Semantics on a State Space

Fine starts with a domainPof possible and actual states whose role is akin to that of the domain of possible worlds in SPW semantics. Fine insist that ‘state’ is, for him, “a term of art and it is used to cover not just states in the ordinary sense of the word but facts, events, conditions or whatever else may legitimately be regarded as a truthmaker (Fine 2015, p.2).”Pthen consists of states such as the actual state of this ball being red, the possible state of my being full, given that I am hungry, and the state of the current weather in Amsterdam being both rainy and windy. This conception of states renders thempartialin a way possible worlds are not: it is not the case that the truth or falsity of any statement will be settled by any state whatsoever, while it is the case it will be settled by any world. For example, the state that it is currently both rainy and windy in Amsterdam does not settle whether this ball is red, while any possible world in which this ball exists settles whether it is red or not.

A natural way for those raised within the possible worlds tradition to think of states is as parts of worlds. By way of an introduction to the notion of a state, characterising states as parts of worlds may be of heuristic value and moreover, when working with extended models, the characterisation can even be formally substantiated. Nevertheless, it is not a characterisation one should read into. The idea Fine is building towards is that if we were to need worlds, then we could have them, for example by defining them as sets of states satisfying certain conditions of completeness, consistency and the like, or as total and consistent statessimpliciter. However, as will soon become evident, “possible worlds completely drop out of the picture (Fine 2015, p.18)”, for we have no need for them.

Be that as it may, already driving the legitimate if misleading conception of worlds as state- conglomerations is an intuition that our set of statesPenjoys a certain mereological structure thet the domain of possible worlds does not, in virtue of the fact that states are partial – the intuition here being that in some sense, the state that it is rainy is Amsterdam is part of the state that it is both rainy and windy in Amsterdam. Indeed, the basis for Fine’s semantics is the notion of astate spaceS which he defines as a tuple(P,v)whereP is a non-empty set – to be thought of as our domain of possible states from above – andvis a partial order onP – to be thought of as the mereological relation that intuitively holds between certain states – that satisfies the following condition:3

Bounded Completeness: Any subset ofPthat has an upper bound has a least upper bound.

3_{To remind the reader, a partial order⊆on a setXis a binary relation onXthat satisfies the following conditions:}

Reflexivity: (∀x∈X)(x⊆x);

Anti-symmetry: (∀x,y∈X)(x⊆y∧y⊆x→x=y);

Now, an upper bound for a set T ⊆ P is ans ∈ Psuch that(∀t ∈ T)(t v s), and a least upper bound forTis an upper boundsforTsuch that for any upper bounds0 of Twe have thats v s0. Where statesis the least upper bound for a set of states T we say that sis the fusion of theT’s and writes=tT. Given that we are working with a set ofpossiblestates only, it would be unreasonable to assume that the fusion of arbitrary states exist, since for example, any state having as parts the state that this ball is red and the state that this ball is not red would be an inconsistent or else impossible state. However, if there is a possible statessuch that it has as parts all states in T ⊆ P then this is indication enough that the states in Tare jointly compatible and thus it is reasonable to assume that such a state as the fusion ofTexists inP. This is then why Fine requires that a state space be bounded complete, and does not require that it be complete – that any arbitrary states inPhave a fusion.

Other than the question of the nature of truthmakers, as soon as one adopts an truth conditional, objectual approach towards content there is a related issue to be tackled: what is the relation of truthmaking/ false making that truthmakers/ falsemakers and statements stand in? Fine is interested inexactverification and falsification according to which a state “should bewholly relevant, and not just relevant in part to the truth or falsity of the statement (Fine 2016b, p.5).” Granted, this is as of yet no real analysis of exact verification, but we return to the issue later, when constructing our notion of a proposition. Leaving the issue aside for now, the state space and flavour of verification/ falsification being in place, we can already state the semantic clauses for a propositional languageLthe formulas of which are constructed from a countably infinite setPof sentence lettersp,q,r, ... and the connectives¬,∧,∨.

Definition 9(Exact Verification/ Falsification). Given a valuation function

[·]:P→ P2

x7→([x]+,[x]−)

taking propositional letters to their exact verification-falsification conditions – the pair of sets of states consisting of the set of all their possible verifiers and the set of all their possible falsifiers – we may recursively define what it is for a statesto exactly verify/ falsify a sentence AofL, writtens+/− A: Atomic A∈ P • s+ A ⇐⇒ s∈[A]+ • s− A ⇐⇒ s∈[A]− Negation A≡ ¬B • s+ A ⇐⇒ s− B • s− A ⇐⇒ s+ B Conjunction A≡B∧C • s+ A ⇐⇒ (∃t,u∈ P)(t+ B∧u+ C∧s=ttu)∗ • s− A ⇐⇒ s− B∨s−C Disjunction A≡B∨C • s+ A ⇐⇒ s+ B∨s+C

• s− A ⇐⇒ (∃t,u∈ P)(t− B∧u− C∧s=ttu)∗

Note that the clauses marked with a∗for conjunction and disjunction place an existential re- quirement onPwhich is not guaranteed to be met.4