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Knowledge/Logic/Field: logical knowledge: when logic is confined to the if-then form: no knowledge about what does not follow. - Solution: differentiated deflationism: two parts: i) Knowledge, which mathematical statement follows from other mathematical statements. (ii) additional knowledge about the consistency of mathematical statements (and other fundamental). - ((s) About that was no conclusion of something). - consistency/(s): is itself not a conclusion. - Field E.g. a knowledge about all models is not a logical knowledge. - Syntactically: E.g. "There is a derivative of B from A": is not a logical knowledge, but knowledge about existence. - Deflationism: both is logical knowledge. - VsDeflationism: the fundamental is metalogical.
Logical knowledge/Field/(s): knowledge about the fact that something is logically true (e.g. that axioms are consistent), but not the axioms themselves. - FieldVsKripke: we then introduce a non-Kripkean concept of logical truth, according to which some non-trivial assertions about possibility are part of the logic. - Then the consistency of axioms becomes a logical truth. - Induction/Field: extra-logical means: empirical, because we find no contradiction.
Knowledge/Possibility/Field: there is knowledge of possibility that is not only based on knowledge of necessity. - Only by thinking about the logical form. - Problem: E.g.: "There are at least 10 to the power of 10 to the power of 10 apples": every statement of the same logical form as this is also a logical truth. - (But in terms of content, it is wrong) - Then one no longer had to rely on the actuality. - Then it would be categorical knowledge. - E.g. apples/Field: here we have stronger reason to believe in the possibility than in the actuality. - Field: but there are infinitely many physical entities: namely, space-time regions.
Logical Knowledge/Frege: Thesis: Problem, whereby do I know that it is logically possible that the axioms of quantum theory are true: by asserting that I know that there are actually entities asserted by the axioms. - FieldVsFrege: if these entities existed, how could one know then that they are in this relationship and not in another?
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980