In a series of recent papers, Kit Fine [e.g., Fine 2016; 2017a,b] has developed and explored several forms of “truthmaker semantics” (TMS). Narrowly understood, TMS is just a collection of mathematical definitions of certain models for certain formal languages. While such models might be interesting in many different ways, one might be interested in them because one took them to provide an accurate account of reality, in the sense that every sentence true in every model in the class is really true.

One simple language of particular interest to metaphysicians is the language of propositional logic supplemented with propositional quantifiers and identity: call it $\mathcal{L}_{\forall=}$. Although Fine’s papers mostly treat less expressive languages, TMS can readily be extended to interpret $\mathcal{L}_{\forall=}$; and on any way of doing this meeting certain very minimal desiderata, sentences like $\forall p(p=(p\lor q))$ and $\forall p\forall q\forall r((p\land (q\lor r))=((p\land q)\lor(p\land r)))$ will be true in all truthmaker models. We take these sentences to express very general and potentially controversial claims about reality (see Dorr 2016), to which those who accept the truthmaker models as metaphysically accurate will be committed.One of our goals in this paper is to map out the metaphysical commitments of various versions of truthmaker semantics expressible in $\mathcal{L}_{\forall=}$.

Of course, the project of extracting metaphysics from TMS need not be limited to accepting sentences of this spare language. If one takes the models to be good pictures of the structure of reality, one will be tempted to start using the characteristic terminology of truthmaker semantics— ‘state’, ‘verify’, ‘obtain’...—not just for characterizing arbitrary set-theoretic models, but in an unrelativized way, to talk about the supposed structure in reality corresponding to the structure of such models. (Such a person might ask, e.g., not ‘How many states in this model are verifiers for the sentence “Snow is white”?’, but unrelativized questions like ‘How many states are verifiers for snow being white?’) Opponents, however, might question the very intelligibility of the new primitives (‘state’, etc.). One way to defuse such worries is to provide definitions of the novel terminology in terms of less controversial vocabulary that is common currency between truthmaker metaphysicians and their opponents, such as that of $\mathcal{L}_{\forall=}$.

Our paradigm for this investigation is work done in the 1970s on possible worlds semantics, in particular Fine 1970 and Fine 2005 (1977). Here, too, technical terms like 'world' first appear as nothing more than suggestive labels for certain ingredients of some mathematical models. But these models suggest a certain metaphysical theory, one that is 'coarser-grained' than the one suggested by truthmaker semantics in that it includes identities like $\forall p \forall q (p = p \lor (q \land \neg q))$. In spelling out this theory, it is natural to use expressions like 'world' and 'true at' in an unrelativized way. But thanks in large part to Prior and Fine, we know that in the context of the theory, adequate definitions of these theoretical terms can be given in $L^{\forall=}$. For example, '$w$ is a world' can be defined as '$w$ is a disjunctive atom', i.e. $\forall p (p \leq q \to (p = w \lor \forall q (p \leq_\lor q)))$ (where $p \leq_\lor q$ means that $p$ is a disjunct of $q$, i.e. $q = q \lor p$). Given these definitions, the whole metaphysical theory of worlds can be derived from a few rather simple and independently appealing principles stated in $L_{\forall=}$.

This paper carries out a similar program for some of the main versions of TMS. We begin with a 'unilateral' version of the semantics where propositions are modeled as sets of states meeting certain well-behavedness conditions (nonempty and regular). We show that in these models, the disjunctive atoms are exactly the singletons of states, so that one can define '$s$ is a state' as '$s$ is a disjunctive atom', and '$s$ verifies $p$' as '$s$ is a disjunct of $p$'. We formulate somewhat natural principles in $L_{\forall=}$ from which, with these definitions, one can derive that $p = q$ iff $p$ and $q$ have the same verifiers, and further derive principles about the verifiers of conjunctions, disjunctions, and negations corresponding to the semantic clauses of the relevant family of models.

Having explored the version of TMS with unilateral, nonempty, regular propositions, we turn more briefly to two other versions: first, a version with unilateral propositions that may be empty or non-regular, and second, a bilateral version where propositions are modeled as ordered pairs of (nonempty and regular) sets of states. The former case is fairly straightforward, but the latter case turns out to be much more complicated, since states can no longer be identified with disjunctive atoms (since disjunctive atoms can have more than one verifier). We formulate more complex definitions of 'state' and 'verify' in $L_{\forall=}$ which work for a class of bilateral models where the state space and domain of proposition are subject to certain further constraints.

• Dorr, C. (2016), “To be F is to be G”, Philosophical Perspectives, 30: 39–134.
• Fine, K. (1970), “Propositional Quantifiers in Modal Logic”, Theoria, 36: 336–46.
• Fine, K. (2005), “Prior on the construction of possible worlds and instants”, in id., Modality and
• Tense, (First published as postscript to Worlds, Times and Selves (with A. N. Prior), pp. 116-168, London: Duckworth, 1977) (Oxford University Press), 133–75.
• Fine, K. (2016), “Angellic Content”, Journal of Philosophical Logic, 45: 199–226.
• Fine, K. (2017a), “A Theory of Truthmaker Content I: Conjunction, Disjunction and Negation”, Journal of Philosophical Logic, 46: 625–74.
• Fine, K. (2017b), “A Theory of Truthmaker Content II: Subject-matter, Common Content, Remainder and Ground”, Journal of Philosophical Logic, 46: 675–702.