Intersubstitutivity principles and the generalization function of truth

The Final Cut

“…While operational classicism would seem to constitute a fairly appealing principled, orderly conception of kindred correlations enjoyed by logical operations, structural classicism would seem to constitute a rather unappealing unprincipled, disorderly list of sundry properties and relations enjoyed by structural objects, 19 and so, in this respect, deviations from structural classicism would seem less problematic than deviations from operational classicism, to the extent that a system deviating only in the former way can reasonably be taken to be a superficially nonclassical system. It is then interesting to note that, contrary to the deeply nonclassical K3LP, there are systems for solving the semantic paradoxes and the paradoxes of vagueness that manage to get rid of structural classicism while upholding operational classicism, and are therefore superficially nonclassical.…”

“…By contrast, given a couple of plausible assumptions, the antideflationistic principle of dependence of truth on reality stated in the second last paragraph can account for that fact: since 'Snow is white or grass is blue' is true, by the dependence of truth on reality 'Snow is white or grass is blue' is true because snow is white or grass is blue; given the plausible assumption that, if [snow is white or grass is blue] and grass is not blue, [snow is white or grass is blue] because snow is white (and since the assumption's antecedent holds), [snow is white or grass is blue] because snow is white; given the plausible assumption that the relevant instance of transitivity of dependence holds, 'Snow is white or grass is blue' is true because snow is white. 38 Gupta and Standefer [19] interestingly argue that what is only required is a weaker version of uniform intersubstitutability, according to which, roughly, in a sentence, all occurrences of 0 , 1 , 2 en bloc are intersubstitutable with the corresponding occurrences of 0 , 1 , 2 . .…”

“…Notice that the fact that the principles are admissible in K3LP is neither here nor there, for sheer admissibility in a logic does not entail that a principle characterises (nor even that it is compatible with) the logic: for example, the metaentailment that, if is valid, is valid is admissible in classical logic, but it is in a very natural sense incompatible with it, as it is incompatible with classical logic's sound application to many correct target theories. Notice also that, starting from the fact that, in the same system, for certain s, is valid, an analogous argument is available that focuses on disjunction and (the exhaustivity of) negation rather than on conjunction and (the exclusivity of) negation 19. See however Zardini[53] for an attempt at bringing in some order into that mess by imposing on it a certain architecture.…”