Notes on an Overview of Modal Logics and Possible Worlds
March 26, 2015
The following are notes prepared for a presentation for Peter Wolfendale’s class “Reintroduction to Metaphysics”:
I’m going to attempt to provide a short overview of Modal logics, using the Possible Worlds Semantics frameworks to illustrate some of the points. I owe the majority of this discussion to Rod Girle’s book “Possible Worlds”, which is great introduction to the topic and many of the issues surrounding it. Surely any mistakes that follow are of my own devising.
Modal logic refers to a broad family of qualificatory logics, but most strictly refers to Alethic modality, which qualifies the truth of judgement statements with the operator “necessary”. Modal logic is achieved by adding qualifying operators such as “necessary”, most often represented by a box, to propositional logic.
Modal logics were developed to deal with ambiguities in classical logics, particularly as concerns the material implications of the procedure, “if… then”. For this brief discussion it will only be necessary to know that in basic logic, implication is combined with symbols for negation, or “not”, “or”, and “and”. So, for example the statement:
“If A then B.” This may be negated to arrive at “Either not A and/or B”, which itself can be negated to arrive at “Neither Not A nor B”, which is itself equivalent to “A and not B”. So, “If A then B” is equivalent to “A and not B”, but this can be problematic if someone claims in ordinary language [the counterfactual claim that] “It’s false that if the US had stayed in Iraq, then ISIS would not be taking over Syria”, which according to this logic would be equivalent to “The US stayed in Iraq and ISIS took over Syria”, which clearly did not happen. This example is not exhaustive of the problems with implication in classical logic, but does illustrate how possibility becomes a concern which is picked up by modal logics. [Note: This example may cause difficulty due to its political controversy. For some clarity, when I say, “the US left Iraq”, I am referring to the draw-down between 2007-2011 that was completed under the Obama administration. Anyway, a lesson in using controversial current event examples. This statement could be replaced with one less controversial than this, such as one Pete Wolfendale mentioned, and retain the same sense: “It’s false that if the Boy had not thrown the ball, then my window would not be broken”, which would be equivalent to “The Boy had not thrown the ball, and my window was broken.” ]
The most basic system of modal logic is K, named in honor of Saul Kripke. K is a weak system of Alethic modal logic, which is not to say that it is not powerful in its own right – weak in this instance just means it lacks additional axiomatics which could more determinately qualify the logics of the system, allowing additional statements to be proven. The symbol system of K, as mentioned, is similar to basic propositional logic, in that it includes the symbols “~” for “not”, “→” for the implication “if…then”, “&” for “and”, and “v” for or, and “□” for necessary, where the box operates somewhat similarly to the quantifiers “∀” “for all” and “∃” for “some”, under propositional logic, but means “necessary” in modal logic. I’ll try and briefly return to quantifiers later. K adds two principles to propositional logic:
The Necessitation Principle – If A is a theorem in K, then A is necessary.
The Distribution Principle – If it is necessary that (if A then B), then (if it is necessary that A then it is necessary that B).
Further operators are derivable from “necessary”, such as “possible” and “contingent”. So for example, given the proposition P:
- P is possible is definable in terms of: it is not necessary that not P. [represented by “◊”]
- P is impossible is definable in terms of: it is necessary that not P.
- P is contingent is definable in terms of: it is not necessary that not P and is not necessary that P; or put another way P is contingent if it is possible that P and not-necessary that P.
As was mentioned, K is a relatively weak system, so, while it would be desirable that from K we could say If it is necessary that theory A then A is true in the system, this is not possible without adding this as an axiom to the system, which produces a self-reflexivity. When this axiom is added to K the resulting system is often referred to as T. What this means will hopefully become clearer later, when we look at CI Lewis’ modal logics next.
CI Lewis developed a number of the initial systems that strengthened modal logics in various ways. He produced the systems which are known simply as S1, S2, S3, S4, and S5. I’ll not go into all of these systems in detail, but will look at S4 and S5 as a matter of comparison to show how the imposition of additional axioms revise the ways in which these systems have distinct access to possibility.
First, a word about Possible Worlds. [As Pete mentioned, the term derives from Leibniz] Possible Worlds are a tool with which we can discuss modal logic. They are not necessarily a logic themselves, and are often construed in terms of Possible Worlds Semantics — this is because while logics themselves are a useful tool to make explicit the implications of ordinary language usage, ordinary language semantics often remain the standard upon which we judge the applicability and flexibility of a logic. As you can see from the statement above regarding the ambiguities of material implication and the phrase on the US and ISIS, translating from ordinary semantics to logic frequently results in puzzling and seemingly wrong conclusions. While this may be mitigated by carefully assessing how statements are translated, it often is the case that, due to the restricted and artificial nature of logic systems, they simply are not adequate to capturing all of the richness of ordinary semantics. Possible Worlds allow us to picture the access relations between modal statements. This is not to say that Possible Worlds are real, though there are philosophers, such as David Lewis, who do make this claim. Thinking about Possible Worlds places true statements relative to the world in which they are stated, so if I say, “Its possible that this presentation is informative”, that would be true for the world in which it is stated, if there is some world where this statement is true — so, to put this a bit more formally:
“A statement is possibly true in some world, say W, just in case it is true in at least one possible world, say U.”
and for necessity:
“A statement is necessarily true in some world, say W, just in case it is true in every possible world.”
But, to return to our examination of S4 and S5, let us think about them in terms of possible worlds. For Alethic Modal logic, necessarily true and possibly true qualify the relationship of access with which the logic has access to the worlds. By modifying the qualifying axioms of the logic, we can change the ways in which the logic has access to the worlds, thus altering the structure of the logic. Modal logic may be construed beyond the Alethic form by replacing the “necessary”, or box operator with other operators. So, for instance:
- Epistemic, or “it is known that”.
- Doxastic (Belief), or “it is believed that”.
- Temporal, or “it has been the case that”, for a modal logic of pasts, or “it is going to be the case that” for a modal logic of the futures.
This list is not exhaustive, and each of these logics may take a variety of forms depending upon how they are constructed and qualify access with their given axiomatics. But, consider these possible worlds: j, k, and n . We can say, somewhat arbitrarily, that each world has access to the other in sequence like this:
So that j has access to k, and k has access to n. Given this set up, this would describe the conditions similar to the logic known as K. If we move to the logic T, then this becomes modified as follows:
So, not only does each world access the other, but also has reflexive access to itself. Now if we look at S4, S4 is not only reflexive but also has transitive access to the the worlds, such that, since j has access to k, which has access to n, j also has access to n:
In S5, access is not only reflexive and transitive, but also symmetrical, like so:
So that each of the worlds has access to one another and itself, so accessibility is universal. Contrasting this with S4, we can see that S4 is asymmetrical, which means that the relationship is only one way — so most modal logics dealing with temporality are themselves variations upon S4.
To understand a little further how this relationship functions in modal predicate logics, I’ll return to the much simpler logic of T. Consider the statement “It is possible for some x to be P”, this means that, in at least one possible world, something is P, and the statement “It is necessary for x to be P’, which translates as, in at least every possible world there is something that is P. The third statement that will illustrate our point is “It is possible that every x is P”, which means Its possible that everything is P. With the abbreviated sequence of worlds, n and k, we can see how these two statements function in relation to accessibility:
So, while all of the above statements are true for both worlds, in world k, it is not possible for some x to be non P, since world k does not have access to world n. So it is not possible for anything to not be P in k. There is a lot more that can be said about these relationships, and further scenarios would give further clarity, but for the sake of time, I hope this is somewhat illustrative regarding how distinctions between access affect the relationships between worlds.
Finally, I want to turn to the quantifier issue I’ve been stressing all along. We have been using the quantifiers “some” and “every” to denote how necessary and possible items range across the worlds. Quantifiers, in ordinary language often imply an existential import, but it is a question as to whether or not the objects of possible worlds actually exist. David Lewis, who is a modal realist about Possible World Semantics believes that existence is basic, and thus accepts a metaphysically realist interpretation of the possible worlds. His four theses are:
- Possible worlds exist.
- Other possible worlds are things of the same sort as the actual world – “I and all my surroundings”.
- The indexical analysis of the adjective “actual” is the correct analysis.
- Possible worlds cannot be reduced to something more basic.
There are two ways of tackling this. One is by simply by denying statement two, that other possible worlds are of the same sort as the actual world. This is still a form of modal realism, but one that makes a strong distinction between the actual world and possible worlds. On Lewis’s account, the only thing distinguishing the actual from the other possible worlds, is the indexical, and since indexicality is available from any world in which the account is presented from, it seems to lack any sufficient distinctions aside from perspective — so one would need a stronger definition of actual. The other way to attack modal realism is through following a reductionist account, or what Lewis perjoratively calls ersatz accounts. This involves the development of either epistemic and/or doxastic logics to overcome the inflation of existing entities, and the reduction of those entities through belief revision. Lewis would of course reject these positions, but we need not accept the existential import as basic as Lewis does, and it is unclear whether or not accepting the existential basis is any more virtuous, given that it leads to the inflation of metaphysical entities.
1. Girle, Rod (2014-12-18). Possible Worlds. Taylor and Francis. Kindle Edition.
2. All of the above diagrams derive directly from Girle’s book, and I owe much of the discussion to him, however I have selected, reworked, and truncated it for brevity. Credit where credit is due. Again, mistakes are my own.
3. Lewis, On the Plurality of Worlds, 227.