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I have been reading Bueno and Colyvan's Logical Non-apriorism and they mentioned that Tarski has the following argument for logic apriorism:

If, in the sentences of the class K and in the sentence X, the constants - apart from purely logical constants - are replaced by any other constants (like signs being everywhere replaced by like signs), and if we denote the class of sentences thus obtained from K by K', and the sentence obtained from X by X', then the sentence X' must be true provided only that all sentences of the class K' are true.

Since we are concerned here with the concept of logical, i.e. formal, consequence, and thus with a relation which is to be uniquely determined by the form of the sentences between which it holds, this relation cannot be influenced in any way influenced in any way by empirical knowledge, and in particular by knowledge of the objects to which the sentence X or the sentences of the class K refer. The consequence relation cannot be affected by replacing the designations of the objects refereed to in these senteces by the designations of any other objects.

Bueno and Colyvan commented,

The requirement that logical consequence be formal is expressed by guaranteeing that extra-logical/empirical considerations are in a clear sense irrelevant for logical consequence. If X already follows from K, a complete reinterpretation of X and the sentences in K won't change this feature. What matters for the relation of logical consequence is that the form of the argument is preserved.

I think this is what's going on: if we have a logical consequence Γ⊨A, then even if we replace all the sentences in Γ with Γ', and A with A' (while leaving the logical constants such as conjunction or implication alone), if every member of Γ' is true, A' must also be true - i.e. Γ'⊨A'.

If we take sentences to have empirical components, then this means that our conception of logical consequence should not be affected by empirical consideration, as the substitution above shows.

What I don't understand is Bueno and Colyvan's rebuttal:

But why should we apply the substitution requirement across the board? Because if we are concerned with a formal account of logical consequence, we simply cannot tolerate the intrusion of empirical factors - what matters for logical consequence is the form of the arguments. But what if, by disregarding the role of empirical factors, we simply obtain the wrong results about a given domain? This is exactly the question that the quantum logician insists on asking. (Referring to how classical logic provides wrong results when applied to quantum mechanics)

Our proposal is then to restrict the application of the substitution requirement: one selects a domain, and the substitution requirement is then applied only to a sentences/objects of that particular domain. By restricting the application of the requirement in this way, we can capture an important aspect of logic: in the context of the particular domain, the systematic substitution of non-logical terms does preserve the consequence relation.

By restricting the substitution requirement in this way, we allow the introduction of non-logical factors in logic selection: the parameters used in the determination of the domain bring in extra-logical factors to logic.

I don't understand what it is exactly that they are proposing to do. Are they proposing that, the substitution as depicted above can only be done with other sentences from the same domain? e.g. If I have a logical consequence Γ⊨A where both Γ and A are sets of sentences describing quantum mechanics, then when I substitute Γ and A with Γ' and A', the latter two must also be strictly about QM?

I fail to see how this will ensure that a correct conclusion that is consistent with empirical observations, unless the logical consequence is custom to that specific domain.

Could anyone help me understand this please?

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    The idea is to make logic itself dependent on the domain of application. Customised logic. Different logics for different domains of application. The principle of substitution is necessarily then restricted to work only within each of the various domains and not necessarily across different domains. So, essentially the end of logic as we know it. All based on a profound misunderstanding of the QM problem. – Speakpigeon Nov 11 '20 at 15:38
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    The equality sign "=" is a non-logical constant, but we would typically not want to allow having it replaced by whatever binary relation when deciding what is or is not a logical consequence. So we would want to exclude "=" from the domain subject to substitution and keep it "universal", on a par with logical constants. – Conifold Nov 12 '20 at 1:18
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Logic is commonly, though not universally, understood as separating form from content. On this understanding, the form is the logical part, and is traditionally taken to be a priori, while the content is the empirical part. If we can completely and successfully separate the parts of a sentence that are formal from the parts that are not, then we have grounds for saying that logic is a priori. One of Tarski's contributions to this effort was to lay the foundations for model theory, under which a sentence expressed in first-order predicate logic can be divided up between the logical constants (and, or, not, forall, exists, etc.) on one hand, and the names, predicates, functions and propositions on the other. A sentence is a logical truth if it holds under all interpretations, which is to say we must keep the logical constants the same, but we may vary the rest. In this way, logical truth, and by extension logical consequence, abstracts away from the empirical components of a sentence or argument.

Bueno and Colyvan's objection is that there is no a priori guarantee that the specific separation between formal and empirical factors envisaged by Tarski will work always and everywhere. They are not objecting to the formal/empirical distinction per se, just to the specific choice of logical constants and their corresponding implicational relationships. The constants used by Tarski are the constants of classical logic. How can we know a priori that those logical constants will work everywhere, especially when there seem to be examples where they give the wrong result?

As an aside, Bueno and Colyvan use the word 'domain' somewhat loosely. In the context of Tarski's model theoretic approach to logic, a domain usually means the set of all individuals in the universe of discourse. Changing domain in this sense does not change whether a sentence is a logical truth or whether an argument is valid. B&C are using the word to mean something more like the logical subject area, or the modality. I will use the word 'modality' here, though it is quite possible they intend something broader.

In the case of quantum logic, the problem with applying the classical logical constants is that we find cases where the distributivity rules fail. Broadly speaking, we have two ways of dealing with this. We can change the logic itself and adopt a non-classical logic. Or we can introduce some kind of modal operator that sits between the logic and the underlying propositions, such as 'it is observable that...'. This allows us to block the implications that give the wrong result by having rules for how observables combine. Classically, P together with Q entails P&Q, but if this fails to hold for observables, we could save the classical logic by requiring that Obs(P) together with Obs(Q) does not entail Obs(P&Q).

I think it is unhelpful to concentrate too much on quantum logic, if only because the merits of quantum logic are highly technical. There are many other logics for other modalities. For example, there are logics of obligation, imperative logics, constructive logics, provability logics, probability logics, etc. These also obey different rules from classical logic, though as above, we may be able to save the classical logic by introducing modal operators. This is what modal logic attempts to do, though there is still no guarantee that it will always work.

To return to your question, B&C are proposing that there are extra-logical factors that determine which is the appropriate logic to use for a given modality. For a chosen logic, the substitution rules will correctly characterise validity for that modality, but not for others. Your observation is correct that what follows from this is that the relation of logical consequence is custom to the modality. That may sound weird, but it comes with the territory if you want to apply logic to modalities that lie outside the bounds of truth and falsehood.

A simple example would be that the argument: ¬(∀x)Fx therefore (∃x)¬Fx is valid in classical logic, but not in intuitionistic logic. We can choose to live with intuitionism, or we can understand the intuitionistic version of the argument as a modal argument about constructive assertability and translate it into a classical modal logic (S4) in such a way that the argument is not valid.

  • Thank you so much for your detailed answer; am I understanding correctly that, when B&C are suggesting that one selects a domain (e.g. quantum mechanics) and the substitution requirement is applied only to sentences of that domain, what they really mean is pick a domain and a logic for it (e.g. QM and quantum logic)... – Daniel Mak Nov 28 '20 at 8:47
  • ...then it follows that the substitution requirement is applied only to sentences of that domain because quantum logic's logical constants mean/work differently from another domain, e.g. constructive maths and intuitionistic logic? – Daniel Mak Nov 28 '20 at 8:47
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    That's pretty much it. Those are not really different options though, since the substitution rules and the logic go hand in hand. – Bumble Nov 28 '20 at 19:39
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Long comemnt

The key point in Tarski'approach is in the clause: "apart from purely logical constants".

The logical constants (aka: syncategorematic terms) are... constants: they are not "reinterpreted" when we change the interpretation of the signs:

[they are not] affected by replacing the designations of the objects refereed to in these senteces by the designations of any other objects.

This cab be called "logical apriorism": the meaning of the logical constant is independent from the experience.

But Quantum Logic is a challenge to this view: to match with QM, Quantum Logic needs a different meaning of some connectives (e.g. disjunction) and this suggest:

that quantum mechanics requires a revolution in our understanding of logic per se. According to Putnam, “Logic is as empirical as geometry. [...] We live in a world with a non-classical logic” [1968].

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