486 research outputs found
Logic of Intuitionistic Interactive Proofs (Formal Theory of Perfect Knowledge Transfer)
We produce a decidable super-intuitionistic normal modal logic of
internalised intuitionistic (and thus disjunctive and monotonic) interactive
proofs (LIiP) from an existing classical counterpart of classical monotonic
non-disjunctive interactive proofs (LiP). Intuitionistic interactive proofs
effect a durable epistemic impact in the possibly adversarial communication
medium CM (which is imagined as a distinguished agent), and only in that, that
consists in the permanent induction of the perfect and thus disjunctive
knowledge of their proof goal by means of CM's knowledge of the proof: If CM
knew my proof then CM would persistently and also disjunctively know that my
proof goal is true. So intuitionistic interactive proofs effect a lasting
transfer of disjunctive propositional knowledge (disjunctively knowable facts)
in the communication medium of multi-agent distributed systems via the
transmission of certain individual knowledge (knowable intuitionistic proofs).
Our (necessarily) CM-centred notion of proof is also a disjunctive explicit
refinement of KD45-belief, and yields also such a refinement of standard
S5-knowledge. Monotonicity but not communality is a commonality of LiP, LIiP,
and their internalised notions of proof. As a side-effect, we offer a short
internalised proof of the Disjunction Property of Intuitionistic Logic
(originally proved by Goedel).Comment: continuation of arXiv:1201.3667; extended start of Section 1 and 2.1;
extended paragraph after Fact 1; dropped the N-rule as primitive and proved
it derivable; other, non-intuitionistic family members: arXiv:1208.1842,
arXiv:1208.591
A proof of strong normalisation using domain theory
Ulrich Berger presented a powerful proof of strong normalisation using
domains, in particular it simplifies significantly Tait's proof of strong
normalisation of Spector's bar recursion. The main contribution of this paper
is to show that, using ideas from intersection types and Martin-Lof's domain
interpretation of type theory one can in turn simplify further U. Berger's
argument. We build a domain model for an untyped programming language where U.
Berger has an interpretation only for typed terms or alternatively has an
interpretation for untyped terms but need an extra condition to deduce strong
normalisation. As a main application, we show that Martin-L\"{o}f dependent
type theory extended with a program for Spector double negation shift.Comment: 16 page
Conditionals and modularity in general logics
In this work in progress, we discuss independence and interpolation and
related topics for classical, modal, and non-monotonic logics
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