Abstract
In standard modal logic, □ ≡ ∼◊ ∼ and ◊ ≡ ∼□∼. I will, first, examine why in tense-logic, Arthur Prior thinks that ∼ ◊ ∼ is weaker than □ and ∼ □ ∼ is weaker than ◊. I will, then, examine whether there are similar motivations in modal logic to take ∼ ◊ ∼ to be weaker than □ and ∼ □ ∼ to be weaker than ◊. The upshot will be that, just as certain metaphysical views within the philosophy of time (e.g., Presentism) motivate one to deny the standard tense equivalences, certain metaphysical views within the metaphysics of modality (e.g., Contingentism, nonmodalism) motivate one to deny the standard modal equivalences.
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Notes
“What makes it the case that vixens are female foxes is whatever makes it the case that female foxes are female foxes” (Williamson 2007, 59). “To count as analytic, a sentence must first be true—true on its own steam, so to speak. To say that a sentence is analytic is to say that it is a certain kind of truth; it is not to explain why it is true” (Sider 193).
What are the non-trivial modal properties? The trivial possibilities are the actualities. If x is F, then it is trivial that x is ◊F; if x is not F, then it is non-trivial that x is ◊F. The trivial necessities are the necessities had by all objects. The non-trivial necessities are the necessities which are had by some, but not all, objects. F is a non-trivial modal property of x if x is a non-trivial possibility of x or a non-trivial necessity of x.
□ ≡ ∼◊∼
□ → ∼◊ ∼ & ∼ ◊ ∼ → □ by biconditional elimination
∼ ∼ ◊ ∼ → ∼ □ & ∼ □ → ∼ ∼ ◊ ∼ by contrapositive
◊ ∼ → ∼ □ & ∼ □ → ◊ ∼ by negation elimination
∼□ ≡ ◊ ∼ by biconditional introduction
Given the standard definition of Realism, viz. Realism is true of x if (i) x exists, and (ii) x is constitutively independent of humans, I, thus, qualify as a Realist about physical objects.
Prior’s motivation for taking ∼ □ (not at all times) to be weaker than ◊ ∼ (at some time not) is that he thought non-existent objects could satisfy the former without satisfying the later (Prior 1957, 37). According to Prior, Queen Victoria is not now female because she does not now exist (and, hence, does not now have any properties), but it is not the case that Queen Victoria is possibly not female because, for this to be true, there would have to be some time at which she had the property being not female and there is not any time at which she has this property. Intuitively ∼ □ does seem to be weaker than ◊∼. An object can satisfy the former simply by not being anything, whereas to satisfy the later, it has to (actively) be not something. So, there are two ways Queen Victoria can fail to be female: (i) when she has properties and being female is not amongst them, or (ii) when she has no properties at all. Although Queen Victoria never satisfies (i)—there’s never a time at which she’s not female; she does satisfy (ii) she’s not at all times female.
See, also, Carnap’s Principle of Tolerance, Carnap (2010).
Although most metaphysicians are deeply conservative with regard to logic, there are a few notable exceptions. In The Structure of Objects, Kathryn Koslicki (2008) argues that, although classical extensional mereology is well and fine as a system of mereology, it is a failure in so far as getting at the correct metaphysics of ordinary objects is concerned. In “Personal Identity,” Derek Parfit (1971) argues that, although identity is well and fine as a philosophical relation, it is a failure in so far as getting at the correct persistence conditions of people is concerned.
References
Carnap, R. (2010). The logical syntax of language. New York: Routledge.
Koslicki, K. (2008). The structure of objects. Oxford: OUP.
Parfit, D. (1971). Personal identity. Philosophical Review, 80(1), 3–27.
Prior, A. N. (1957). Time and modality. Oxford: Clarenden Press.
Prior, A. N. (1968). Worlds, times, and selves. London: Duckworth.
Sider, T. (2011). Writing the book of the world. Oxford: Oxford University Press.
Williamson, T. (2007). The philosophy of philosophy. Oxford: Blackwell Publishing.
Acknowledgments
A debt of gratitude is owed to the Logic Group at the University of Melbourne, which read Prior’s Time and Modality with me, especially to Lloyd Humberstone, who first introduced me to Prior and who has continually elucidated him to me; and to an anonymous referee at Acta Analytica.
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Goswick, D. Why Being Necessary Really Is Not the Same As Being Not Possibly Not. Acta Anal 30, 267–274 (2015). https://doi.org/10.1007/s12136-014-0244-6
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DOI: https://doi.org/10.1007/s12136-014-0244-6