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Second, taking sparse facts seriously does not mean taking them to be basic.
Maybe there is something to be said for a Tractarian ontology, according to
which the world is, ultimately, all that is the case (the totality of sparse facts),
and objects and properties alike are in some sense abstractions from the facts.
That view has the merits of sidestepping a familiar putative difficulty with views
that take objects and universals as the two basic kinds of entity for how, it is
46 Peter Smith
asked, do they get combined into a fact? Not, for familiar reasons, by standing
in a relation: so we have to postulate some kind of irreducible non-relational
tie between objects and the universals they instantiate. And this, some say, is
difficult to understand. Others will feel that taking facts as basic and treating
objects as some kind of abstraction from the facts in which they feature does
not give chunky physical objects the right kind of status (as if, perhaps, we are
in danger of assimilating the being of all objects to that of other abstracta like
Frege s directions). But we just do not need to tangle with that kind of debate
now. Which is fortunate, as the rules of engagement for debates about what
is metaphysically basic are obscure, to say the least.
Thus, suppose we agree with Mellor that causation involves facts (the sparse,
worldly facts facta as he calls them), and that it will not do, for example,
to treat causation as relating tropes, for tropes do not have enough structure.
That in itself does not rule out analysing facts as complexes of (sparse) objects
and properties, and then treating those in turn as each constituted in different
ways by suitably structured collections of tropes. Maybe, then, tropes are the
alphabet of being, and the facta are (so to speak) rather long paragraphs. Or
changing tack, perhaps some will prefer to treat sparse properties as elite sets
of objects drawn from many possible worlds, and ultimately construct facta
from cross-world collections of objects. For the present, it really does not matter
what our favoured metaphysical story is, so long as it has the resources to make
sense of talk of sparse worldly objects and properties (if only by construction),
and thus make sense of talk of sparse facts (if only by further construction).
Because once we do have facts in our story of the world, however they are to
be further analysed, surely they should enter into our story about truth? That
is the basic challenge to the metaphysical deflationist about truth.
Third, it certainly is not settled how best to frame a theory of truth that
is both formally competent yet also uncontentiously deserving of the label
metaphysically deflationist . Again, it turns out that the details mostly do not
matter for our problem, but it is worth pausing to say more about this.
To help fix ideas, take the theory PA, i.e. first-order Peano Arithmetic, whose
language is L. Extend L to L+ by adding both a construction )#& *#, which forms
terms from wffs (well-formed formulae) of L (with the intended interpretation
that «)#Õ*#» denotes Õ), and also a new predicate Tr . Let MT be the set of
instances of the T-schema
Tr)#Õ*# a" Õ,
where Õ is a closed wff of the original L. PA + MT might thus be advertised as
arithmetic plus a theory of truth that captures in a minimal way the thought
that there is no more to the idea of (arithmetic) truth than is given by the
requirement that arithmetic instances of the T-schema hold.
PA + MT indeed involves a very modest theory of truth. For example, as
you would expect if the truth-predicate really is just akin to a disquotational
device, PA + MT is conservative over PA (i.e. no L-wff, not already provable
Deflationism: the facts 47
from PA, is provable from PA + MT). The trouble is that MT looks too modest.
For any closed L-wff Õ, we have both
MT g Tr)#Õ*# a" Õ
MT g Tr)#¬Õ*# a" ¬Õ,
and hence
MT g Tr)#¬Õ*# a" ¬Tr)#Õ*#.
However, while we can prove each instance of Tr)#¬Õ*# a" ¬Tr)#Õ*#, we cannot
yet even express, let alone prove, the generalization that a negated wff of L
is true if and only if the original wff is not true. Now, the expressive lack is
easily repaired. Extend L+ by adding the functor neg , where, for each wff Õ
of L, we have as a syntactic axiom
neg)#Õ*# = )#¬Õ*#
and add too, perhaps, a predicate sen , where for each closed wff Õ of L we
have the syntactic axiom
sen)#Õ*#.
And we can now, in this extended language, frame a generalization N about
negation thus:
"x(sen x ’! (Tr neg x a" ¬Tr x)).
But even with the syntactic axioms S in play, we do not have
PA + MT + S g N.
Why so? The basic idea is to take a natural model for PA + MT + S, add a
rogue element ± to the domain and extend the interpretations in the natural
model so that ± is in the new extension of sen while the new interpretation
of neg maps ± to itself, and this model will still satisfy PA + MT + S while
falsifying N.
It is sometimes said that the truth-predicate is just a formal device of
disquotation and that a major point of having such a truth-predicate is to be
able to frame generalizations (such as that a negation is true just so long as its
un-negated counterpart is not) which it would otherwise need infinite conjunc-
tions to express. But now we can see that the two halves of this claim do not
quite chime together. For the minimal rules governing a mere disquotational
48 Peter Smith
device (even given the needed syntactic resources) do not by themselves entitle
us to make the desired generalizations.
This shortcoming of MT was long ago noticed by Tarski, and we have learnt
from him one way of doing better namely replace MT (plus the syntactic
extras) with a full Tarskian theory of truth, TT. This certainly allows us to
derive the laws of truth like N. But, from a deflationist perspective, the price
is high. To take a dramatic example, it is familiar that
Not [PA g G],
where G is a standardly constructed Gödel-sentence for the given version of
PA. However, we also have1
PA + TT g G.
So TT is not conservative over PA. But a theory that enables us to deduce new
truths in an old domain can hardly be said to be unsubstantial, minimal or
fully deflationary.
Still, it might perhaps be said that the Tarskian theory remains metaphysi-
cally deflationary, even if not maximally deflationary in other ways. But is that
entirely right? To be sure, a Tarskian truth-theory is blind to any metaphysical
difference between the truth-conditions for Caesar is dead and Gwyneth is
beautiful and or 7 is prime (thus, the base clauses for the predicates & is
dead , & is beautiful , & is prime treat them exactly on a par). And the
truth-theory does not balk either at delivering, in the same indiscriminate
way, T-biconditionals for This emerald is grue and Caenyth is gappy
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