This is a reflection on Kit Fine’ (2017) survey of truthmaker semantics. I will describe the basic set-up presented by Fine, aiming to clarify the logical relations between exact and inexact truthmakers.
1 Exact truthmakers p. 1
2 Sentences and truthmaking p. 2
3 Propositions and logic: an isomorphism p. 3
1. Exact truthmakers
Exact truthmakers are entities of some sort (‘states’, ‘facts’, ‘events’) which combine in just one way (‘fusion’, I’ll use symbol +). Kit Fine gives as examples wind, rain, and their fusion wind and rain. That fusion is associative is assumed:
- x + (y + z) = (x + y) + z
So we can just write e.g. wind and rain and snow without ambiguity. In mathematical terms, this property makes the exact truthmakers a semigroup. But it has two additional properties:
- x + x = x idempotency
- x + y = y + x commutativity
So the exact truthmakers form an idempotent, commutative semigroup. That structure is at the same time a semilattice by the following:
Definition. x ≤ y iff x + y = y and x ≥ y iff x + y = x
So wind is part of wind and rain, for the fusion of wind and rain is just wind and rain. That the exact truthmakers thus form a a semilattice means first of all that ≤ , and its converse ≥, are partial orders.
I find it convenient to focus attention on the converse order. For that will make it easier to see the relation between exact and inexact truthmakers, when we discuss that below.
- x ≥ x reflexivity
- if x ≥ y and y ≥ x then x= y anti-symmetry
- if x ≥ y and y ≥ z then x ≥ z transitivity
In addition, a fusion is the least upper bound of its parts in the ≤ ordering (as Fine has it). That is the same as the greatest lower bound in the ≥ ordering:
- x + y ≥ x and x + y ≥ y
- if z ≥ x and z ≥ y then z ≥ x + y
By anti-symmetry it follows quickly that x + y is the unique element for which (7) and (8) hold in general, hence
Lemma 1. {z: z ≥ x + y} = {z: z ≥ x} ∩ {z : z ≥ y}
For by (7), if z ≥ s + t then z ≥ s and z ≥ t. And by (8), if z ≥ s and z ≥ t then z ≥ s + t.
Idempotent commutative semigroup and semilattice are really one and the same. For starting with the semilattice and defining x + y to be the least upper bound of x and y in the ≤ ordering, we arrive at properties (1)-(3) above for the defined notion.
I have followed Fine here in defining the ≤ order so that wind and rain looks in the mathematical representation like a disjunction rather than like a conjunction. So this semilattice <S, ≤ >is a join semilattice, while equivalently <S, ≥ > is a meet semilattice, in which the operator selects the greatest lower bound. (Ignore the visual connotation of symbol “≥” in the latter case.)
Kit Fine added to his definition of a state space that the family of exact truthmakers with the fusion operation forms a complete semilattice. That is, each set of exact truthmakers (and not just pairs or finite sets) has a least upper bound. That is audacious, and may introduce difficulties if probabilities are eventually introduced into this framework. So I will just stay with the finitary operation here.
I will make the definition redundant, so as to display the two equivalent ways of looking at this structure.
Definition. A state space is a triple S =(S, +, ≥), with S is a non-empy set (the exact truthmakers), + is a binary operator on S, and ≥ is a binary relation on S such that (S, +) is an idempotent commutative semigroup, (S, ≥) is a semilattice, and for each x, y in S, x + y is the greates lower bound of x and y in that semilattice.
Although this is more elaborate, to make some details explicit, this definition is equivalent to Fine’s definition of a state space (S, ≤) as a semilattice in which the fusion of x and y is the least upper bound of x and y in the ≤ ordering.
2. Sentences and truth-making
Let language L have sentence connectors &, v, to be read as ‘and’ and ‘or’. An interpretation of L in state space S is a function |.|+ assigns to each sentence A the set |A|+ (the exact truthmakers that verify A, that make A true), subject to the conditions
- |A & B|+ = {t + u : t is in |A|+ and u is in |B|+}
- |A v B|+ = |A|+ ∪ |B|+
Kit Fine defines (page 565) the notion of an inexact truthmaker for sentence A so that e.g. wind and rain inexactly verifies “It is raining” just because rain is an exact truthmaker of that sentence. That is, if s exactly verifies A then s + t inexactly verifies A.
Definition. s inexactly verifies A if and only if there is some element t of |A|+ such that t ≤ s.
I will use the notation || . ||+ for the set of inexact truthmakers of a sentence:
|| A ||+ = {s: s ≥ t for some t in |A|+}
that is, the set of inexact truthmakers of A is the upward closure of its set of exact truthmakers in the ≤ ordering.
This notion is so far defined only in terms of the relation to language. But there is obviously a corresponding language-independent notion, and the two go well together. For each exact truthmaker engenders a specific set of inexact truthmakers:
Clearly u ≥ t exactly if 𝜙(u) ⊆ 𝜙(t), and so t is in |A|+ exactly if 𝜙(t) is part of || A ||+. In fact,
|| A ||+ = ∪{ 𝜙(t): t is in |A|+}
This is an upward closed set in the ≤ ordering.
3. Propositions and logic : an isomorphism
What should we take to be the proposition expressed by sentence A, should we take it to be |A|+ or || A ||+ ? We can just say there are two, the ‘exact proposition’ and the ‘inexact proposition’.
Just how different are the families { |A|+: A a sentence of L} and {|| A ||+: A a sentence of L}? They are certainly different; for example be |A & B|+ is not in general part of |A|+, while || A & B ||+ ⊆ || A ||+. So as far as exact truthmakers are concerned, A & B does not entail A, while for the inexact truthmakers A & B does entail A.
But the relationship that we saw just above shows that from a structural point of view there is an underlying identity.
Lemma 2. For s, t in S, 𝜙(s) ∩ 𝜙(t) = 𝜙(s +t)
This is just Lemma 1 transcribed for function 𝜙.
Lemma 3. The system Φ(S) = < {𝜙(t): t in S} , ∩, ⊆ > is a (meet) semilattice.
The Lemma follows Lemma 2 and the informal discussion at the end of section 2. In fact, by our earlier definition, this system is also a state space.
Theorem. The function 𝜙 is an isomorphism between state space S = <S, +, ≥ > and state space Φ(S) = < {𝜙(t): t in S} , ∩, ⊆ >
So the inexact truthmakers provide a set-theoretic representation of the semilattice of exact truthmakers. The proof is standard textbook fare for semilattices.
Sketch of the proof.
To begin, the ordering in S is mirrored in the range of 𝜙 because ≥ is also a partial ordering:
- if x ≥ y then by transitivity, if z is in 𝜙(x) then z is in 𝜙(y); hence 𝜙(x) ⊆ 𝜙(y).
- x is in 𝜙(x) so if 𝜙(x) ⊆ 𝜙(y) then x ≥ y
- 𝜙 is one-to-one by the anti-symmetry of ≥
This establishes an order isomorphism between S and the range of 𝜙. In addition:
𝜙(x + y) = {z: z ≥ x + y} = {z: z ≥ x and z ≥ y} = 𝜙(x) ∩ 𝜙(y)
by Lemma 2.
How much should we conclude from this?
Perhaps not much. As Fine shows, there are many distinctions to be made in terms of exact truthmakers that can be exploited in different ways, and might be obscured by turning to the inexact truthmakers alone. And many examples can be illuminated by focusing just on the exact truthmakers of atomic sentences, with propositions, in either sense, built up from there. In addition, we have only been discussing the most basic set-up, and there are many interesting complications when negation and modal operators are introduced, as Kit Fine shows.
But at the same time, it may help to reflect that the state space of exact truthmakers does have a ‘classical’ structure, with a set-theoretical representation.
REFERENCES
Kit Fine (2017) “Truthmaker Semantics”. Pp. 556-577 in A Companion to the Philosophy of Language (eds B. Hale, C. Wright and A. Miller)
Bas van Fraassen's philosophy blog 