Notes on Timothy Williamson’s George Myro Memorial Lecture:

Logics as Scientific Theories

Notes taken by Jason Zarri, 11/1/12

Is logic a “referee” for substantive disputes in metaphysics? If it is a referee it cannot be a player and be substantive itself if it is to remain neutral, and thus many think that logic must be conceived as a minimalist discipline in order to preserve its neutrality. However, this belief has not gone unchallenged; many principles of logic have been challenged on metaphysical and substantive grounds. The Law of Excluded Middle has been rejected by intuitionists and logicians who advocate certain many-valued logical systems. The principle of Non-Contradiction has been challenged by dialetheists, who think that some sentences, the Liar and its kin being prime examples, are both true and false. A group of non-analytic philosophers in Hamburg argued that Hegel refuted the law of identity. Distribution has been rejected by advocates of quantum logic. Finally, in some accounts of vagueness the transitivity of entailment has been given up. Williamson thinks that these challenges show that the disputes about the laws of logic are genuine ones; hence the conception of logic as a neutral arbiter is hopeless.

One proposed remedy to the problem raised by the multitude of logics is that the “correctness” of a logic must be relativized to specific contexts or disputes. To Williamson this suggestion seems a bit like a hypothetical situation in physics in which physicists try to settle their disputes by setting aside the disputed claims and constructing a “neutral physics” which would contain only those claims that were accepted by all parties. Would there be nothing left of the core of logic? Is a transcendental logic necessary to settle disputes?

Williamson says that there is a tradition going back to W. V. Quine and Rudolf Carnap that to adopt an alternative logic is simply to alter the meaning of the logical constants of the language, summed up by Quine in the slogan “Change of logic is change of subject”(Philosophy of Logic, 2nd ed., pp. 80-94). He there asked, rhetorically, if there could be a logic in which all the logical properties of conjunction and disjunction were switched, and suggested that this would amount merely to a different but  equivalent notation in which the connectives for conjunction and disjunction had switched their meanings (p. 81). Furthermore, Quine argued in Word and Object and other writings that there could not be a “pre-logical” tribe—a tribe whose members had unorthodox beliefs about logical laws. If there were a certain translation of the tribe’s language which committed them to the belief in the literal truth of some contradictory claims, for example, that would only show that something was wrong with the translation. Quine thus accepted a principle of charity: One should not attribute to people beliefs which explicitly contravene elementary logical laws.

Williamson rejects these views of Quine’s. If one looks at actual debates between logicians, it is evident that one can indeed have rational discussion with those who reject certain logical laws. For instance, there is really no way to make sense of what dialetheists such as Graham Priest say without attributing explicitly contradictory beliefs to them. But while Williamson could have a rational conversation with someone who denied the Law of Non-Contradiction, he could not have a rational conversation with someone who had orthodox beliefs about logical laws but was a holocaust denier. There are also positive reasons to take the phenomenon of logical disagreement seriously.

That logical disputes cannot be what they seem does not seem to jive with interpretive practice. Take intuitionists as an example, who do not accept the Law of Excluded Middle. If the intuitionistic logical constants mean something different from the corresponding connectives of classical logic, one should be able to have both sets of connectives in a single language. But a result of Karl Popper’s shows this to be impossible.

There is also a more general reason to take logical disagreement seriously: These disputes are conducted in natural language. We have strong reason to think that in a natural language meaning is in part socially determined. Call this ‘social externalism’. Williamson raises, but does not answer, the question of whether social externalism extends to logical connectives. Also, in modal logic the status certain logical principles–e.g., if something is possibly true, is it necessarily true that it is possibly true?—is left unsettled by ordinary linguistic practice. In the rest of the talk, Williamson proposes to argue that logic is much more methodologically similar to science than many philosophers have assumed, and the methodological principles that hold sway in science are applicable to logic, too.

Williamson next gives a rough outline of Alfred Tarski’s account of logical consequence and logical truth. Williamson says his approach resembles a model-theoretic approach, but he will not use such an approach himself. The general picture is this: First, we choose a set of terms of the language that are to be taken as primitive—that is, undefined. Call these logical constants. One then replaces all of the non-logical constants by variables of the appropriate type. Once we’ve finished that replacement, we determine whether a sentence is a logical consequences of a set of premises by checking to if every assignment of objects to the variables that makes all of the premises true also makes the conclusion true. If so, it is a logical consequence of that set of premises. If a sentence is true on every assignment whatsoever of values to its variables, it is a logical truth, and can be thought of as a logical consequence of an empty set of premises. (Personal Note: The clause to the effect that “every assignment of objects to the variables that make all of the premises true also makes the conclusion true.”  is then vacuously satisfied because in that case there are no premises, and hence “all of them” are made true by the assignments in question.) Williamson gives an example of this. Consider the following sentences:

1a. All furze is gorse.

1b. All furze is furze.

2a. All F is G.

2b. All F is F.

(1a) corresponds to (2a), and (1b) to (2b). The former two are not logical truths, but the latter two are. This is because (1a) and (1b) are instances, respectively, of (2a) and (2b), and some assignments to the predicate variables F and G in 2a) make it false, while on the other hand all assignments to the predicate variable F in (2b) make it true.

The general proposal Williamson wants to defend is that logic is by far the science best equipped to determine what is true in general (by which I think he means that logic is best equipped to determine what must be true, and what follows from what, by using a method similar to that of Tarski’s).

Williamson’s says that, whatever we choose to regard as logical constants, there will always be some sentences that are true on all replacements of their variables, and similarly for logical consequence. Which terms are to be regarded as logical constants is to a large extent arbitrary. Williamson himself is interested in modality, so he decides to treat modal operators as logical constants. One who doesn’t share that interest is free not to treat them as logical constants. One must give an interpretation to all the logical constants, including quantifiers. All questions of interpretation must be settled.  Furthermore, the domains of the quantifiers must be absolutely universal.

If Williamson is right, what kind of methodology is necessary for logic as a discipline? In the long run, Williamson thinks, one must be concerned not just with the interaction of different logics with each other but also in the interaction of different logics with other theories. Also, principles like simplicity, consistency with what is already known, and (non-causal) explanatory power will play a role in one’s choice of logic. This is an abductive methodology, just as (Williamson says) some areas of mathematics also have an abductive methodology. Williamson has no pretensions to logic’s neutrality. It is not necessary to have a “transcendental” logic to investigate the questions about logic; one merely needs the much more standard scientific abductive methodology. And though logic is generally an abductive discipline, it is a mistake to raise its status to that of a natural science.

What light does this cast on classical logic? Some object that classical logic is too strong—it proves too much. But if the proper method for assessing logics is abductive, the strength of classical logic is a strength rather than a weakness. Williamson advocates classical logic, but not because he thinks it can be known to be true a priori, but rather because of the same kinds of abductive considerations that are in effect in scientific methodology.

Bibliography

Quine, W. V. Philosophy of Logic, 2nd ed. Harvard University Press, 1986

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