Reply to Quine's & Restall's Meaning Objection to logical pluralism

Introduction

This post is a little different from some of my others. It started as a review of Greg Restall's paper "Carnap's Tolerance, Meaning, and Logical Pluralism", but a few factors convinced me to shift my focus towards replying to a specific point, rather than reviewing the paper as a whole. 

First, a great deal of what I had to say about the paper related to this particular argument against Carnapian logical pluralism (a view that's quite close to my own). I largely either agree or have no present thoughts on the rest of the paper. In particular, Restall aims to set out how his view (and that of Beall) is different from Carnap's. He does that quite successfully and I'm certainly more clear on the nature of his position as a result of reading the paper.

Second, I had a paper R&R'd by a journal a few weeks back. The result was that my timetable was suddenly thrown off balance and I needed to find a way both to work on my current paper (a paper on logical pluralism) and do a review. Given that my reply to Restall's and Quine's argument is a part of that logical pluralism paper but also stands alone, I decided to write it up informally as a post here.

I'm not aware of any current published literature that explicitly deals with this argument, though the impression I get is that most people working on the philosophy of logic are not convinced by it, though this is purely anecdotal.

I'd also like to thank the attendees at the DocQWIP event at the University of Vienna for their thoughts, following a talk I gave on this material.

The review proceeds in three parts. Part one outlines Carnapian logical pluralism, part two explains Quine's and Restall's objection and part three explains where I think it goes wrong.

Carnapian Logical Pluralism

As I use the term, logical pluralism is the view that there is more than one correct logic. As I explained in my review of Blake-Turner's paper, the term "logic" is used in different ways across the literature. Some use it as a success term, where all logics are, by definition, correct logics. Some, myself included, use it as a neutral term. Certain logics might be wrong, unhelpful or even bad but they are, nevertheless, logics. Consequently, people who use "logic" as a success term understand logical pluralism to be the claim that there are many logics. I won't use that terminology here.

But what are logics and how can they be correct? One can model logics relatively simply as sets of statements about which inferences are valid/acceptable and which are not. Perhaps with the additional requirement that this set be generated in some systematic manner. A logic is correct iff it gets the facts about what inferences are valid/acceptable right (A correct logic might not predict all the valid inferences. See, for instance, issues regarding incompleteness in higher order logics).

What are the facts about valid/acceptable inferences? The Carnapian answer is that these facts are not language-independent. One might construct a language that models classical logic just as easily as one might construct a language that models non-classical logics. For example, one might allow gappy predicates into a language and this forces it to become at least a three-valued logic. Carnap's own example is between languages with our without types.

For the Carnapian, what logic is correct for a language is a result of the kinds of semantic objects present in that language. It is just a matter of convention if a language has, for example, empty names, fuzzy predicates, higher-order quantifiers, etc. One can alter one's language (or construct a new language) to contain widely varied combinations of the relevant sorts of semantic objects.

But this means that a kind of logical pluralism is true. Facts about entailment are language relative meaning that facts about logical correctness are language relative. Consequently, there are many logics that are correct for some languages. Logical pluralism is true.

There's a sense in which logical nihilism is also true on a Carnapian view. There are no logics that are absolutely and universally correct. Really, what's gone on here is that the naive term "logical pluralism" is ambiguous for the Carnapian. For them, logical correctness is language relative. This means that asking "How many correct logics are there?" is ambiguous between three non-ambiguous questions:

• Wide scope: How many logics are there that are correct for all languages?
• Medium scope: How many logics are correct for some language?
• Narrow scope: For any given language, how many logics are correct?

About the wide scope question, the Carnapian is a nihilist. About the medium scope question, the Carnapian is a pluralist. Though, I believe the view as a whole has more the flavour of a pluralist view than a nihilist one, hence why I refer to the Carnapian as a logical pluralist. 

What the Carnapian should think about the Narrow scope question is a question for another day. Carnap was a narrow scope monist. I personally disagree with that conclusion, but won't argue for that here.

Quine's & Restall's meaning objection

Consider a disagreement between a classical logician and a paraconsistent logician (in particular a defender of LP) about the truth of disjunctive syllogism. 

The classical logician argues that the truth conditions for the "or" connective are as follows: "X or Y" is true iff at least one of X and Y is true. So if "X or Y" is true, then at least one of X and Y is true. But suppose Y is false, it follows that X is true. Hence disjunctive syllogism holds.

The paraconsistent logician responds that whilst that is the case for the classical "or" connective, the "or" that they use is the "or" connective of LP (the logic of paradox). LP has the truth values T, F and B (for "both"), with designated (informally, "truth-like") values of T and B. An argument is valid iff there is no assignment upon which all the premises are designated values but the conclusion is not. Consider the order T>B>F. The value of X or Y is the maximum of the values of X and Y. Negation works classically for T and F and takes B to B (notice this means that both X and its negation can be a designated value at the same time). Consider the assignment where X is B and Y is F. The value of X or Y is max(B, F)=B. This is a designated value. The value of not X is B, also a designated value. Yet the value of Y is F, not a designated value. Hence disjunctive syllogism is invalid in LP.

It seems, on the face of it, that there is a disagreement about the validity of disjunctive syllogism.

However, Quine and Restall argue that this is not the case. If one assumes that a truth-conditional theory of meaning is true (i.e. the meaning of a sentence is its truth conditions), then the meaning of the classical "or" is different from the meaning of the LP "or" (similarly so for "not"). Let "orc" mean the classical "or" and "orp" mean the LP's "or". The sentence "X or Y" is ambiguous between "X orc Y" and "X orp Y". The disagreement is purely a linguistic confusion and not substantive. Everyone agrees that disjunctive syllogism holds for "orc" but not for "orp". But if there is no substantive disagreement, so goes the argument, then there's no logical pluralism.

Quine and Restall take a logical pluralist to be committed to the claim that there are inferences about which there are a plurality of correct answers as to if the inference is valid. But as soon as the linguistic confusions are clarified and it's clear which system the connectives, etc in the sentence belong to, then, on the Carnapian view, there is no disagreement about validity to be found.

Restall puts this in quite a clear way in his paper. He states that what the logical pluralist should want is a genuine plurality of correct answers as to if a certain conclusion is entailed by certain premises. What the Carnapian view offers is ambiguity about the content of the premises (e.g. if "or" should be understood as "orc" or "orp") followed by monism once the ambiguity is resolved.

Putting this together, the argument is as follows:

P1: If Logical Pluralism is true, then there are some validity statements with a plurality of correct answers.

P2: If a validity statement has a plurality of correct answers, then this is because of a truth-conditional ambiguity either in one of its premises or in its conclusion.

P3: If a strict truth-conditional theory of meaning is true, then there are no truth-conditional ambiguities in sentences. (Any apparent ambiguity is really just an ambiguous sentence, i.e. an inscription that fails to uniquely pick out a sentence to express)

P4: Just such a strict truth-conditional theory of meaning is true

C: Logical pluralism is false

As a brief side note, Restall is a logical pluralist, despite advancing this argument against the Carnapian logical pluralist. His argument is that his view allows him to reject P2 whereas the Carnapian is committed to it. I have some reservations as to whether his view escapes the argument, but I certainly don't think the Carnapian is committed to P2.

My response

This argument is, I think, unconvincing. I reject P1,2 and 4. I believe my responses against P1 and P2 are relatively strong. My response against P4 is more speculative.

Starting with P1, suppose one accepts that Quine and Restall are correct and, on the Carnapian view, any apparent plurality in the correctness of validity statements is really just the result of linguistic ambiguity. Suppose that once it's clear how each symbol is to be interpreted, there is no remaining validity-pluralism. It does not then follow that logical pluralism is false.

(medium scope) Logical pluralism is the claim that multiple logics are correct for some language. If Quine and Restall are correct, then there is no disagreement (on the Carnapian view) about the correct inference rules for particular connectives. This does not mean that every language has every connective. Languages with different connectives might still have different correct logics.

The classical logician, for instance, can entirely concede that were they to add "orp" to their language, it wouldn't satisfy disjunctive syllogism, but simply contest that their language does not contain "orp". Similarly so for the paraconsistent logician. They accept that disjunctive syllogism is valid for "orc" but do not have "orc" in their language. Because of the respective languages choices of connectives, different logics are correct for each of them.

In other words, Quine's and Restall's criticism show that differences in logical correctness are just a matter of which connectives a language has. But, for the Carnapian, this is a feature, not a bug. They are quite happy if logical correctness is really just a matter of semantic convention and disagreements relating to logic-choice are really just highly general meta-semantic negotiations, rather than more substantive disagreements.

Moving to P2, not all differences in the validity of inferences come directly from differences in the truth conditions of the premises or the conclusion, as is the case with connectives. It's clear to see why one might think that "orc" and "orp" are not the same connective and hence analogue sentences containing them are not translations of one another, but the same is not true for all logics objects. 

For example, consider identity in negative free and classical languages. In classical logic, every term is paired with exactly one object in the domain. A classical interpretation applied to terms is a proper function from terms into the domain of the model. In free logics, this is relaxed and there are allowed to be empty terms. Negative free logics are logics where empty terms yield false atomic propositions (as opposed to positive free logics, where they yield true ones, or many-valued free logics where they yield some other value).

In classical languages "a=b" is true iff a and b are assigned the same object in the domain. In free languages "a=b" is true iff a and b are assigned the same object in the domain. The truth conditions of identity are the same in both classical and free languages and hence the inscription "a=b" expresses the same sentence in both languages, yet the languages differ in what inference rules they allow for identity.

Identity introduction is classically valid but negative-free invalid. In classical logics, every term must be assigned an object, so every term is always assigned to the same object in the domain as itself. In negative free logics, if a term a is not assigned to anything, then "a=a" will turn out false.

This means that the disagreement between classical and free logicians over if identity introduction is a valid inference is not merely a confusion over an ambiguity. "a=a" means the same thing in classical and free languages, yet the languages differ on if "a=a" is a theorem or not.

A comparable example also holds for quantifier rules in free and classical logics, depending on how one defines one's variable assignments.

Lastly to P4. My reply here is a great deal more speculative than the other two. I suspect many readers will disagree with some of the details of what I have written here (I suspect my future self will disagree with some of the details as well!). What I'm trying to convey is more of a broad strategy for approaching this premise in a non-classical context.

In any setting that involves many-valued logics, a strict truth-conditional theory of meaning will be overly conservative. I say strict, because it's still likely the case that truth conditions play an important role in determining meaning, just not quite the role that two sentences mean the same thing iff they have the same truth conditions.

Consider, for instance, a fuzzy and a classical language both with "is bald" predicates. Let "Bf" be the fuzzy bald predicate and "Bc" be the classical bald predicate. Classical baldness will be given an epistemicist treatment such that even though there is a strict cut off between baldness and non-baldness, it is unknown to speakers. Even though all classical language users are confident that everyone is either bald or they are not, there are borderline cases where they will only offer a degree of confidence in the claim that someone is bald. "Bf" is a fuzzy predicate and be true-to-an-extent of various individuals. Let the extent to which "Bf" is true of some x equal the degree of confidence in which a classical speaker asserts "Bc(x)".

In my opinion, "Bf" and "Bc" are the right translations of one another. They play the same conversational role and are asserted in all the same cases (though the consequences of those assertions are different). Yet on a strict truth-conditional theory of meaning, they mean different things as they have different truth conditions. There is a great deal more can could be said here which I don't want to go into but suffice to say this: I think a more appropriate way of looking at meaning in the context of non-classical logics, where truth can behave so differently across different languages making translation difficult, is to have a kind of truth-conditional counterpart view of meaning/translation. For two sentences to mean the same thing it suffices that they are one another's nearest truth-conditional match in the respective languages. But if this is the case, P4 is false.

On a truth-conditional counterpart view, the analogue of P3 is false. Even though "orc" and "orp" have different truth conditions, they are one another's truth-conditional counterparts and express the same content (though that content behaves differently in different languages). The sentence "X or Y" is then not ambiguous between "orc" and "orp" but rather expresses this common content shared by "X orc Y" and "X orp Y".

A counterpart view of meaning would have some disadvantages. Synonymy would become a non-transitive relation. If this is too stark a consequence, a weaker option would be to argue that "Bf" and "Bc", whilst not synonymous, have some common core to their meaning; and it's about this common core that advocates of different logics can non-ambiguously disagree.

In summary, I think there are pretty strong reasons to reject P1 and P2 of the meaning objection and some speculative reasons to reject P4. Rejecting P1 is the strongest response, in my opinion. Quine & Restall seem to miss what the commitments of the logical pluralist need to be. At worst, the crux of their objection is a feature, not a bug.

Summary

A well-known objection to Carnapian logical pluralism is the meaning objection, as advocated first by Quine and later by Restall. It argues that there is no real disagreement between advocates of rival logics, on the Carnapian view. Everyone agrees that classical inference rules are correct for classical connectives and that (for example) LP inference rules are correct for LP connectives. As there is no disagreement about what inferences are valid, there is no pluralism. 

However, this argument fails in several ways. First, the logical pluralist is not committed to the kind of validity pluralism that Quine & Restall take them to be. Second, not all differences in validity can be reduced to differences in the content of the premises or conclusion. Sometimes they come down to the more holistic semantic features of the language, such as rules governing how terms function. Quine & Restall's objection doesn't generalise to all differences between logics. Lastly, I've given some speculative reasons to reject the strict truth-conditional view of meaning that Quine and Restall rely on.

The meaning argument can be found in the following places:

• Quine, W. V. O. (1970). Philosophy of Logic. Harvard University Press.
• Restall, G. (2002). Carnap’s tolerance, meaning, and logical pluralism. Journal of Philosophy, 99(8):426–443.

Quine needs no introduction!

Greg Restall is Professor of Philosophy at the University of St Andrews. His personal website can be found here: https://consequently.org/. He is most known in the philosophy of logic for his joint work with J.C. Beall on logical pluralism.

The picture is by Frank Vassen from Brussels, Belgium, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0>, via Wikimedia Commons. Chosen not because of its relevance to the subject matter, but because birds are pretty.

I welcome comments or friendly criticism below, or you're welcome to email me at gareth.pearce@univie.ac.at

New reviews are advertised over Twitter so follow me @GarethRPearce for updates