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I am having trouble understanding what 'extra-logical' actually means in the context of philosophical logic.

Case in point: Bueno and Colyvan argues in their paper Logical Non-Apriorism and the ‘Law’ of Non-Contradiction that logic is non-apriori, and that logic is revisable on 'extra-logical' grounds:

The idea is that it is possible to revise logical principles (or logical rules) on the basis of extra-logical considerations—which include empirical considerations. In other words, extra-logical considerations play a role in the selection and evaluation of logical principles (or rules).

...by ‘logical nonapriorism’ we simply mean that extra-logical considerations come to play in theory choice in logic. As it turns out , we also think that logic is non-apriori in a stronger sense (in that empirical considerations come to play). Our main purpose in this chapter, however, is to defend an account of theory change in logic that allows, and makes sense of, debates about the law of non-contradiction. It’s important for our case that the role of extra-logical considerations in these debates is appreciated. Some of these considerations are empirical, while others aremerely extra-logical.We find it convenient to use the term‘non-apriori’ to include both, but nothing hangs on this admittedly non-standard usage.

Obviously this term literally means 'something other than logic'; but exactly what is this referring to? It seems to be a catch-all phrase that literally means just that, and includes but not limited to empirical factors.

Some of the examples they gave are:

  1. Distributivity failing in quantum mechanics
  2. T-schema
  3. Natural language being semantically closed, an example they gave: 'The Earth is round' being grammatical in English is an empirical fact

The first (and perhaps to a certain extent, the third) seems to be empirical, but it is not clear exactly why the second is 'extra-logical'?


As far as I understand, this distinction is usually used to distinguish between logical constants, such as material conditional and conjunction, and other variables such as propositional constant. And this is a distinction that goes back to Tarski.

The problem is, even Tarski seems to admit in his paper On the concept of logical consequence that this distinction is problematic:

Underlying our whole construction is the division of all terms of the language discussed into logical and extra-logical. This division is certainly not quite arbitrary. If, for example, we were to include among the extra-logical signs the implication sign, or the universal quantifier, then our definition of the concept of consequence would lead to results which obviously contradict ordinary usage.

On the other hand, no objective grounds are known to me which permit us to draw a sharp boundary between the two groups of terms. It seems to be possible to include among logical terms some which are usually regarded by logicians as extra-logical without running into consequences which stand in sharp contrast to ordinary usage. In the extreme case we could regard all terms of the language as logical. The concept of formal consequence would then coincide with that of material consequence. The sentence X would in this case follow from the class K of sentences if either X were true or at least one sentence of the class K were false.

In any case, as far as I am aware Tarski never provided a precise definition for what it means to be 'extra-logical' either.

So what does 'extra-logical' actually mean? (Especially in the context of Bueno and Colyvan's paper)

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  • Example: physical theories e.g. Quantum Mechanics. Jun 11 at 10:46
  • IF we assume that "logic" is a priori, based on intuition alone, extra-logical means grounded on empirical evidence. Jun 11 at 10:47
  • See e.g. the debate about Russell's Axiom of Infinity and Reducibility axiom that were originally motivated in the construction of (High-order) logic as "needed" and then rejected because based on extra-logical considerations. Jun 11 at 10:50
  • @MauroALLEGRANZA I also thought that extra-logical just means empirical; but could it be that extra-logical means more than that? I can't see how the T-schema is empirical, yet Bueno and Colyvan also count that as extra-logical Jun 11 at 11:21
  • Maybe we have to start from their definition of "logic" (if any)... "We provide a defence of non-apriorism in logic." Thus, if not a priori, logic must be a posteriori, i.e. grounded on empirical facts and our knowledge of them. Jun 11 at 11:27
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Regarding your "It seems to be a catch-all phrase that literally means just that, and includes but not limited to empirical factors.", it's correct as your reference clearly states:

The idea is that it is possible to revise logical principles (or logical rules) on the basis of extra-logical considerations—which include empirical considerations.

Tarski's T-schema is the inductive definition of truth of his semantic theory of truth for languages, so it's considered as extra-logical not part of the underlying logic as inference rules. And we know in philosophy of language, this context-free compositional inductive schemes to define truth or meaning may fail under the context of necessary hermeneutic circle interpretation in a priori manner for some texts. So this is a case which is extra-logical but not empirical (have to come from sensual experience like the other 2 examples).

Regarding your general question about the definitional boundary of logical and extra-logical, your concern is shared by many philosophers such as Hartry Field that there's no a priori epistemic criterion to demarcate them as referenced here:

Tarski’s (1936) thesis that there is no principled division of concepts into the logical and the nonlogical, and the related view that there is no principled division between logical truths and truths that don’t belong to logic. This seems plausible: there seems little point to a debate between a person who takes first-order logic with identity to be logic and someone who thinks that only first-order logic without identity is really logic. Well, there might be a point if the second person were to claim that some of the axioms of identity that the first person was proposing aren’t true, but I’m imagining that the two parties agree on the truth of the axioms of identity, they just disagree as to whether they should count as part of logic... In general, then, I’m inclined to agree with any pluralism based on the arbitrariness of the demarcation between logic and nonlogic.

So we can say the current classic first order logic with identity is just by convention to be the best suited logic system for most classic use cases, and perhaps empirical experiences cannot help much for this demarcation as Gödel once confessed even natural laws may be a priori determined as depicted in Rebbeca Goldstein's Incompleteness: The Proof and Paradox of Kurt Gödel:

The linguist Noam Chomsky, too, reported being stopped dead in his linguistic tracks by the logician. Chomsky asked him what he was currently working on, and received an answer that probably nobody since the seventeenth-century's Leibniz had given: `I am trying to prove that the laws of nature are a priori.'

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To keep things simple, think of logic as (ideally) the manipulation of symbols according to certain rules that are designed to maintain the self-consistency of class property relationships. 'Extra-logical' then, is anything that does not constitute a logical rule or the manipulation of symbols within the constraints of logical rules. Premises, assumptions, empirical evidence, speculations, analogies and metaphors, etc., are all extra-logical.

There are some things that are clearly matters of logic, some things that are clearly extra-logical, and some things that fall within a gray area of interpretation and debate.

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The idea is that it is possible to revise logical principles (or logical rules) on the basis of extra-logical considerations—which include empirical considerations. In other words, extra-logical considerations play a role in the selection and evaluation of logical principles (or rules)

The distinction between logical and extra-logical requires another, prior, distinction, between the logic of human deductive reasoning and formal logic. Nobody is going to revise the principles of human deductive reasoning. These principles may perhaps conceivably change over time but nobody is going to revise them.

In contrast to that, the principles admitted in any formal logic system are of course highly revisable, in the same sense as the principles admitted in any theory are revisable depending on the empirical evidence available.

Given this, to revise the principles of a formal logic system according to logical considerations would mean in effect that you revise these principles according to considerations issued from, based on or related to the logic of human deductive reasoning.

Given this, the idea that we can revise the principles of a formal logic system according to non-logical considerations is perhaps the idea that it is possible to improve on the logic of human deductive system. This is an idea which seems already widespread and even prevalent in mathematical logic. Non-logical considerations would include empirical considerations about the physical world as well as pragmatic considerations such as convenience, effectiveness, economy etc.

This sort of revisionism is inherent to mathematical logic and started with George Boole in 1847. The material implication itself is a very good example. It does not correctly represent the logical implication and yet it was adopted in mathematical logic. The same can be said about every notion of implication used in mathematical logic as it is practised today.

That being said, the idea that logical distributivity fails in quantum mechanics only demonstrates that many people don't really understand what is logic. The T-schema seems also irrelevant in this context.

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