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**Some Neglected Work in General Metamathematics**

**David Miller**

*University of Warwick*
The logic of deductive theories (including theories that may not be
finitely axiomatizable) was initiated by Tarski in the double paper `Foundations
of the Calculus of Systems' of 1935/1936. In effect he showed that
if the meet and join operations on theories are extensions of the
same operations on sentences, then deductive theories form a Brouwerian
lattice under the relation of implication. That is, the logic of
deductive theories is dual to intuitionistic logic. Although the theory
of Brouwerian lattices is quite well developed, its purely logical
applications have received little attention.

Starting from frankly verificationist assumptions in the theory of meaning,
Dummett has argued that intuitionistic logic is the correct logic not only
in mathematics but in all discourse. Luntley and Milne have separately
claimed that the same result is obtained even if the premises are falsificationist.
But Dummett himself later reached a different & more congenial conclusion,
that falsificationist premises lead to something like dual-intuitionistic
logic. These writers were not helped by their assumption that logic
must support a material conditional (an operation that does not exist
everywhere in a Brouwerian lattice).

It is suggested in the paper that Dummett's later speculations were
on the right lines, though little sympathy is shown for his doctrine that
matters of meaning are important in determining which logic should be used.
If we represent our state of knowledge not by what has been proved but
by what has not been falsified, then we can provide in an obvious way a
Kripke-style semantics for dual- intuitionistic logic. This logic,
which is paraconsistent, is not a logic of truth, for which classical logic
suffices at the level of sentences, but a logic of what is falsified, or
of what must be regarded as falsified. It is shown how these ideas
connect naturally with the logic of deductive theories with which the paper
began.