THE UNDERDETERMINATION OF THE MEANING OF
LOGICAL WORDS BY RULES OF INFERENCE
Elia Zardini
LanCog
Philosophy Centre
University of Lisbon
eliazardini@gmail.com
Inferentialism about a class of expressions C is the view that the meaning of any expression
member of C is constituted by the rules of inference associated with that expression. Global
inferentialism is inferentialism about all the atomic expressions of the language. Even
though with its own defenders (see Brandom [1994]), global inferentialism is wildly
implausible: in the 16th century, the meaning of the English word ‘gold’ was arguably
already different from the meaning the word would have had if used by a population whose
inferential dispositions were exactly the same as those of the English population at that
time, but whose environment were occupied by a gold-like substance whose atomic number
is not 79. Hence, many inferentialists have retreated to the weaker claim (henceforth,
‘logical inferentialism’) that it is the meaning of the logical words that is constituted by the
rules of inference associated with them (Tennant [1987], Read [2000]). While more
plausible, logical inferentialism is still a very controversial view, which has crucially served
as the meaning-theoretic foundation for a variety of recent attempts at accounting for our
apparent a priori knowledge of logic (see e.g. Boghossian [1996]).
The paper will present and discuss what I call ‘the underdetermination challenge’ to
logical inferentialism. The challenge consists in arguing that some logical words are such
that no set of formal, recursively enumerable rules of inference can completely fix their
meaning. I will substantiate the challenge by looking at four broad classes of logical words.
The first class concerns monadic intensional operators. For example, the modal
operator ‘it is necessary that’ has actually been shown by philosophical reflection to be
ambiguous among different readings (e.g. conceptual necessity, metaphysical necessity,
nomological necessity), yet no plausible reading of it can be uniquely characterised by a set
of rules of inference that may be associated with that operator, for, in all these cases, the
associated rules of inference will yield a logic in the vicinity of S5. Also, every rule of
inference that is sound with respect to (any reading of) it will also be sound with respect to
the non-synonymous temporal operator ‘it will always be the case that’, and vice versa. A
similar situation arises for epistemic operators like ‘it is certain that’ and ‘it is provable
that’ and for determinacy operators like ‘it is definitely the case that’ and ‘it is settled that’,
which pairwise share exactly the same rules of inference in spite of their being nonsynonymous.
The second class concerns conditionals. I will briefly rehearse the reasons for
thinking that indicative conditionals, just like subjunctive ones, are not captured by either
material implication or by any standard “relevant” implication (since even the weakest
standard “relevant” implications validate for instance transitivity). In the framework of
conditional logics (see Chellas [1980]), I will then argue that no rule of inference actually
seems to discriminate between indicative and subjunctive conditionals: if a rule of inference
is clearly valid for the subjunctive conditional (like reflexivity), it also is clearly valid for
the indicative conditional; if a rule of inference is clearly not valid for the subjunctive
conditional (like monotonicity), it also is clearly not valid for the indicative conditional; if a
rule of inference is neither clearly valid nor clearly not valid for the subjunctive conditional
(like conjunction in the consequent), it also is neither clearly valid nor clearly not valid for
the indicative conditional.
The third class concerns quantifiers. I will argue that no inferentialistically
acceptable rule of inference can distinguish objectual quantifiers from substitutional ones.
Moreover, more complex, generalised quantifiers (for example, cardinality quantifiers such
as ‘there are infinitely many’ and ‘most’), as well as higher-order quantifiers, demonstrably
lack a set of sound and complete recursively enumerable rules of inference. I will show
how these incompleteness phenomena can be used to generate another yet another version
of the underdetermination challenge.
The fourth class concerns the identity predicate. Drawing from examples from
graph theory and symmetric universes, I will argue that, for many languages and domains,
the usual rules of inference based (among others) on the indiscernibility of identicals fail
uniquely to pick out identity (understood as the strongest universally reflexive relation—
that is, as the relation that everything has to itself and nothing else) as opposed to weaker
congruence relations.
I will close by offering a tentative diagnosis of why logical inferentialism is subject
to the underdetermination challenge. Just like non-logical words, logical words often have a
quite specific content that is primarily grounded in and grasped by either paradigmatic
applications to particular cases or by informal elucidations of the property they express. In
both cases, the point of entry of meaning is situated on the world-word side rather than on
the word-word side which is favoured by logical inferentialism and which is constituted by
inferential transitions relying on formal, recursively enumerable rules. Contrary to what
logical inferentialism requires, there is thus no reason to expect that a word that is primarily
endowed with content by paradigmatic applications to particular cases or by informal
elucidations of the property they express will exhibit any general inferential connection to
other parts of language, let alone uniquely characterising ones.
REFERENCES
Paul Boghossian [1996], ‘Analyticity Reconsidered’, Nous 30, pp. 360–391.
Robert Brandom [1994], Making It Explicit, Harvard University Press, Cambridge MA.
Brian Chellas [1980], Modal Logic. An Introduction, Cambridge University Press,
Cambridge.
Stephen Read [2000], ‘Harmony and Autonomy in Classical Logic’, Journal of
Philosophical Logic 29, pp. 123–154.
Neil Tennant [1987], Anti-Realism and Logic, Clarendon Press, Oxford.