Abstract
In recent years, there has been a growing interest in truthmaker semantics as a framework for understanding a range of phenomena in philosophy and linguistics. Despite this interest, there has been limited study of the various logics that arise from the semantics. This paper aims to address this gap by exploring numerous ‘truthmaker logics’ and proving their compactness and decidability. This is in continuation with the inquiry of Fine and Jago (2019), who proved compactness and decidability for a particular kind of truthmaker logic. The key results going into this are (1) ‘standard translations’ into first-order logic; (2) a truthmaker analogue of the finite model property; and (3) a proof showing that truthmaker consequence on semilattices coincides with truthmaker consequence on complete lattices. Finally, the connection with modal logic is examined. Specifically, it is illustrated how endowing truthmaker semantics with classical negation results in modal information logics.
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Notes
‘Fusion’ is synonymous with ‘supremum’ or ‘join’.
Among more, on Fine’s account, for propositions P and Q (of a certain form): (i) P weakly grounds Q iff P truthmaker entails Q; (ii) P weakly partially grounds Q iff \((P \wedge Q)\vee Q\) is truthmaker equivalent to Q; and (iii) P contains Q iff \(P\wedge Q\) is truthmaker equivalent to P.
That is, \(\le \) is reflexive, transitive, anti-symmetric and for all \(s, s' \in S\), there is some \(s'' \in S\) s.t. \(s'' = \sup \{s, s'\}\). I.e., our semilattices are join-semilattices.
Another option would be semantics where the disjunction is defined in terms of infimum—mirroring how conjunction is defined in terms of supremum. Under this definition, it might be natural to require the frame to not only be a join-semilattice but also a meet-semilattice, or in other words: a lattice.
Special thanks to a reviewer for suggesting the inclusion of ‘convex truthmaking’. Convex, inclusive semantics with non-vacuous valuations closed under binary joins – i.e., so-called ‘replete’ semantics – is discussed in, among more, Fine and Jago (2019); Korbmacher (2022) and, on complete lattices, has been shown to be complete for Angell’s logic for Analytic Containment (see (Angell, 1977, 1989)) in Fine (2016).
Caution: Our formalization of ‘convex truthmaking’, as expressed by ‘\(\Vdash ^{+,c}\)’, differs from the definition of ‘\(\Vdash _{\textit{cvx}}\)’ found in Fine and Jago (2019). There are different methods for enforcing convexity while yielding the same consequence relation. We have opted for \(\Vdash ^{\pm ,c}\), as it makes for a clearer presentation of our results.
Note that decidability of the distributive entailment has as a special case decidability of the collective entailment, which is where \(\Gamma \Vdash ^+\varphi \) holds iff whenever \({\mathbb {M}}, s\Vdash ^+\bigwedge _{\gamma \in \Gamma }\gamma \), \({\mathbb {M}}, s\Vdash ^+\varphi \) (this is, of course, only defined for finite \(\Gamma \) [which suffices for decidability], but the definition can be extended to the infinite case).
It is worth noting that the proofs of these two theorems given in Fine and Jago (2019) somewhat generalize to other truthmaker logics and are of independent interest.
If one also deals with infima and, e.g., requires the underlying frames to be lattices, one shall assume \((S_1, \le _1)\) to be a sublattice of \((S_0,\le _0)\).
The intuition for \(T(\gamma , s)\) (resp. \(F(\gamma , s)\)) is that it is a set of states by virtue of which \(s \Vdash ^+ \gamma \) (resp. \(s \Vdash ^- \gamma \)).
For other semantics (e.g. inclusive), we modify this definition in the obvious way.
Notice that this proposition implies the analogous proposition stated in terms of falsitymaking, qua negating all formulas.
In case we require, e.g., all \(V^+(p)\ne \varnothing \), the proof goes through by simply adding a state \(s_{p^+}\in V^+_0(p)\) to the set of generators for all propositional letters occurring in the formulas \(\Gamma _F\cup \{\varphi \}\).
A poset \((S,\le )\) has a bottom element :iff there is some \(s\in S\) s.t. \(s\le t\) for all \(t\in S\). So \(C_2\) is the class of complete lattices, or alternatively, it is the restriction of \(C_1\) to all and only the members with a bottom element.
Once more, while our notation is \(\Vdash _X^{+}\), it will become apparent that this theorem likewise applies to the convex consequence relations \(\Vdash _X^{+, c}\).
This lemma corresponds to a weaker version of Lemma 3.5 in Fine &Jago (2019), which describes all such conjunctions of literals corresponding to a formula \(\varphi \).
From a mathematical perspective, we can also think of these as corresponding to whether \(p^T\) [resp. \(p^F\)] was even [odd] in an enumeration of the propositional variables of \({\mathcal {L}}_M\).
For inclusive semantics, the translation modifies canonically.
‘Complementary’ as in the logics being defined on the same class of structures with the same admissible valuations. For instance, the modal information logic on semilattices is ‘complementary’ to the truthmaker logic first specified in Sect. 2, as well as to the truthmaker logic that uses inclusive semantics instead.
For truthmaker logics with inclusive semantics, we restrict the fragment of \({\mathcal {L}}_M\) so that ‘\(\vee \)’ only occurs as ‘\(\varphi \vee \psi \vee \langle \sup \rangle \varphi \psi \)’ and modify the translation accordingly.
Symmetric results for falsitymaking are achieved by a symmetric translation.
References
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Acknowledgements
This paper builds upon Appendix B from my Master’s thesis (Knudstorp (2022)), overseen by Johan van Benthem and Nick Bezhanishvili. I am grateful to both for their support, extending from my thesis to the preparation of this manuscript. I would also like to thank Tibo Rushbrooke for proofreading a draft of this manuscript, and a referee for many helpful comments. This work received support from Nothing is Logical (NihiL), an NWO OC project (406.21.CTW.023).
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This work received support from Nothing is Logical (NihiL), an NWO OC project (406.21.CTW.023).
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Knudstorp, S.B. Logics of truthmaker semantics: comparison, compactness and decidability. Synthese 202, 206 (2023). https://doi.org/10.1007/s11229-023-04401-1
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DOI: https://doi.org/10.1007/s11229-023-04401-1