Abstract
In this paper, I set out a semantics for identity in relevant logic that is based on an analogy between the biconditional and identity. This analogy supports the semantics that Priest has set out for identity in basic relevant logic and it motivates a version of the Routley–Meyer semantics in which identities can be viewed as constraints on the ternary relation that is used to treat implication.
This chapter is dedicated to Graham Priest. I first met Graham in the late 1980s, when I was a postdoc at the Australian National University. I owe Graham a substantial personal debt. He has always been supportive of my work and my career, and a friend. But I would like to say here that every philosophical logician owes Graham a sizeable debt. When I got into philosophical logic in the 1980s, the situation was not good. Logicians were leaving philosophy and going to computer science. One reason was that there were more jobs for computer scientists, but that wasn’t the only reason. There were more exciting things to do in computer science. Graham, more than anyone else, changed that. With the publication of In Contradiction in 1987, a clear, sustained, and coherent case for paraconsistent logic came into being. That book also gave us a new paradigm of how to weave logic, epistemology and metaphysics together to the benefit of all three disciplines. David Lewis had published Plurality of Worlds a year earlier, but his logical views were much more conservative. Plurality argued that classical first-order logic is all we really need to talk about modality and other supposedly intentional phenomena. For those of us working in non-classical logics, Graham’s work was an exemplar of how to motivate and apply our logical systems. The clarity of his work and the strength of his arguments (not just in that book) have made dialetheism a going concern in philosophy, and they have also brought the problem of alternative logics into the centre of contemporary philosophical debate. Now philosophical logic is a very exciting field and it is no longer isolated from the rest of philosophy. It is now really fun to be a philosophical logician. And that is due in no small part to Graham.
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Notes
- 1.
I discuss the differences between worlds and situations in Sect. 16.5.
- 2.
I have heard predicate logic called ‘one-half order logic’, but I don’t think that name really connotes anything.
- 3.
Both of these logics have the depth relevance property. The depth of a subformula in a formula is the number of times it is within the scope of nested implications. If MidT were derivable, then we would be able to derive from the axiom \(((p\rightarrow p)\rightarrow (p\rightarrow p))\rightarrow ((p\rightarrow p)\rightarrow (p\rightarrow p))\) and the axiom \(p\rightarrow p\) to \(((p\rightarrow p)\rightarrow (p\rightarrow p))\rightarrow (p\rightarrow p)\), which would violate depth relevance.
- 4.
- 5.
Having non-normal worlds of this sort is useful in representing various sorts of misinformation.
- 6.
This argument is only meant to show why we need to reject FullSub if we reject pseudo modus ponens. It is not meant to show that pseudo modus ponens is not intuitive from the perspective of an informational interpretation of the semantics. The latter issue is not easily settled. But given the view that the R relation is supposed here to represent application, it is at least plausible that on some understanding of application, not every situation is closed under the application of itself to itself. A complete world is so closed, but it is not clear that every informational part of the world need be closed under self-application.
- 7.
In Mares (1992) I gave an argument due to Belnap that we should reject FullSub for R. I no longer find this argument very persuasive.
- 8.
- 9.
Although Robert Goldblatt does add it to Fine’s semantics for quantified relevant logic for technical reasons (Goldblatt 2011, Chap. 6).
References
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Mares, E. (2019). From Iff to Is: Some New Thoughts on Identity in Relevant Logics. In: Başkent, C., Ferguson, T. (eds) Graham Priest on Dialetheism and Paraconsistency. Outstanding Contributions to Logic, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-25365-3_16
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