...although our ‘logical’ statements seem to depend on the world — at a minimum, on our linguistic choices, our derivation rules, etc. — they don’t seem to depend on there being a literal worldly correlate for what they assert. Truth-conditions and representational content come apart radically.
What’s missing from this part, to keep it from adequately addressing the question (combined with the earlier post on the nature of logic)?
To compare a mental image of high-level apple-objects to physical reality, for it to be true under a correspondence theory of truth, doesn’t require that apples be fundamental in physical law. A single discrete element of fundamental physics is not the only thing that a statement can ever be compared-to. We just need truth conditions that categorize the low-level states of the universe, so that different low-level physical states are inside or outside the mental image of “some apples on the table” or alternatively “a kitten on the table”.
...And thus “The product of the apple numbers is six” is meaningful, constraining the possible worlds. It has a truth-condition, fulfilled by a mixture of physical reality and logical validity; and the correspondence is nailed down by a mixture of causal reference and axiomatic pinpointing.
I. ‘Valid’ is a bad word for what Eliezer’s talking about, because validity is a property of arguments, proofs, inferences, not of individual assertions. For now, I’ll call Eliezer’s validity ‘derivability’ or ‘provability.’
II. Strictly speaking, is logical derivability a kind of truth, or is it an alternative to truth that sometimes gets confused with it? Eliezer seems to alternate between these two views.
III. Are some statements simply ‘valid’ / ‘derivable’? Or is validity/derivability always relative to a set of inference rules (and, in some cases, axioms or assumptions)?
IIII. If derivability is always relativized in this way, then what does it mean to say that “The product of the apple numbers is six” is true in virtue of a mixture of physical reality and logical derivability? A different set of logical or mathematical rules would have yielded a different result. ‘Logical pinpointing’ is meant to solve this — there is a unique imaginary, fictional, mathematical, etc. image that we’re reasoning with in every case, and ‘intuitionistic real numbers’ simply aren’t the same objects as ‘conventional real numbers,’ and there simply is no such thing as ‘the real numbers’ absent the aforementioned specifications. Should we say, then, that truth is bivalent, whereas derivability/validity is trivalent?
Here’s an example of where this sort of reasoning will lead us: First, there simply isn’t any such thing as a ‘continuum hypothesis;’ we must exhaustively specify a set of inference rules and axioms/assumptions before we can even entertain a discrete logical claim, much less evaluate that claim’s derivability. Once we have fully pinpointed the expression, say as the ‘conventional continuum hypothesis’ or the ‘consistent Zermelo-Frankel continuum hypothesis,’ we then arrive at the conclusion that the hypothesis is not true (since it is logical and not empirical); nor is it false; nor is it valid/derivable; nor is its negation valid/derivable. It is thus ‘invalid’ in the weak sense that it can’t be derived, but is not ‘invalid’ in the strong sense of being disprovable. So, again, we have reason to speak of three properties (perhaps: provable, unprovable, disprovable), rather than of a bivalent ‘validity.’
V. Supposing correspondence-conditions fix truth-conditions, what fixes the correspondence-conditions? Relatedly, what makes assertions have the particular contents and referents they do, what supplies the ‘semantic glue’? And do logical or mixed-reference truths refer to anything in the world? If so, to what?
I especially like point/questions V. If we abandon the correspondence theory of truth, can we duck the questions? Because answering them seems like a lot of work, and like Dilbert and his office mates, I love the sweet smell of unnecessary work.
What’s missing from this part, to keep it from adequately addressing the question (combined with the earlier post on the nature of logic)?
I. ‘Valid’ is a bad word for what Eliezer’s talking about, because validity is a property of arguments, proofs, inferences, not of individual assertions. For now, I’ll call Eliezer’s validity ‘derivability’ or ‘provability.’
II. Strictly speaking, is logical derivability a kind of truth, or is it an alternative to truth that sometimes gets confused with it? Eliezer seems to alternate between these two views.
III. Are some statements simply ‘valid’ / ‘derivable’? Or is validity/derivability always relative to a set of inference rules (and, in some cases, axioms or assumptions)?
IIII. If derivability is always relativized in this way, then what does it mean to say that “The product of the apple numbers is six” is true in virtue of a mixture of physical reality and logical derivability? A different set of logical or mathematical rules would have yielded a different result. ‘Logical pinpointing’ is meant to solve this — there is a unique imaginary, fictional, mathematical, etc. image that we’re reasoning with in every case, and ‘intuitionistic real numbers’ simply aren’t the same objects as ‘conventional real numbers,’ and there simply is no such thing as ‘the real numbers’ absent the aforementioned specifications. Should we say, then, that truth is bivalent, whereas derivability/validity is trivalent?
Here’s an example of where this sort of reasoning will lead us: First, there simply isn’t any such thing as a ‘continuum hypothesis;’ we must exhaustively specify a set of inference rules and axioms/assumptions before we can even entertain a discrete logical claim, much less evaluate that claim’s derivability. Once we have fully pinpointed the expression, say as the ‘conventional continuum hypothesis’ or the ‘consistent Zermelo-Frankel continuum hypothesis,’ we then arrive at the conclusion that the hypothesis is not true (since it is logical and not empirical); nor is it false; nor is it valid/derivable; nor is its negation valid/derivable. It is thus ‘invalid’ in the weak sense that it can’t be derived, but is not ‘invalid’ in the strong sense of being disprovable. So, again, we have reason to speak of three properties (perhaps: provable, unprovable, disprovable), rather than of a bivalent ‘validity.’
V. Supposing correspondence-conditions fix truth-conditions, what fixes the correspondence-conditions? Relatedly, what makes assertions have the particular contents and referents they do, what supplies the ‘semantic glue’? And do logical or mixed-reference truths refer to anything in the world? If so, to what?
I especially like point/questions V. If we abandon the correspondence theory of truth, can we duck the questions? Because answering them seems like a lot of work, and like Dilbert and his office mates, I love the sweet smell of unnecessary work.