44 research outputs found
A Galois connection between classical and intuitionistic logics. II: Semantics
Three classes of models of QHC, the joint logic of problems and propositions,
are constructed, including a class of subset/sheaf-valued models that is
related to solutions of some actual problems (such as solutions of algebraic
equations) and combines the familiar Leibniz-Euler-Venn semantics of classical
logic with a BHK-type semantics of intuitionistic logic.
To test the models, we consider a number of principles and rules, which
empirically appear to cover all "sufficiently simple" natural conjectures about
the behaviour of the operators ! and ?, and include two hypotheses put forward
by Hilbert and Kolmogorov, as formalized in the language of QHC. Each of these
turns out to be either derivable in QHC or equivalent to one of only 13
principles and 1 rule, of which 10 principles and 1 rule are conservative over
classical and intuitionistic logics. The three classes of models together
suffice to confirm the independence of these 10 principles and 1 rule, and to
determine the full lattice of implications between them, apart from one
potential implication.Comment: 35 pages. v4: Section 4.6 "Summary" is added at the end of the paper.
v3: Major revision of a half of v2. The results are improved and rewritten in
terms of the meta-logic. The other half of v2 (Euclid's Elements as a theory
over QHC) is expected to make part III after a revisio
Sheaf Logic, Quantum Set Theory and the Interpretation of Quantum Mechanics
Based on the Sheaf Logic approach to set theoretic forcing, a hierarchy of
Quantum Variable Sets is constructed which generalizes and simplifies the
analogous construction developed by Takeuti on boolean valued models of set
theory. Over this model two alternative proofs of Takeuti's correspondence,
between self adjoint operators and the real numbers of the model, are given.
This approach results to be more constructive showing a direct relation with
the Gelfand representation theorem, revealing also the importance of these
results with respect to the interpretation of Quantum Mechanics in close
connection with the Deutsch-Everett multiversal interpretation. Finally, it is
shown how in this context the notion of genericity and the corresponding
generic model theorem can help to explain the emergence of classicality also in
connection with the Deutsch- Everett perspective.Comment: 34 pages, 2 figure
Topos Semantics for Higher-Order Modal Logic
We define the notion of a model of higher-order modal logic in an arbitrary
elementary topos . In contrast to the well-known interpretation of
(non-modal) higher-order logic, the type of propositions is not interpreted by
the subobject classifier , but rather by a suitable
complete Heyting algebra . The canonical map relating and
both serves to interpret equality and provides a modal
operator on in the form of a comonad. Examples of such structures arise
from surjective geometric morphisms , where . The logic differs from non-modal higher-order
logic in that the principles of functional and propositional extensionality are
no longer valid but may be replaced by modalized versions. The usual Kripke,
neighborhood, and sheaf semantics for propositional and first-order modal logic
are subsumed by this notion
First steps in synthetic guarded domain theory: step-indexing in the topos of trees
We present the topos S of trees as a model of guarded recursion. We study the
internal dependently-typed higher-order logic of S and show that S models two
modal operators, on predicates and types, which serve as guards in recursive
definitions of terms, predicates, and types. In particular, we show how to
solve recursive type equations involving dependent types. We propose that the
internal logic of S provides the right setting for the synthetic construction
of abstract versions of step-indexed models of programming languages and
program logics. As an example, we show how to construct a model of a
programming language with higher-order store and recursive types entirely
inside the internal logic of S. Moreover, we give an axiomatic categorical
treatment of models of synthetic guarded domain theory and prove that, for any
complete Heyting algebra A with a well-founded basis, the topos of sheaves over
A forms a model of synthetic guarded domain theory, generalizing the results
for S