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Finitary Upper Logicism
- Bruno Jacinto
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- Journal:
- The Review of Symbolic Logic / Accepted manuscript
- Published online by Cambridge University Press:
- 31 May 2024, pp. 1-68
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Necessitism, Contingentism, and Theory Equivalence
- Part of
- Bruno Jacinto
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- Journal:
- Bulletin of Symbolic Logic / Volume 27 / Issue 2 / June 2021
- Published online by Cambridge University Press:
- 16 September 2021, pp. 217-218
- Print publication:
- June 2021
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- Article
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Necessitism, Contingentism, and Theory Equivalence is a dissertation on issues in higher-order modal metaphysics. Consider a modal higher-order language with identity in which the universal quantifier is interpreted as expressing (unrestricted) universal quantification and the necessity operator is interpreted as expressing metaphysical necessity. The main question addressed in the dissertation concerns the correct theory formulated in this language. A different question that also takes centre stage in the dissertation is what it takes for theories to be equivalent.
The whole dissertation consists of an extended argument in defence of the (joint) truth of two seemingly inconsistent higher-order modal theories, specifically:
1. Plantingan Moderate Contingentism, a theory based on Plantinga’s [1] modal metaphysics that is committed to, among other things, the contingent being of some individuals and the necessary being of all possible higher-order entities;
2. Williamsonian Thorough Necessitism, a theory advocated by Williamson [3] which is committed to, among other things, the necessary being of every possible individual as well as of every possible higher-order entity.
Part of the case for these theories’ joint truth relies on defences of the following metaphysical theses: (i) Thorough Serious Actualism, the thesis that no things could have been related while being nothing, and (ii) Higher-Order Necessitism, the thesis that necessarily, every higher-order entity is necessarily something. It is shown that Thorough Serious Actualism and Higher-Order Necessitism are both implicit commitments of very weak logical theories. The defence of Higher-Order Necessitism constitutes a powerful challenge to Stalnaker’s [2] Thorough Contingentism, a theory committed to, among other things, the view that there could have been some individuals as well as some entities of any higher-order that could have been nothing.
In the dissertation it is argued that Plantingan Moderate Contingentism and Williamsonian Thorough Necessitism are in fact equivalent, even if they appear to be jointly inconsistent. The case for this claim relies on the Synonymy account, a novel account of theory equivalence developed and defended in the dissertation. According to this account, theories are equivalent just in case they have the same commitments and conception of logical space.
By way of defending the Synonymy account’s adequacy, the account is applied to the debate between noneists, proponents of the view that some things do not exist, and Quineans, proponents of the view that to exist just is to be some thing. The Synonymy account is shown to afford a more nuanced and better understanding of that debate by revealing that what noneists and Quineans are really disagreeing about is what expressive resources are available to appropriately describe the world.
By coupling a metatheoretical result with tools from the philosophy of language, it is argued that Plantingan Moderate Contingentism and Williamsonian Thorough Necessitism are synonymous theories, and so, by the lights of the Synonymy account, equivalent. Given the defence of their extant commitments made in the dissertation, it is concluded that Plantingan Moderate Contingentism and Williamsonian Thorough Necessitism are both correct. A corollary of this result is that the dispute between Plantingans and Williamsonians is, in an important sense, merely verbal. For if two theories are equivalent, then they “require the same of the world for their truth.”
Thus, the results of the dissertation reveal that if one speaks as a Plantingan while advocating Plantingan Moderate Contingentism, or as a Williamsonian while advocating Williamsonian Thorough Necessitism, then one will not go wrong. Notwithstanding, one will still go wrong if one speaks as a Plantingan while advocating Williamsonian Thorough Necessitism, or as a Williamsonian while advocating Plantingan Moderate Contingentism.
On the basis of a conception of the individual constants and predicates of second-order modal languages as strongly Millian, i.e., as having actually existing entities as their semantic values, in the appendix are presented second-order modal logics consistent with Stalnaker’s Thorough Contingentism. Furthermore, it is shown there that these logics are strong enough for applications of higher-order modal logic in mathematics, a result that constitutes a reply to an argument to the contrary by Williamson [3]. Finally, these logics are proven to be complete relative to particular “thoroughly contingentist” classes of models.
STRONGLY MILLIAN SECOND-ORDER MODAL LOGICS
- BRUNO JACINTO
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- Journal:
- The Review of Symbolic Logic / Volume 10 / Issue 3 / September 2017
- Published online by Cambridge University Press:
- 19 June 2017, pp. 397-454
- Print publication:
- September 2017
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The most common first- and second-order modal logics either have as theorems every instance of the Barcan and Converse Barcan formulae and of their second-order analogues, or else fail to capture the actual truth of every theorem of classical first- and second-order logic. In this paper we characterise and motivate sound and complete first- and second-order modal logics that successfully capture the actual truth of every theorem of classical first- and second-order logic and yet do not possess controversial instances of the Barcan and Converse Barcan formulae as theorems, nor of their second-order analogues. What makes possible these results is an understanding of the individual constants and predicates of the target languages as strongly Millian expressions, where a strongly Millian expression is one that has an actually existing entity as its semantic value. For this reason these logics are called ‘strongly Millian’. It is shown that the strength of the strongly Millian second-order modal logics here characterised afford the means to resist an argument by Timothy Williamson for the truth of the claim that necessarily, every property necessarily exists.
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