Logic and Philosophy

Once the meaning theory is settled there will be no further, properly metaphysical, question to be determined. Michael Dummett

Choosing a non-classical logical basis for set theory does not amount to selecting a collection of validities which is smaller than, or in anyway different from, that of classical logic.

One should [instead] specify a logical consequence relation that is alternative to the classical one.i Francesco Paoli

Towards the end of a recent class on Cantor’s Diagonal Argument a student said ‘It seems like some cheating is going on – one simply introduces some non-standard formal machinery that will allow the result one wants. How is that not just as biased as relying on the law of bivalence to get a realist conclusion?’

More generally the student was wondering about the relationship between logic and philosophy. Obviously – this is not a trivial question. We begin by briefly recalling Michael Dummett’s contribution to the question, then consider the details of Graham Priest’s resuscitation of an Aristotle argument with respect to future contingents [where our interest will be in the justification in the introduction of a new logic to adjudicate the question at hand]. Finally – we will say a brief word about how Luis Estrada-Gonzalez stretches the relation under investigation under consideration.

I. Michael Dummett on Bivalence, Models and Realism

Dummett is known to have argued that the assumption of bivalence guarantees a realist analysis. But he was also an early critic of the bias of model-theoretic analyses and one who anticipated that Gentzen’s work might benefit

Philosophical analysis.ii

II. Graham Priest resuscitates Aristotleiii

Clearly then, it is not necessary that of every affirmation and opposite negation one should be true and the other false. de Interpretatione (19a 39-4)iv

Our primary interest is not on future contingents - nor even on Aristotle’s views on LEM. Rather – our focus is on how logic effects our appreciation of a venerable philosophical argument.

A certain logic made Aristotle’s argument look foolish – but then a more subtle more expressive logic resuscitated Aristotle’s game

Aristotle has a well-known objection to the law of excluded middle [LEM] based on an interpretation of future contingent statements. The standard response is that his argument is guilty of a modal fallacy. Graham Priest resorts to a more expressive modal system to fend off that objection. This provides a lovely example of how a logic that lacks sufficient expressive power can operate as a straightjacket on our philosophical investigations.v

We combine [relational] FDE relevant logic with the basic Kripke modal system K. We then consider combining

K modality with specifications of FDE – most notably the three-valued [strong Kleene] K3 which invalidates LEMvi

The modalised K3 system is called: KK3. A variant of this system provides a possibility for Aristotle to evade the standard objection to his future contingents argument.

I. Aristotlevii and Future Contingents 1. Aristotle’s argument from future contingents against LEM:

… if a thing is white now it was true before to say that it would be white, so that of anything that has taken place, it was always true to say