insisting that the accessibility relation must be an equivalence relation will vindicate the R$\mathrm{𝟓}$ rule:

The rule says that if $𝒜$ is not necessary, i.e., false in some accessible world, it is also not necessary in any accessible possible world, i.e., we have $\neg \mathrm{\square }𝒜{⊢}_{\mathrm{𝐒𝟓}}\mathrm{\square }\neg \mathrm{\square }𝒜$.

To see this, just imagine trying to cook up a counter-interpretation to this claim:

We would need to construct a world, $w_1$, at which $\neg \mathrm{\square }𝒜$ was true, but $\mathrm{\square }\neg \mathrm{\square }𝒜$ was false. Now, if $\neg \mathrm{\square }𝒜$ is true at $w_1$, then $w_1$ must access some world, $w_2$, at which $𝒜$ is false. Equally, if $\mathrm{\square }\neg \mathrm{\square }𝒜$ is false at $w_1$, then $w_1$ must access some world, $w_3$, at which $\neg \mathrm{\square }𝒜$ is false. Since we are now insisting that the accessibility relation is an equivalence relation, and hence symmetric, we can infer that $w_3$ accesses $w_1$. Thus, $w_3$ accesses $w_1$, and $w_1$ accesses $w_2$. Again, since we are now insisting that the accessibility relation is an equivalence relation, and hence transitive, we can infer that $w_3$ accesses $w_2$. But earlier we said that $\neg \mathrm{\square }𝒜$ is false at $w_3$, which implies that $𝒜$ is true at every world which $w_3$ accesses. So $𝒜$ is true and false at $w_2$. Contradiction!

In the definition of ${\models }_{\mathrm{𝐒𝟓}}$, we stipulated that the accessibility relation must be an equivalence relation. But it turns out that there is another way of getting a notion of validity fit for $\mathrm{𝐒𝟓}$. Rather than stipulating that the accessibility relation be an equivalence relation, we can instead stipulate that it be a universal relation:

In English: every world accesses every world. Or in terms of relative possibility: every world is possible relative to every world. Using this restriction on the accessibility relation, we could have defined ${\models }_{\mathrm{𝐒𝟓}}$ like this:

If we defined ${\models }_{\mathrm{𝐒𝟓}}$ like this, we would still get the same soundness and completeness results for $\mathrm{𝐒𝟓}$. What does this tell us? Well, it means that if we are dealing with a notion of necessity according to which every world is possible relative to every world, then we should use $\mathrm{𝐒𝟓}$. What is more, most philosophers assume that the notions of necessity that they are most concerned with, like logical necessity and metaphysical necessity, are of exactly this kind. So $\mathrm{𝐒𝟓}$ is the modal system that most philosophers use most of the time.

Modal logic is a large subfield of logic. We have only scratched the surface. If you want to learn more about modal logic, here are some textbooks you might consult.

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  George E. Hughes and Max J. Cresswell, A New Introduction to Modal Logic, Oxford: Routledge, 1996.

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  Graham Priest, An Introduction to Non-Classical Logic, 2nd ed., Cambridge: Cambridge University Press, 2008.

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  James W. Garson, Modal Logic for Philosophers, 2nd ed., Cambridge: Cambridge University Press, 2013.

None of these authors formulate their modal proof systems in quite the way we did, but the closest formulation is given by Garson.