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Metaphysical Considerations on State Space Semantics

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Meaning and Proscription in Formal Logic

Part of the book series: Trends in Logic ((TREN,volume 49))

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Abstract

In this chapter, we review elements of Kit Fine’s project of truth maker semantics , in which models are constructed on spaces of states—fine-grained semantical devices that can stand in for many objects, such as facts, truthmakers, situations, and so forth. Fine’s framework has rapidly borne fruit, providing very natural semantics for many logics and providing elegant solutions to many thorny semantical problems.

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Notes

  1. 1.

    It would not be unreasonable to include the contemporary work of Richard Sylvan and Val Plumwood in [62] within this group. However, Sylvan and Plumwood’s interpretation of the novel semantical invention of [62]—the set-up—bears more of a likeness to possible worlds than their parts.

  2. 2.

    See [55, 65] for a discussion of how fine-grained an analysis is possible from semantics using possible worlds.

  3. 3.

    N.b. that this presupposes that A and B are distinct.

  4. 4.

    The appearance of notions of compatibility and incompatibility is especially interesting due to the role of these notions in the development of modal logic. In early presentations of Lewis’ systems of strict implication, entailment ‘’ is not primitive, but is defined in terms of the primitive, binary compatibility or co-consistency connective ‘\(\circ \).’ Lewis goes so far as to refer to the Survey System as the ‘Calculus of Consistencies’ in [45].

  5. 5.

    N.b. that the canonical model Fine gives in [35] is a term model with \(\sqsubseteq \) construed as set inclusion. Hence, valuations in the canonical model are complete and Semi-Regularity can be assumed without loss of generality. We sacrifice a modest amount of the flexibility of Fine’s models, but nothing upon which anything in the sequel turns.

  6. 6.

    Fine also defines a notion of content \(|A|\) as the set of exact verifiers without the constraint that \(|A|\) is complete. As we are considering complete valuations, however, completeness will be inherited by the content-sets of complex formulae and the definitions will coincide, that is, for any A, \(|A|=\lceil A\rceil \) in any model.

  7. 7.

    There is a parallel between how models are employed by Fine and Correia and how models are employed by van Fraassen and Suszko . van Fraassen was concerned with providing fine-grained accounts of entailment while Suszko was concerned with an appropriate account of identity. A similar divide occurs between Fine and Correia: Correia is concerned with the equivalence of two formulae in a model while Fine is concerned with representing consequence in a model.

  8. 8.

    Kit Fine has objected that Correia’s “Sam” example does not demand the existence of models in which atoms have neither verifiers nor falsifiers but only that there are models in which an atom lacks one or the other. Indeed, the “Sam” example does not require this. But in an intermediate state space semantics in which atoms must have at least a verifier or a falsifier, every formula \(A\vee \lnot A\) would have a verifier and a falsifier, in which case a stronger logic in which distribution holds for all instances of excluded middle would be generated.

  9. 9.

    While using “\(\vDash \)” both as a relation between formulae and as a relation between a model and a formula might be considered an abuse of notation, this is standard in model theory.

  10. 10.

    Recall that Bochvar’s internal nonsense logic \(\mathsf {\Sigma }_{0}\) was also introduced by Stephen Kleene in [44] and is thus frequently referred to as Kleene’s “weak” three-valued logic, whence the description of \(\mathsf {K}_{3}\) as “strong.”

  11. 11.

    Precluding inconsistent situations is analogous to removing the “both” value \(\mathfrak {b}\) from the semantics for \(\mathsf {E}_{\texttt {fde}}\) described in Definition 3.1.17. Removing this value leads to the many-valued semantics for \(\mathsf {K}_{3}\) in Definition 3.3.5.

  12. 12.

    It is worth noting that there exists an intriguing connection between Dunn’s semantics for \(\mathsf {RM}_{\texttt {fde}}\) and the more recent swap structure semantics introduced by Walter Carnielli and Marcelo Coniglio in [21]. In many cases, swap structures are isomorphic to finite collections of logical matrices, enabling Carnielli and Coniglio to show that a number of logics of formal inconsistency (cf. [22]) that are not characterizable by a finite matrix have characterizations by collections of such matrices.

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  1. Cycorp, Inc., Austin, TX, USA

    Thomas Macaulay Ferguson

  2. Saul Kripke Center, New York, NY, USA

    Thomas Macaulay Ferguson

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Ferguson, T.M. (2017). Metaphysical Considerations on State Space Semantics. In: Meaning and Proscription in Formal Logic. Trends in Logic, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-319-70821-8_3

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