Wyatt Payette 2018 Le A
Wyatt Payette 2018 Le A
Wyatt Payette 2018 Le A
Abstract
Disputes about logic are commonplace and undeniable. It is sometimes argued that
these disputes are not genuine disagreements, but are rather merely verbal ones.
Are advocates of different logics simply talking past each other? In this paper we
argue that pluralists (and anyone who sees competing logics as genuine rivals),
should reject the claim that real disagreement requires competing logics to assign
the same meaning to logical connectives, or the same logical form to arguments.
Along the way we argue that ascriptions of logical form, as well as connective
meaning, are always theory-relative.
Keywords: logical form, logical pluralism, meaning variance
1. Introduction
The classical logician and the intuitionist dispute excluded middle, while
the relevant logician and the classical one dispute ex falso quodlibet (i.e.,
explosion). But are these disputes genuine disagreements, or to put the
same point differently, are these logics genuinely rivals to one another?
Quine is notoriously taken to think otherwise, writing that “[this] is the
deviant logician’s predicament: when he tries to deny the doctrine he only
changes the subject” (Quine 1986, 81).1 If correct this appears to be a
significant challenge for those forms of logical pluralism which require that
there be at least two logics that are rivals and correct.2 But if changing one’s
logic changes the subject, it seems that the first of these conditions can
never be met. Rivals, it is natural to assume, must be rivals with respect to
something, that is, they must genuinely disagree.
In this paper we argue that the pluralist should not concern themselves
with this objection. Perhaps surprisingly, however, the reason the pluralist
1
The famous slogan, “change of logic, change of subject” appears as the section heading
on page 80 of Quine (1986).
2
See Restall (2014) and Hjortland (2013) for examples of pluralists who both take
seriously and attempt to meet this challenge.
It is worth noticing that Quine, while ultimately dismissing so-called deviant logics,
3
does not himself think that the mere fact that they involve a change of subject settles
the important logical questions: “whoever denies the law of excluded middle changes the
subject. This is not to say that he is wrong in so doing” (Quine 1986, 83).
4
See Haack (1978) and Hjortland (2013) for discussion
5
This explication of ‘verbal dispute’ comes from Jenkins (2014).
Logical Pluralism and Logical Form 27
6
Even those who reject the idea of applying logic to natural language can just take T as
the empty relation. But this discussion cannot get started unless one takes logic to have some
application.
7
See part 2 of Béziau (2007).
28 Nicole Wyatt & Gillman Payette
One way in which logical pluralists have attempted to resist the claim
that logical disputes are merely verbal is by arguing against the charge that
change of logic requires a change in meaning. One strategy is to advocate
minimalism as to the meaning of the logical connectives.10 On this approach,
agreement as to the so-called meaning-constituitive properties for two con-
nectives c1 and c2, is sufficient to block the claim that the subject has been
changed. Two logics that agree on all the meaning-constituitive properties
of a pair of connectives, are thus in a position to be genuine rivals. If both
logics are also correct, pluralism has been established.
The difficulty for minimalism, as Putnam (1979) and Field (2008) have
discussed, is in distinguishing the meaning-constitutive properties of a
11
This is an idea that goes back at least to Dummett (1991).
12
An important difference between Hjortland and Paoli’s approach is that Paoli counts
the rules governing logical constants other than the one of interest — c, say — towards the
global meaning of c, whereas in Hjortland’s example the systems agree on the rules govern-
ing all the constants. See also Paoli (2003).
13
In LP the value i is usually interpreted as meaning that a sentence with that value is
true and false.
14
This treatment of these two logics was originally developed in Baaz et al. (1993).
30 Nicole Wyatt & Gillman Payette
Let’s call this argument Finn. What toy examples don’t display, but argu-
ments in an actual C should display, is the context, etc., that is necessary
for understanding the argument.15 But toy examples are useful for illustrat-
ing the problem. What we need to do is specify what is relevant in relating
15
As a brief philosophical aside, this is what made mathematics such an easy target
for the development of formal logic in the late nineteenth century. Little context is needed
to make sense of a mathematical argument. And it is what has made applying formal logic
to social philosophy so difficult: e.g., the implicit understanding underlying our many and
varied moral judgments.
Logical Pluralism and Logical Form 31
P ∨ Q P DS
Q
formulas are governed by the same rules and given the same semantics. No
amount of disambiguation would settle their dispute over the validity of
that argument: the meaning of ‘or’ and ‘not’ are not disputed. Genuine
disagreement between correct (adequate, etc.) logics requires a common
core of agreement, and a shared account of the logical form of natural lan-
guage arguments is intended to provide that core.
If successful, what Hjortland’s intra-theoretic pluralism accomplishes is
threefold. First, it offers a robust account of logical form that is taken to be
shared between two rival logics. Second, this shared account of logical
form is used to block the charge of merely verbal disagreement. And finally,
it provides a way to locate the source or foundation of the disagreement
between two logics. In the example above the disagreement concerns the
structure of the consequence relation. Hjortland allows that this disagreement
over consequence is a form of meaning-variance, but maintains that it does
not collapse into a merely verbal dispute.
Notice, however, that the argument against meaning variance requires
that the account of logical form be independent of the conceptions of logical
consequence. If the logical forms depended on the notion of logical conse-
quence, one could not claim that the forms assigned by the two systems to,
say, Finn were the same.
What we argue in the sequel is that the apparent robustness of the shared
account of logical form in intra-theoretic pluralism is a technical mirage;
the shared account is mathematically robust, but not philosophically robust.
However, we ultimately agree with Hjortland that the charge of merely verbal
disagreement is misplaced. To make this argument we must digress at some
length on the topic of logical form before returning to the possibilities of
pluralism and disagreement. In the next section we will discuss the relation-
ship between mathematical models and the concept of logical consequence
for natural language. The upshot of this discussion is to argue that there is
no independent notion of logical form of the sort intra-theoretic pluralism
needs.
4. Logical form
On this view what matters to consequence is all and only what is repre-
sented in Frege’s formal language. Church also captures the idea that we
must isolate those parts of the argument that are relevant to consequence
succinctly in his Introduction to Mathematical Logic:
The argument, it may be held, is valid from its form alone, independently
of matter … The reasoning may be right though the facts be wrong, and
it is just in maintaining this distinction that we separate the form from the
matter. (Church 1956, 2)
16
Frege has historical priority, but we name it after Church for two reasons. First, as a
practical matter, two Frege’s Principles seemed to us to be sufficient. Second, as we will
see subsequently, Church is particularly insightful as to the philosophical significance of CP
for logic.
34 Nicole Wyatt & Gillman Payette
17
Getting the intuitive consequences ‘right’ is also central to Russell’s critique of
Meinong’s treatment of sentences like (1), though of course metaphysical issues are at play
there as well.
Logical Pluralism and Logical Form 35
The upshot of this quote from Church is that formal languages, under-
stood as representing logical forms, and systems of analysis cannot be
separated. This holism has the consequence that there is no such thing as a
logical form independent from a logical system.19 We will return to this
point in the last section of the paper.
Returning to our main argument, we can put the general situation roughly
as follows. What one needs for pluralism without meaning variance is that
the common logical forms assigned to utterances by rival logics must be
compatible with the logics’ different consequence relations. It follows that
there must be something that makes a difference to whether u is a logical
consequence of u which isn’t in the logical form. In contrast, CP requires
that all the things that can make a difference to consequence are included
in the logical form.
In short, the pluralist who wishes to deny meaning variance must reject
CP as inconsistent with their claims about logical form. However it’s not
sufficient to reject CP — the pluralist must provide us with an alternate
adequacy condition for ascriptions of logical form to natural language utter-
ances and arguments.
18
The idea that the ‘formality’ of formal logic consists essentially in the adoption of a
particular system of logical analysis is in itself notable, and suggests that Church, despite
his earlier invocation of the form/matter distinction, might ultimately be conceiving of the
formal as pertaining to rules. See Dutilh Novaes (2011) for a discussion of this idea of
formality and its various subtypes.
19
Related observations concerning the holism of logical analysis have been made by
others. Cf. Cargile (2010).
36 Nicole Wyatt & Gillman Payette
definition of consequence. That is, you are given the three-sided sequent
calculus, and the three valued model theory, along with a mapping between
an appropriate fragment of natural language and the system. This system,
you are told, captures the logical forms of these these natural language
sentences. On what basis could you assess this claim?
One might think that logical forms can be read in some way off the
grammatical properties of natural language sentences. This approach comes
in both a naive form, in which logical form can be read off the surface
grammar, and a more sophisticated version in which syntactic theory will
eventually deliver the logical forms of utterances.
The problem with either approach is that there is nothing about the
syntax or grammar of natural language which can distinguish the logical
operators from the non-logical ones. The concept of logical form at play
in intra-theoretic pluralism depends upon this distinction. More precisely,
it depends upon the assumption that only some of the syntactic properties
of natural language are logical properties. To give a simple example,
standard first-order languages do not allow us to distinguish between ‘Finn
loves the tree-house’ and ‘The tree-house is loved by Finn’, despite their
grammatical differences. What makes this acceptable is the (no doubt cor-
rect!) assumption that the differences between the sentences are not logical
ones.
What motivates the division between the logical and non-logical parts of
a language are precisely convictions about what matters to consequence—
that is, we are implicitly relying on Church’s Principle when we make this
division. So even if the defender of intra-theoretic pluralism is in possession
of a complete syntactic theory for the natural language, they do not have
independent criteria for ascriptions of logical form unless they also have a
theory of which syntactic properties are also logical properties which itself
does not rely on CP.
But how could we recognize the logical properties of an utterance
independently of a notion of logical consequence? This is, we admit, argu-
ment by rhetorical question. But it is genuinely unclear to us what response
could be given here, and given the dominance of CP as the criterion of
adequacy for theories of logical form, the burden is clearly on the defender
of intra-theoretic pluralism to provide an alternate.
Another strategy for someone who wants to avoid merely verbal
disagreement would be to find a way of describing consequence which
doesn’t invoke logical forms or connective meaning at all. A ready option
is available to us: necessary truth preservation. On the necessary truth
preservationist account there is no privileged set of logical connectives.
Thus, it might seem that CP’s role as an adequacy criterion can be rejected
by advocates of this approach. We will consider this strategy in the next
section.
Logical Pluralism and Logical Form 37
20
We can see this in the work of Etchemendy (1990) among others.
21
In her case a formal language would simply be a free algebra of some sort.
22
A limited example would be: Bachelor(John) entails Unmarried(John) iff in all
models M where I(Bachelor) ⊆ D, I(Unmarried) ⊆ D, I(John) ∈ D and I(Bachelor) ⊆
I(Unmarried) are satisfied, if Bachelor(John) is true, then so is Unmarried(John).
38 Nicole Wyatt & Gillman Payette
of a logical system. The formulas in her system do not carry any burden of
representing logical structure since they only have structure because of the
semantic constraints the models impose.23
If the model theory is now the representation of logical form, we can ask
our basic question about adequacy conditions, which caused problems for
the Tarskian approach. What makes a model theory adequate? This will
again be whether it gets the logical relations in natural language right.
And as we saw above, the only plausible adequacy condition on an account
of logical relations is whether we have captured all that matters to conse-
quence. But that is just Church’s Principle in a slightly different guise:
model theory includes everything relevant to consequence.
On the Tarskian approach it was more plausible that logical forms qua
formal languages could play the role of an independent mediator between
logical systems because the formal languages were thought to be autono-
mous of the rest of the system. Here the representations of logical form
and the thing used to determine the consequence relation are one and the
same.
For similar reasons the necessary truth preservationist approach implic-
itly relies on a relative of CP as an adequacy condition. Preservationist
accounts give a theory of truth and truth preservation for a given class of
utterances C.24 This usually proceeds by offering a mathematical framework
where the truth conditions of utterances and the consequence relation are
represented in a formal system. But regardless of the framework used, the
system’s adequacy is a function of its explanatory power with respect to
class C.25 In particular, a preservationist theory is adequate to the extent that
it captures everything relevant to necessary truth preservation, i.e., consequence.
But again, this is just CP in a slightly different form.
Recall that what intra-theoretic pluralism needs is some common ground
between two logics that is determined independently of the consequence
relation. On a Tarskian approach logical form is the natural candidate, but
it turns out that our adequacy conditions for logical form are not independ-
ent of a particular consequence relation. On a preservationist approach,
which eshews the fundamentality of logical constants for consequence,
it is the theory of truth (often given in terms of model theory of some kind)
which plays this role. Preservationist accounts in general define the conse-
quence relation as something which preserves some property. In truth theo-
retic cases, that property is truth. However, what counts as truth can also
be determined by what it is that a truth preservationist account preserves.
She does go on to show how one can develop a theory of argument schema which is
23
Think of the example of SK3 and LP above. They are both truth preserva-
tionist accounts of consequence, but what counts as truth in the underlying
semantics is different. Thus, the adequacy conditions for those truth theories
also depend upon a particular account of consequence.
Suppose we want to evaluate the claim that f captures the logical form
of u. The upshot of the discussion so far is that, regardless of the nature of f,
we need two other pieces of information. First, we need some pre-theoretic
idea of the logical relations between the utterances, and second we need a
theory of the relationships between the f s. The claim that f captures u will
be satisfactory to the extent that the theory matches with the pre-theoretic
intuitions.
What emerges from our discussion is that evaluation and justification of
systems of logical analysis must be understood holistically: the adequacy
of each part of a logical system L, ,T with respect to a reference class
C depends upon the interrelations of the parts. At the end of section 4 we
observed that Church shares this view of logic.
Whatever the meanings of the operators in LP, they are the same no mat-
ter what mathematical treatment of the logic we use. What makes one logic
different from another logic is that the extensions of the consequence rela-
tions are different (up to isomorphism in some sense). A symbol has a
particular meaning in a system because it represents something which matters
to consequence in that system. Minimalism and intra-theoretic pluralism
justify claims about sameness of meaning on the basis of overlap between
the mathematical treatments of logical operators combined with agreement
on the assignment of formulas to utterances. But systems are holistic and
logical form only makes sense within the context of a system of analysis.
Hence, sub-components of mathematical models cannot reasonably be
taken to have significance in isolation. And ipso facto, the logical forms for
SK3 and LP are different even if they can be given an overlapping math-
ematical treatment.
If intra-theoretic pluralism were the best hope for avoiding the charge of
meaning variance, the outlook for real logical disagreement, and thus for
logical pluralism, might seem bleak. Our arguments support the view that
logical form is a system relative notion, and furthermore, that logical sys-
tems can only be evaluated and compared holistically. Attempts to explicate
these disagreements in terms of shared notions of logical form—and in
particular in terms of agreement about the meanings of logical operators
and the logical forms of natural language utterances—are futile in light of
system relativity and holism.
40 Nicole Wyatt & Gillman Payette
But it seems clear that there are genuine disagreements in logic. Mini-
malism and intra-theoretic pluralism attempt to defend this truism against
the charge that all disagreements are merely verbal in nature. But their
approach makes methodological assumptions that are problematic.
CP is implicit in our logical methodology. Once we make it explicit we
see why connective meaning seems so important. According to CP, logical
forms must contain all the vehicles of consequence. The mathematical
methodology of contemporary logic is to represent logical forms with
formulas, and the consequence relevant parts of formulas are the logical
operators. Formulas are parts of formal languages, and intuitively terms
are the same in languages when they have the same meanings. So in order
to meaningfully disagree about consequence, two logics would have to
give the same meanings to the logical operators, but disagree on their
consequences.
This focus on the logical operators distracts us from seeing the scope and
nature of real disagreement between logics. Suppose, for a moment, that
the classicist and the intuitionist accept the claim that they are talking past
each other (changing the subject!) and allow that there are two kinds of
negation, disjunction, etc. This will not, contra the claim of merely verbal
disagreement, actually dissolve the dispute. The intuitionist will continue
to maintain that classical negation does not matter to consequence, and the
classicist will say the same about intuitionist negation.
This point about the nature of logical disagreement was actually clearly
seen by Quine, if not more generally. He writes “the intuitionist should not
be viewed as controverting us as to the true laws of certain fixed logical
operations, namely negation and alternation. He should be viewed rather as
opposing our negation and alternation as unscientific ideas, and propounding
certain other ideas, somewhat analogous, of his own” (Quine 1986, 87). Of
course Quine was no pluralist—he took it for granted that not both intuition-
ist and classical logic could be properly scientific ideas, and advocated the
cause of the latter strenuously. But there is nothing about this conception of
logical disagreement as involving two or more largely incomparable systems
of logical analysis that precludes the possibility of multiple correct systems.
The minimalist and the intra-theoretic pluralist attempt to meet the charge
of merely-verbal disagreement on its own grounds; accepting the claim that
logical disagreement must, if it is to exist at all, take the form of disagree-
ment with respect to some common pool of logical operators. But framed
this way, as we have seen, the problem is insoluble. On our view we should
instead reject the terms of the debate.
The way to resolve this problem is to view competing logical systems
as competing research programs in science. We will not give a recipe for
how to compare research programs in logic. We will simply say that inso-
far as comparisons between research programs in science are generally
Logical Pluralism and Logical Form 41
Acknowledgements
We would like to thank the referees for this journal and the audience at
the 2015 meeting of the Society for Exact Philosophy at McMaster University
for helpful comments and discussion. We would also like to thank Patrick
Allo for his helpful comments. Gillman Payette would like to acknowledge
the support of this research from the Killam foundation through a Killam
postdoctoral fellowship, and from the Social Sciences and Humanities
Research Council of Canada through a Banting postdoctoral fellowship.
References
Nicole Wyatt
University of Calgary
nicole.wyatt@ucalgary.ca
www.nicolewyatt.net
Gillman Payette
University of British Columbia
gpayette@dal.ca