Logic of Negation-Complete Interactive Proofs (Formal Theory of Epistemic Deciders)

We produce a decidable classical normal modal logic of internalised negation-complete and thus disjunctive non-monotonic interactive proofs (LDiiP) from an existing logical counterpart of non-monotonic or instant interactive proofs (LiiP). LDiiP internalises agent-centric proof theories that are negation-complete (maximal) and consistent (and hence strictly weaker than, for example, Peano Arithmetic) and enjoy the disjunction property (like Intuitionistic Logic). In other words, internalised proof theories are ultrafilters and all internalised proof goals are definite in the sense of being either provable or disprovable to an agent by means of disjunctive internalised proofs (thus also called epistemic deciders). Still, LDiiP itself is classical (monotonic, non-constructive), negation-incomplete, and does not have the disjunction property. The price to pay for the negation completeness of our interactive proofs is their non-monotonicity and non-communality (for singleton agent communities only). As a normal modal logic, LDiiP enjoys a standard Kripke-semantics, which we justify by invoking the Axiom of Choice on LiiP's and then construct in terms of a concrete oracle-computable function. LDiiP's agent-centric internalised notion of proof can also be viewed as a negation-complete disjunctive explicit refinement of standard KD45-belief, and yields a disjunctive but negation-incomplete explicit refinement of S4-provability.Comment: Expanded Introduction. Added Footnote 4. Corrected Corollary 3 and 4. Continuation of arXiv:1208.184

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

The price of inscrutability

In our reasoning we depend on the stability of language, the fact that its signs do not arbitrarily change in meaning from moment to moment.(Campbell, 1994, p.82) Some philosophers offer arguments contending that ordinary names such as “London” are radically indeterminate in reference. The conclusion of such arguments is that there is no fact of the matter whether “London” refers to a city in the south of England, or whether instead it refers to Sydney, Australia. Some philosophers have even suggested that we accept the conclusion of these arguments. Such a position seems crazy to many; but what exactly goes wrong if one adopts such a view? This paper evaluates the theoretical costs incurred by one who endorses extreme inscrutability of reference (the ‘inscrutabilist’). I show that there is one particular implication of extreme inscrutability which pushes the price of inscrutabilism too high. An extension of the classic ‘permutation’ arguments for extreme inscrutability allow us to establish what I dub ‘extreme indexical inscrutability’. This result, I argue, unacceptably undermines the epistemology of inference. The first half of the paper develops the background of permutation arguments for extreme inscrutability of reference and evaluates some initial attempts to make trouble for the inscrutabilist. Sections 1 and 2 describe the setting of the original permutation arguments for extreme inscrutability. Sections 3 and 4 survey four potential objections to extreme inscrutability of reference, including some recently raised in Vann McGee’s excellent (2005a). Sections 5 sketches how the permutation arguments can be generalized to establish extreme indexical inscrutability; and shows how this contradicts a ‘stability principle’—that our words do not arbitrarily change their reference from one moment to the next—which I claim plays a vital role in the epistemology of inference. The second half of the paper develops in detail the case for thinking that language is stable in the relevant sense. In section 6, I use this distinction to call into question the epistemological relevance of validity of argument types; Kaplan’s treatment of indexical validity partially resolves this worry, but there is a residual problem. In section 7, I argue that stability is exactly what is needed to bridge this final gap, and so secure the relevance of validity to good inferential practice. Section 8 responds to objections to this claim. An appendix to the paper provides formal backing for the results cited in this paper, including a generalization of permutation arguments to the kind of rich setting required for a realistic semantics of natural language.1 Extreme indexical inscrutability results can be proved within this setting. The first half of the paper shows that the inscrutabilist is committed to extreme indexical inscrutability, which implies that language not determinately ‘stable’. The second half of the paper argues that good inference requires stability. The price of inscrutabilism, therefore, is to sever the connection between the validity of argument-forms and inferential practice: and this is too high a price to pay

Refinement Modal Logic

In this paper we present {\em refinement modal logic}. A refinement is like a bisimulation, except that from the three relational requirements only `atoms' and `back' need to be satisfied. Our logic contains a new operator 'all' in addition to the standard modalities 'box' for each agent. The operator 'all' acts as a quantifier over the set of all refinements of a given model. As a variation on a bisimulation quantifier, this refinement operator or refinement quantifier 'all' can be seen as quantifying over a variable not occurring in the formula bound by it. The logic combines the simplicity of multi-agent modal logic with some powers of monadic second-order quantification. We present a sound and complete axiomatization of multi-agent refinement modal logic. We also present an extension of the logic to the modal mu-calculus, and an axiomatization for the single-agent version of this logic. Examples and applications are also discussed: to software verification and design (the set of agents can also be seen as a set of actions), and to dynamic epistemic logic. We further give detailed results on the complexity of satisfiability, and on succinctness