Here is my latest draft of a paper attempting to give an account of a stronger-than-material conditional which can be adapted to various sentential logics. An abstract is provided below. To see the draft, go here:
Some Strong Conditionals for Sentential Logics (Circulation Draft)
Please keep in mind that the paper is a work in progress, and is still in a fairly rough state. That being said, I appreciate any comments and / or criticism that those who are interested in the subject have to offer.
In this article I define a strong conditional for classical sentential logic, and then extend it to three non-classical sentential logics. It is stronger than the material conditional and is not subject to the standard paradoxes of material implication, nor is it subject to some of the standard paradoxes of C. I. Lewis’s strict implication. My conditional has some counterintuitive consequences of its own, but I think its pros outweigh its cons. In any case, one can always augment one’s language with more than one conditional, and it may be that no single conditional will satisfy all of our intuitions about how a conditional should behave. Finally, I will make no claim that the strong conditional is a good model for any particular use of the indicative conditional in English or other natural languages, though it would certainly be a nice bonus if some modified version of it could serve as one.
The basic idea is this: In general, one starts out with with the logic that one wants to define the conditional for, and describes a meta-language for it. The metalanguage contains ┌(q|p)┐, a “conditional designator” which designates the truth value that q takes given that p is true, i.e., given that p has the value 1. It is to be read as ┌ the value of q given p┐ or ┌the value of q conditional on p┐. The stroke, |, is not a connective; it merely serves to separate the letter q from the letter p. The designator works like this: If p never takes the value 1, then ┌(q|p)┐ designates nothing—for q cannot take a value given that p is true if p can never be true—and is said to be empty. It is also empty if the value of q varies when the value of p is 1, for in that case q doesn’t take a unique value given that p is true. If q always takes the value 1 when p takes the value 1, then ┌(q|p)┐ designates 1, and in our meta-language we can say that ┌(q|p)┐ = 1, which is another way of saying that ┌(q|p)┐ designates 1. Similarly, if q always takes the value 0 when p takes the value 1, then ┌(q|p)┐ designates 0, and in our meta-language we can say that ┌(q|p)┐ = 0.
With our meta-linguistic conditional designator ready to hand, one can now define what I call the strong conditional, or strong implication, for which I will use the symbol ‘→’. Its definition is (where ‘v( )’ is the valuation function, which gives the semantic value of an expression):
If ┌(q|p)┐= 1, then v(p → q) = 1
If ┌(q|p)┐= 0, then v(p → q) = 0
If ┌(q|p)┐ is empty, then v(p → q) = 0
I shall begin by exploring some of the disadvantages of the material conditional, the strict conditional, and some relevant conditionals. I proceed to define a strong conditional for classical sentential logic. I go on to adapt this account to Graham Priest’s Logic of Paradox, to S. C. Kleene’s logic K3, and then to J. Łukasiewicz’s logic Ł, a standard version of fuzzy logic.