On Philosophy

September 9, 2006

Possible Possibilities

Filed under: Logic — Peter @ 2:20 am

Often we use possible worlds when discussing logical systems. For example, a necessary truth is often understood as one that is true in all possible worlds (and hence can be known without any investigation into which world you are actually in). In the simplest possible terms a possible world an assignment of truth-values other than the one that occurs in the real world. The assignment is not arbitrary, however; possible worlds may be physically impossible, but they aren’t nonsensical. If q is the statement “pigs are fat and round” and r the statement “pigs are fat” there is no world that makes q true while making r false.

And because possible worlds aren’t “nonsensical” can do more than provide a foundation for discussing necessary and contingent truths in a formal manner; they can also help us shed some light on the notoriously problematical nature of implication. Typically we define the implication “A -> B” as “~A or B”. Such a definition, however, leads to some very unintuitive conclusions. For example, the statement “(A -> B) or (~A -> B)” is a logical truth for every choice of A and B. Even more unintuitively the statement “(A -> B) or (B -> C)” is also a logical truth for every A, B, and C, even when they are totally unrelated.

We can resolve some of the difficulties associated with the implication by defining it in terms of possible worlds instead of logical connectives. Specifically we say that A -> B is true if and only if in every possible world where A is true B is also true. This definition of implication is called strict implication, and can be defined formally through modal logic. Strict implication works well in conjunction with modal or classical logic (although it makes certain constructions more difficult), but when we attempt to use it as implication within a system of paraconsistent logic we run into problems.

It is true that strict implication does resolve the problem of “explosion” (that from a paradox all statements can be deduced), because if some “paradoxes” can indeed be true then statements such as “(A & ~A) -> B” may be false because there are some possible worlds where “A & ~A” is true and, if neither A nor ~A implies B, B is false. However, under strict implication the statement “(A -> (A -> B)) -> (A -> B)”, called absorption is still logically true, and this can be a problem for some systems of paraconsistent logic. One of the reasons to work with a paraconsistent logic system is that it allows (or at least has the possibility to allow) ideas such as the truth of statements to be captured within the system, and to accommodate self-reference, without relying on additional meta-systems. Thus we allow statements such as S: “T(S) -> Q”. This statement should be read as “if this statement is true then Q”. We also know, from the definition of the truth predicate (Tarski) that T(A) A. Thus T(S) (T(S) -> Q), and thus both (a.) T(S) -> (T(S) -> Q) and (b.) (T(S) -> Q) -> T(S). By absorption and (a.) it follows that (c.) T(S) -> Q. By (c.) and (b.) we can conclude that (d.) T(S). Together (c.), (d.), and modus ponens allow us to conclude that Q. And since Q was arbitrary it seems that strict implication still allows any sentence to be deduced, which is clearly undesirable.

There are several possibilities at this juncture. We could give up on self-reference or the ability to express the truth and falsehood of statements within the same language those statements are in, but since this was one of the motivations to develop a paraconsistent system of logic let us set that possibility aside for now. The only other viable possibility is to deny that absorption is a valid logical principle, and the only way to do that is to re-define how implication works. One promising redefinition is to introduce the idea of second order possible worlds (or the notion of accessibility of possible worlds, as it is called in relevance logic systems). The idea is that various possible worlds themselves have different sets of possible worlds that their terms are evaluated in light of. For example our “starting world” has some set of possible worlds, W, which we “quantify over” when addressing questions of necessity and implication. However, under this interpretation of possible worlds, each of the possible worlds in W has their own set of possible worlds, which are “quantified over” when determining implication and necessity in those worlds, and this set does not have to coincide with W. Thus, when analyzing a complicated statement such as “A->(B->A)”, we must first determine the truth of “B->A” separately for each of the possible worlds in question before we can determine the truth of the whole statement. Now, let’s apply this idea to the principle of absorption.

     A    B    Worlds    A->B    A->(A->B)   (A->(A->B))->(A->B)
Q    T    T    {Q, R}    T       T            F
R    F    F    {S}       F       T            T
S    T    F    {Q, R}    T       T            F

As you can see, if our “starting world” was Q then the principle of absorption would be deemed false, and if it can be deemed false in some possible worlds then it isn’t a necessary truth, and hence not a valid logical principle.

Of course this whole approach, with each world having its own set of possible worlds, and the possibility that a given world is not counted among its own possible worlds, may seem arbitrary. And if modus ponens is to hold we must add the additional requirement that a “starting world” must contain itself among its possible worlds, a requirement that may seem even more arbitrary. This raises the possibility that the structure we are describing here is not one of possible worlds, and that it really corresponds to something else. This is certainly an intriguing possibility, but unfortunately at the moment I have no idea as to what else it could be.

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