Teresa Kouri on Logical Instrumentalism

Teresa Kouri came to Ohio State from the University of Calgary, where she completed an MA in 2010, specializing in the philosophy of mathematics. She completed her PhD under Stewart Shapiro in 2016. In her dissertation, Teresa argues that the best version of logical pluralism is logical instrumentalism: the view that norms for deductive reasoning should be evaluated based on one’s aims and goals in reasoning and the domain of investigation.

The following short paper, written especially for Logos, draws upon Teresa's dissertation work. 

Logical Instrumentalism

Logical Instrumentalism is the view that norms for deductive reasoning should be
evaluated based on one's aims and goals in reasoning and the domain of investigation.
In my dissertation, I argue that the best version of logical pluralism is logical instru-
mentalism. Logic and reasoning go hand in hand. We say that someone has reasoned
poorly about something if they have not reasoned logically, or that an argument is bad
because it is not logically valid. To date, research has been devoted to exactly just
what types of logical systems are appropriate for guiding our reasoning. Traditionally,
classical logic has been the logic suggested as the ideal for guiding reasoning (for ex-
ample, in WVO Quine's Philosophy of Logic or Michael Resnik’s Ought There to be but
One Logic). More recently, non-classical logics have been put forward as alternatives
(for example, in Michael Dummett’s Elements of Intuitionism, Alan Ross Anderson and
Nuel Belnap’s Entailment and Graham Priest's Doubt Truth to be a Liar). Even more
recently, it has been suggested that multiple logics are reasoning-guiding (for example,
JC Beall and Greg Restall’s Logical Pluralism and Stewart Shapiro's Varieties of Logic),
or that none should do so. So far, no one has addressed the way natural language
constrains and guides our reasoning. My project fills this gap.

I focus on the relationship between the meaning of the connectives (and, or, not,
etc.) in natural language and logic. By assessing what these connectives mean in
natural language, we can figure out what they must mean in the formal language if the
two sets of connectives are to be similar in meaning. I show that in order for logic to
appropriately guide natural language reasoning, the meanings of natural language and
formal connectives must be similar. Further, I show that the connectives do not have a
single meaning across all contexts in natural language, and thus there can be no single
meaning to the connectives in the formal language which guides our reasoning. This
means that the right logic to guide our reasoning depends on our context.

My dissertation is divided into four chapters. In the first, I examine the classical
and non-classical answers to the question “which logic guides our reasoning?” and find
them all wanting, as they either fail to respect the fact that reasoning takes place in
natural language, or make false predictions about the meanings of the natural language
connectives. In the second chapter, I show that the views which postulate either that
there are multiple logics which guide our reasoning or that there are none are flawed
for the reason that they do not appropriately account for the relationship between
natural language and logic. These two chapters together imply that we need some
third alternative which takes seriously the relationship between natural language and
logic. In the third chapter, I show how we can adapt an old view (from Rudolf Carnap’s
Logical Syntax of Language and Empiricism, Semantics and Ontology) to begin to
appropriately account for the relationship between natural and formal language, in
particular by starting to make steps towards using the formal language to account for
the meaning-variance in natural language. Finally, in the last chapter, I propose a
formal reasoning-guiding system which takes into account natural language and defend
it from some potential objections.

Given what my dissertation shows, we can see how the traditional view that there is one logic to guide our reasoning in all contexts is misguided, as they do not appropriately account for the relationship between natural language and logic. However, if we replace the notion of logic with the formal system I develop, we can still use logic as a reason-guiding tool.

What follows is a summary of the second half of the second chapter of my dissertation, where I address why a particular pluralist view does not meet the requirement to take natural language seriously. In particular, I show that if we take the view seriously, then we cannot have at least one logic which we would like: relevance logic. It appears in full in “Restall’s Proof-Theoretic Pluralism and Relevance Logic,” published in Erkenntnis.

Synopsis

Restall (2014) proposes a new logical pluralism. This is in contrast to the pluralism he and Beall proposed in Beall and Restall (2000) and in Beall and Restall (2006). What I will show is that Restall has not described the conditions on being admissible to the new logical pluralism in such a way that relevance logic is one of the admissible logics.

Though relevance logic is not hard to add formally, one critical component of Restall's pluralism is that the relevance logic that gets added must have connectives which mean the same thing as the connectives in the already admitted logic.[1] This is what I will show is not possible.

The argument presented in this paper relies on three facts: that Restall (2014) takes the logical connectives to be defined by their left and right logical rules in a sequent calculus (and takes these to be the same across logics), that he takes the sequents in such a calculus to be read “every evaluation which takes everything on the left to be true, takes something on the right to be true,” and that the weakening rules must be omitted from any relevance sequent calculus.

Restall's calculus encompasses at least the following rules:

I

For any connective, *, Restall claims that *L and *R constitute the meaning of, or implicitly define, the connective *. The *L and *R are the logical rules for the connective *. For example, Ø: is implicitly defined by ØL and ØR. Any rule which does not explicitly involve a connective is a structural rule. Identity, cut, weakening L and weakening R are such structural rules.[2] Though the meanings of the connectives are constituted by the rules given above, it will become important for our purposes that, according to Restall, the connective meanings are constituted by those rules read in a particular way. The rules must fit with reading the sequent as “every evaluation which makes everything on the left true makes something on the right true.”

In order for Restall to be able to demonstrate that his sequent calculus pluralism
can accommodate a relevance logic, which he suggests in Beall and Restall (2000) we
must do, I argue he must show two things. First, he must demonstrate that one can
generate a relevance sequent calculus by making changes only to the structural rules of
the sequent calculus he works with in his paper. This will ensure that the connective
meanings do not change in the new relevance calculus.

Second, Restall must show that this relevance consequence is a proper consequence relationship in his system. In their 2000 publication, Beall and Restall do this by claiming that all consequence relations are necessary, normative and formal. No such criteria are presented in the 2014 paper, but I argue that the criteria ought to be that the sequent can still be read “every evaluation which takes everything on the left to be true, takes something on the right to be true.” This, I claim, Restall cannot have happen. For, in order to remove the weakening rules, one must also remove 

But since the meta-language is classical in this case, this means that there must be some evaluation where A^B is true, but neither A nor B is, and some evaluation where A and B are true, but AvB is not. Neither of these evaluations are coherent. Thus, I claim, Restall cannot succeed in having a relevance consequence relation in his proof-theoretic pluralism. This means that Restall's new pluralism is a distinct pluralism from the model-theoretic pluralism he gives with Beall, which explicitly includes relevance logic. Only one of them can be right.

In the final sections of the paper, I discuss a number of options available to Restall. I consider the possibilities that he might provide no criteria for something being a proof-theoretic consequence relationship, or that he might make use of something like intensional quantifiers (in “every evaluation which takes everything on the left to be true, takes something on the right to be true”) rather than the usual ones. I find both of these option lacking, the first because it is ad hoc, and the second because we will not be able to produce a classical consequence relationship with these new intensional quantifiers.

References

Beall, J. and G. Restall (2000). Logical Pluralism. Australian Journal of Philosophy 78 (4), 475-493.

Beall, J. and G. Restall (2006). Logical Pluralism. Clarendon Press Oxford.

Restall, G. (2000). An Introduction to Substructural Logic. New York: Routledge.

Restall, G. (2014). Pluralism and Proofs. Erkenntnis 79 (2), 279-291.



[1]                      Classical logic is the logic we all know and love. Intuitionistic logic, which Restall explicitly adds to the system in Restall (2014), is just like classical logic, but one cannot infer a proposition from a double negation of it. Relevance logic is just like classical logic, but the premises must be relevant to the conclusion. This means that we cannot infer just anything from a contradiction.

[2]                      The two weakening rules are not given explicitly in Restall (2014). They have been adapted from Restall (2000).

 

 

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