Prospects for Modest
Inferentialism
Jack Woods
In this paper I will argue that the combination of two insights from Prior,
(1) that there are tonkish sets of natural deduction rules, and (2) that modeltheoretic accounts of meaning should be viewed as models of underlying intuitive
meaning, together show that an otherwise promising form of inferentialism will
not work. In particular, I argue that there is no way to formulate conditions
on the intuitive meaning of the connectives which simultaneously (a) allow the
natural deduction rules to specify the correct particular meanings of connectives
like conjunction, negation, disjunction, and the conditional while (b) not tacitly
specifying the meaning of some of these connectives independently of the natural
deduction rules. The upshot of this is that modest forms of inferentialism are
unworkable. This is no idle result; philosophers have been repeatedly tempted
by this position, even if under slightly different guises (Boghossian 1996; Peacocke 1976; Hodes 2004).
In perhaps his best known paper, Prior formulated the most important challenge to developing an inferentialist account of meaning (Prior 1960). The challenge, posed to inferentialist accounts of the meaning of the logical connectives,
was that there are sets of rules, such as from A to infer A-tonk-B and from
A-tonk B to infer B, from which anything can be inferred from anything. Excluding these sets of rules for principled reasons has become perhaps the main
problem in developing an inferentalist account of connective meaning. In the
wake of tonk, two main strains of inferentialism have developed. One strain,
developing work by Gentzen, has attempted to formulate various conditions
on pairings of introduction and elimination rules which rule out tonkish rules.
This approach has no truck with model-theoretic accounts of meaning, viewing a
model theory as a mere heuristic with which to shed light on the proof-theoretic
meaning of the connectives. My focus is on the other strain, so we will leave
this approach to the side.
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The other strain, developing work by Carnap and Stevenson, retreats from a
pure form of inferentialism to a more modest hybrid view where inference rules
only specify connective meaning against a presumed background account of what
semantic meaning a connective should have (Stevenson 1961; Carnap 1959). So,
modest views accept as given some model-theoretic account of what type of
semantic value—a truth-function, perhaps—a connective should have, as well
as some constraints on the consequence relation—perhaps it must be transitivity, reflexivity, monotonic, and truth-preserving—and, finally, some constraints
on permissible interpretations—the exclusion of the trivial interpretation where
every statement is true, perhaps. Given these background conditions on connective meaning, modest inferentialists derive particular semantic values, in the
forms of constraints on permissible models, for logical expressions from their
constitutive proof rules. To analyze conjunction, for example, we look to see
which interpretations are such that when they assign T to a conjunction, they
assign T to both of its conjuncts and, conversely. This, unsurprisingly, turns
out to be equivalent to those models v where
v(φ ∧ ψ) = T if and only if v(φ) = T and v(ψ) = T1
Tonk is ruled out by the fact that its introduction and elimination rules violate
the background account of meaning: in the presence of reflexivity, the only possible interpretation of tonk marks every statement as T, contra our exclusion of
the trivial interpretation.
This strain of inferentialism has many virtues, but it has been less well developed than its purely proof-theoretic counterpart. Part of the reason for this
is that it is well-known that ordinary natural deduction rules fail to specify the
correct interpretation of familiar connectives like the conditional, negation, and
disjunction (Carnap 1959, 89-94). It is entirely consistent with the characteristic rules for disjunction preserving truth (really, T) to assign a disjunction T
when both disjuncts are assigned F.2 This fact, first demonstrated by Carnap
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use T instead of ‘true’ to emphasize that we need not presume any particular interpretation of what the “truth-values” of an interpretation represent. See discussion below about
what model theory really is.
2 For an easy case of this, consider an interpretation which assigns a formula φ T if and only
if φ is a truth-functional tautology. Since classical deduction rules preserve tautologousness,
they will not take us from T to F. Yet p ∨ ¬p is T while both the atomic p and its negation
¬p are F.
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and evident by inspection of supervaluational or intuitonistic semantics, seemed
to condemn this approach as an interesting form of inferentialism. Various responses to this problem have been developed, but each seems to take for granted
the meaning of one of the relevant connectives—bilateralist views like those developed in (Rumfitt 2000) and (Smiley 1996) take for granted the meaning of
negation, multiple-conclusion views such as that developed in (Shoesmith and
Smiley 1978) take for granted the meaning of disjunction. Whatever the merits
of these projects, they lose some of the initial appeal of the inferentialist position since the meaning of all logical expressions is not given solely by rules of
inference.
Recent work by James Garson has provided a recipe for liberating this modest form of inferentialism (Garson 2001). Garson has showed how to generate
the correct meanings for familiar connectives from an underlying set of plausible
background conditions on connective meaning. The key move is distinguish the
meaning expressed by a set of provable arguments and the meaning expressed
by a set of rules. We can make this distinction once we have moved from determining connective meaning against a single assignment of truth and falsity
to determining connective meaning against sets of assignments. When we then
focus, as we should, on the meaning expressed by a set of rules, the familiar
inference rules for the conditional and negation express the proper and more
or less expected meanings. The conditional, for example, turns out to have
the expected intuitionistic meaning; expected because we expect the weakest
possible meaning to be expressed by a set of rules. That is, the set of sets of
interpretations on which the rules for the conditional preserve validity—here
the property that if every premise is true throughout the set of interpretations,
then the conclusion of the rule holds throughout the model—is exactly the set
of sets S of interpretations where for all v ∈ S:
v(φ → ψ) = T if and only if for every u ≥ v, if u(φ) = T, then u(ψ) = T
where u ≥ v if and only if v(χ) = T only if u(χ) = T. Similar results hold for
conjunction, negation, and the biconditional. And, analogously with the earlier
approach, Prior’s tonk preserves validity on no set of sets of intepretations.
This is a welcome development which seems to hold the potential to skirt
Prior’s tonk without abandoning the virtues of the inferentialist program or
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moving to purely proof-theoretic account of connective meaning. However, there
are some serious difficulties with the approach. One problem I will not address
is that the account of how we come to grasp the meaning of the logical connectives in this way is problematic for reasons well-articulated in (Quine 1936).
For my purposes here, I will assume, contra my intuition, that these problems
can be answered. There remain other serious problems. In previous work, I exposed one problem having to do with the meaning of disjunction (Woods 2012).
However, I also noted that a classical modest inferentialist could attempt to
solve this problem by imposing an additional side-condition on permissible sets
of interpretations. The side-condition demands that if an interpretation assigns
a formula false, then there is an extension of that interpretation—an extension
of an interpretation, again, being another interpretation which agrees with the
first on all formulas assigned T—such that every succeeding extension assigns
that formula F. That is, if a formula is F somewhere in the set of interpretations, then eventually it is settled as F somewhere in the set extending where it
is false. Formally:
(LF) v(φ) = F ⇒ ∃v 0 [v ≤ v 0 ∧ ∀v 00 (v 0 ≤ v 00 ⇒ v 00 (φ) = F)
The problem with this approach—which stems from (Humberstone 1981) and
has been defended in (Rumfitt 2012) and (Garson Forthcoming)—is that it
seems to place constraints directly on the meaning of the logical expressions
and so violates the spirit of the modest inferentialist program. In other words,
LF is not plausible as an independent constraint on connective meaning.
We can put the problem in terms of a compelling thought of Prior’s. Modeltheory is just that: a way of modeling meaning. It is not itself an account of
meaning in any substantive sense (Prior 1964). The background framework on
top of which Garson’s solution is generated is a more-or-less standard modeltheoretic semantics. Exclusion of the trivial interpretation, the idea that a
connective should find a condition in this framework, and the imposition of the
structural rules can all be motivated as part of a background understanding
of the meaning of logical expressions. And, the resulting bare model-theory
can be seen as representing the structure of this minimal account of connective
meaning. But it turns out that LF cannot be motivated in any obvious way as
representing bare features of the meaning of the logical expressions. Perhaps
we could motivate a weakened condition pLF which is defined only over atomic
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sentences. If we could motivate pLF as part of the meaning of the non-logical
expressions—here propositional atoms—and argue that this should constrain
our account of the meaning of the logical expressions, we would be in the clear.
However, in the my earlier work I showed that pLF is insufficient to solve my
problem with disjunction. LF is thus not derivable from pLF and the natural
deduction rules, being an essentially stronger side-condition.
My aim is to formulate and evaluate the best version of this side-condition
approach. LF and pLF do not seem to do the work required, but this does not
show that there is no close cousin of LF or, better, pLF which is capable of being
motivated as a part of bare connective meaning and which would also suffice
to rule out my problem with disjunction. If such a plausible side-condition
could be motivated, then we would have a tempting package: an account of
background meaning sufficiently robust so as to permit the particular meaning
of the logical expressions to be entirely derived from natural deduction rules
without vindicating pernicious sets of rules. In my view, this would almost
entirely vindicate the modest inferentialist’s position from problems with tonk.
What was missing was a plausible account of what the meaning of a connective
should look like. Augmented with this, there is no problem. However, I conclude
in the paper that Prior’s attack was just. There is no account of what connective
meaning should look like which is sufficiently strong to rule out problematic cases
which does not also involve tacit stipulations about the particular meaning of
connectives like disjunction. The modest inferentialist position, tempting as it
is, is simply not capable of correctly generating meaning in a way which does
not involve presupposing the meaning of the connectives already.
References
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Garson, J. 2001. Natural Semantics: Why Natural Deduction is Intuitionistic.
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Garson, J. Forthcoming. What Logic Means: From Proof to Model-Theoretic
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