Abstract
I follow Jago by extending the state space with disjunctive states and consider what might then justify the existing or alternative semantic clauses for disjunction and conjunction.
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Notes
- 1.
Such a structure is commonly known as a complete bi-lattice.
- 2.
Distribution in this sense should not be confused with the principle in which \( \bigvee \) is replaced by the operation ⊓ for the greatest lower bound (with respect to ⊑). Also, in a fully general formulation of Distribution, we should allow for there to be infinitely many disjunctive conjuncts.
- 3.
Some further issues concerning the correspondence between the operational and relational approaches are discussed in Fine (2017).
- 4.
Curiously, I adopted an operational approach in Fine (2012), though one which still took fusion to apply to sets rather than to indexed sets. One might also consider dropping some of the other algebraic conditions. But this is not a line of enquiry that I shall pursue.
- 5.
A theory Δ is said to be prime if \({\text{B}} \in \Delta {\text{ or C}} \in \Delta\) whenever \(({\text{B}} \vee {\text{C}}) \in \Delta\). One naturally thinks of a prime theory as corresponding to a definite state. But it corresponds, at best, to a definite state in the weaker, not the stronger, sense.
- 6.
An assumption, I might note, that has been questioned by Wilson (2013).
- 7.
I should add that it is only ‘flat’ or non-hierarchical versions of truthmaker semantics that can be obtained in this way, not the hierarchical versions of the semantics—considered, for example, in de Rossett and Fine (2023). Closure semantics is discussed in the context of substructual logic in Chap. 12 of Restall (2000) and is further discussed in my response to Correia.
- 8.
- 9.
To turn this into a formal example, one might appeal to a closed extended propositional space. The underlying states are subsets of {s1, s2, s3} (which we interpret conjunctively). Given a set T of such states (which we interpret disjunctively), the closure operation * puts u ∪ {s2} into T* whenever u ∪ {s1}, u ∪ {s3} ∈ T. This operation satisfies Upward Transfer and so we obtain a regular space in this way.
- 10.
The operation * may be shown to satisfy Upward Transfer and so will give rise to a regular propositional space.
- 11.
We should note that if Conjunctive Closure fails then we can no longer represent the states within the extended state space as propositions whose definite states are the singletons. For \(\{ s\} \wedge \{ t\} = \{ s \wedge t\}\) and so will always be a definite state.
- 12.
This is a problem which arises in providing a semantics for relevance logic, discussed in my response to Bimbo and Dunn. For the deductive closure of the union of two prime theories for a given relevance logic will not in general be itself a prime theory.
- 13.
- 14.
- 15.
A more general formulation would allow us to derive conjuncts and conjunctions from the basis; and we might also require that each possible world should be derivable in this way.
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Fine, K. (2023). ‘To Be or Not to Be Disjunctive’: Response to Mark Jago’s ‘Conjunctive and Disjunctive Parts’. In: Faroldi, F.L.G., Van De Putte, F. (eds) Kit Fine on Truthmakers, Relevance, and Non-classical Logic. Outstanding Contributions to Logic, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-031-29415-0_10
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