4.2 Thick Propositions
4.2.2 Recursive Truthmaking
Clearly, thick propositions are more than thin propositions; they are thin propositions coupled with overall subject matter. Hence we can expect that if we take propositions to be thick, then they are individuated more finely than if we take them to be thin. This is the simple, underlying thought behind examining Yablo’s account with an eye on hyperintensionality. In this section I zoom in, looking at the data from §1 to see whether thick propositions succeed in making the hyperintensional distinctions simple SPW propositions – thin proposition in Yablo’s terms – fail to deliver. However, I cannot really do so until I have a semantics for truthmakers. Yablo does not provide one. He provides two ‘pictures’, and admits that “[t]he models representtendencies in truthmaker assignment that pull at times in different directions” (Yablo 2014a, p.62). The pictures in question are therecursiveand the reductiveapproach to truthmakers. Here I only have space for one, and I adopt the recursive approach; Yablo’s worries about the recursive approach – discussed in §4 of Yablo 2014a – do not interfere with the data I wish to examine, recursive truthmaking is the more common way to go (see Fine, Jago etc – citation needed), and Fine ( find which paper) has argued it is superior to the reductive one.
Before going into the semantics some exegetical points are due. What concerns Yablo is to de- velop a semantic conception of truthmaking, as contrasted to a metaphysical one. Under the metaphysical conception, truthmaking is seen as a relation between entities in the world – be those metaphysical chunks such as facts and worlds or even objects – and truths. Here we care not for developing a correspondence theoryá laRussell 1918 or Armstrong 2004, rather we seek an understanding of truthmaking “whereby it can play a foundational role in semantics” (Yablo 2014a, p.54). ‘Truthmakers’ might thus be a misleading term for Yablo to use, for truthmakers have often been supplied to remedy Tarski-style deflationary accounts of truth, in order to ex- plain how the true sentences of a language are made true by reality.6 But this is precisely what Yablo does not want to do: Yablo’s truthmakers care about what a statement says about reality, but not about whether reality accords with what the statement says. This is unusual, for truth- makers have customarily been used to help with the latter – as for example in correspondence theories –but it is no damning conviction. Perhaps one finds Yablo’s theory of truth incomplete, but as MacBride 2016 notes “[..] pointing out that a Tarski-style theory of truth doesn’t tell us this, doesn’t establish that the theory is lacking in this respect – not unless it has already been established that a theory of truth for a language that fails to explain how its sentences are made true fails to articulate in some critical respect how reality conspires with meaning to deliver their mutual upshot,viz.truth.”
So far so good, there is nothing the matter with a semantic conception of truthmakers. But Yablo also seems to suggest that we adopt an instrumentalist stance towards facts and truth-
makers, nicely summarised in the motto “[t]ruthmakers are as truthmakers do [...]” (Yablo 2014a, p.54), whilst requiring that the truthmaking relation be not vertical – between metaphys- ical entities and truths – but rather horizontal, holding between truths and other truths. And there now seems to be tension between our desiderata: on the one hand, we adopt a purely in- strumentalist stance towards truthmakers, on the other we require that they reside exclusively in the semantical realm and that they play a foundational role there. That makes for rather un- founded semantics, and we cannot avoid concluding that the theory is lacking, in that, at the very least, it is underdeveloped.
Without further ado I present the recursive truthmaker approach I will be coupling with Yablo’s framework. Most of the clauses for it take their cue from Van Frassen 1969, although the quan- tifier clause is modified following a suggestion by Yablo.7 Lastly, it need be noted that Yablo does not endorse the clauses for the atomic sentences, since it is those clauses precisely he finds most objectionable in the recursive picture.8 Yablo’s objections do not come into play in exam- ining the hyperintensional data I am interested in, and I need a full semantics for a first order language in order to examine the data and be consistent with work in previous chapters.
We begin with a first order language Lwith identity. The logical symbols ofLcomprise of the connectives¬,∧,∨, the quantifier∀and a logical relation symbol =. The non-logical vocabulary includes a stock of variables, a stock of constants, and a stock ofn-ary predicates for each n. Forsan assignment of values to variables, Da domain of individuals and δ an
interpretation, we have that: • δs(c)∈D, forca constant.
• δs(c)∈D, forxa variable.
• δs(P,w)⊆Dn, forPann-ary predicate.
For eachn-ary predicateP, we add a symbol ¯P to our language to symbolise then-ary pred- icate Xsuch that(∀w ∈ W)(δs(X,w) = D\δs(P,w)). The interpretation extends to complex
formulas as per usual, and truth is defined as in the SPW framework in the familiar way based on intensions.
However, it is not simply truth/ falsity we are interested in but reasons for being true/ false. Reasons for being true/ false are truthmakers/ false makers, so we now want to find, given a formulaφwhat its truthmakers and falsemakers are, so we may arrive at thick propositions.
Definition 8(Recursive Truthmakers). On the recursive approach, given a formulaφits truth-
makers and falsemakers are given by the following simultaneous recursion, where byTM(φ)I
symbolise the set of truthmakers and byFM(φ)the set of false makers ofφ:
Atomic–1 φ≡Pc1..cn
• TM(φ) = {{|Pc1...cn|}}, where|Pc1...cn| is the intensional content of φ, and thus
{|Pc1...cn|}is thefact thatφ.
• FM(φ) =TM(φ¯) =TM(Pc¯ 1...cn).9
7See Yablo 2014a §4.4.
8Yablo disagrees with the present approach insofar as it implies that simple sentences are true for simple reasons.
Yablo certainly has a point but the matter is irrelevant for the discussion in the following subsection and I thus suppress it. See §4.2 and 4.3 of Yablo 2014a for his objections in detail.
9Note that the truthmaker for an atomic sentence, say,Pa, is the atomic fact{|Pa|}. This contrasts slightly with the
informal discussion in the previous section where Yablo argued that a truthmaker is simply a proposition. Given the latter we would expect that the truthmaker ofPais the thin proposition|Pa|, and it for our technical convenience that
Atomic–2 φ≡c=c0 • TM(φ) ={{|c=c0|}}. • FM(φ) =TM(φ¯) =TM(c6=c0). Complex–1 φ≡ ¬ψ • TM(φ) =FM(ψ). • FM(φ) =TM(ψ). Complex–2 φ≡ψ∧χ • TM(φ) ={τ∪σ|τ∈TM(ψ),σ∈TM(χ)}. • FM(φ) =FM(ψ)∪FM(χ). Complex–3 φ≡ψ∨χ • TM(φ) =TM(ψ)∪TM(χ). • FM(φ) ={τ∪σ|τ∈ FM(ψ),σ∈FM(χ)}.
But how about quantified truths? Let us concentrate here on universal truths, and treat existen- tial ones simply as their dual. In hisLogical Atomismlectures, Russell concedes with certainty that there are general facts over and above the atomic ones which make generals truths true. However, he soon after admits: “I do not profess to know what the right analysis of general facts is” (Russell 1972, pp.71–72). Russell’s conviction about the existence of general facts is rather irrelevant to the current project; as stressed above, we are after a horizontal, semantic conception of truthmakers and not interested in developing a correspondence theory, ground- ing truth to a metaphysical reality of facts. For the same reason, the work of D.M.Armstrong Armstrong 2004 is not entirely relevant either. However, Armstrong, contrary to Russell, does attempt an analysis of facts that make general truths true:
My idea is that the truthmakers for such truths are facts, states of affairs, having the following form: a relation, which I will here call theTot relation, holds between a certain mereological object and a certain property.
The truths Armstrong is alluding to above are universal truths of the form “a,b,c, .... are all the A’s", and the idea is that the object – made out of the mereological fusion of a,b,c, .. – totals the property A. Regardless of whether one agrees with unrestricted mereological fusion, or whether one agrees that such a fact asa,b,c, ... totallingAis a legitimate, non-negative wordly fact, the hint I take from Armstrong is the following. Assume thata,b,c, ... are theA’s. Thatais B, and thatbisB, and thatcisB, .... do not suffice to make the general truth “AllA’s areB’s" true, for in addition we need somehow guarantee thata,b,c, .. are all theA’s that there are. For consider the alternative. According to Van Frassen 1969, the clause for the universal quantifier should be as follows:
TM(∀xφ) =TM(φ(a)∧φ(b)∧φ(c)∧...)
we take the singleton{|Pa|}instead. I have aimed at keeping the clause for the truthmakers for conjunction and false makers for disjunction as is usually found in the literature, i.e. a truthmaker for a conjunction is theunionorfusionof some truthmakers of its conjuncts. It would be okay to stick to|Pa|and change the latter clauses but I have chosen the current ‘fix’.
However, recall that a truthmaker for ∀xφ is a fact, a proposition that, among other things,
implies∀xφ. And now consider that we may be working with worlds with variable domains
such that in w0 there exists an object othat does not exist atw. Then ifφ(a)∧φ(b)∧... is a
truthmaker for∀xφatw, then it is not a truthmaker for it atw0for we would additionally need
thatφ(o). What we are in need of is a fact that guarantees thata,b,c, .. are all the objects that
there are, and of course that they are allφ.
An initial suggestion would be the following: ∀xFxis made true byFa,Fb,Fc, ... combined with the fact that a,b,c, ... are everything. Let us abbreviate the totality fact that a,b,c, .. are everything byT, and for arbitrary Athe fact thata,b,c, ... are all theA’s byTA, and recall that
according to Yablo there’s two sides to truthmaking: necessitation and explanation. And we may ask, is the totality fact explanatorily on par with the set of all instances falling under a uni- versal truth? That is, doesTexplain that∀xFx, or is it rather thatTexplains whyFa,Fb,Fc, ... explains and necessitates that∀xFx? It rather seems that to simply union the totality fact with the totality of instances of a universal truth is to “confuse the issue of what the truthmaker is, with the issue of how it acquires that status” (Yablo 2014a, p.63). However, our semantic conception of truthmakers allows us to make sense of the idea of truthmakers for truthmakers, and this seems to be what the totality fact is to the set ofFa,Fb,Fc, ... But not quite, for truth- makers necessitate and it is beyond the current scope to ask whether the totality fact suffices to necessitate thatFa,Fb,Fc, .. make∀xFxtrue. Yablo’s suggestion is then the following: ∀xFx is made true byFa,Fb,Fc, ..qua complete list of instances. As Yablo notes, this turns the totality fact to something akin to a presupposition: Fa,Fb,Fc, ...qua everythingimplies thataisFfor it fails ifFadoes. However, it does not implyT, for whenTfailsFa,Fb,Fc, ...qua everythinghas a truth value gap. As for the false makers of a universal claim∀xFx, any instance of ¯Fxwill do. So, borrowing notation from the literature on presuppositions, we may state the clauses for the universal quantifier as follows:10
Complex–4 φ≡ ∀xψ
• TM(φ) =TM(ψ(a)∧ψ(b)∧ψ(c)∧...∧∂Tψ) • FM(φ) =FM(ψ(a)∨ψ(b)∨ψ(c)∨...)
And dually for the existential quantifier: Complex–5 φ≡ ∃xψ
• TM(φ) =FM(∀xψ¯)
• FM(φ) =TM(∀xψ¯)
This is then the truthmaker semantics I will be pairing with Yablo’s framework. They are not the semantics Yablo endorses, for Yablo does not endorse a single semantics. Where Yablo would object to the current clauses I have pointed out throughout the exposition, but I believe his worries are largely irrelevant to my current goal, the goal – pursued in the subsequent section – being to examine the data from §1, and find out how finely thick propositions cut. It need be noted that the semantics do not include clauses for the conditional. It would take me too far afield to do any justice to Yablo’s work on the subject matter, and those data that pertain to conditionals will be discussed somewhat more informally, yet rather decisively still.
10The point in common with presuppositions that suggests∂as a good notation here is, very roughly speaking, the
following: ifXpresupposesYin a certain sense, then, ifYis true thenXis subject to certain truth conditions, whilst if
Yis false then –quaFrege 1892, Strawson 1950 and others –Ywe have a presupposition failure andXexhibits a truth value gap. For a longer discussion see §5.2 of Beaver and Bart 2014.
4.3
Conclusion
Allow me to start by noting the rather obvious: thick propositionsà laYablo cut at least as finely as standard possible worlds – or else thin – propositions. This can be readily acknowledged: a thick proposition comprises of the corresponding thin proposition, and its subject matter, so if we have two thick propositions sayA= h|A|,σ(A)i,B= h|B|,σ(B)i– where|X|is the thin
proposition thatX, andσ(X)is the overall subject matter ofX– and|A| 6= |B|, then Ais not
the same asB. Hence we concentrate on the data from §1, propositions which are, as has been extensively argued,wronglyconflated within the SPW framework.