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?