Logic of Non-Monotonic Interactive Proofs (Formal Theory of Temporary Knowledge Transfer)

We propose a monotonic logic of internalised non-monotonic or instant interactive proofs (LiiP) and reconstruct an existing monotonic logic of internalised monotonic or persistent interactive proofs (LiP) as a minimal conservative extension of LiiP. Instant interactive proofs effect a fragile epistemic impact in their intended communities of peer reviewers that consists in the impermanent induction of the knowledge of their proof goal by means of the knowledge of the proof with the interpreting reviewer: If my peer reviewer knew my proof then she would at least then (in that instant) know that its proof goal is true. Their impact is fragile and their induction of knowledge impermanent in the sense of being the case possibly only at the instant of learning the proof. This accounts for the important possibility of internalising proofs of statements whose truth value can vary, which, as opposed to invariant statements, cannot have persistent proofs. So instant interactive proofs effect a temporary transfer of certain propositional knowledge (knowable ephemeral facts) via the transmission of certain individual knowledge (knowable non-monotonic proofs) in distributed systems of multiple interacting agents.Comment: continuation of arXiv:1201.3667 ; published extended abstract: DOI:10.1007/978-3-642-36039-8_16 ; related to arXiv:1208.591

A Galois-Connection between Cattell's and Szondi's Personality Profiles

We propose a computable Galois-connection between, on the one hand, Cattell's 16-Personality-Factor (16PF) Profiles, one of the most comprehensive and widely-used personality measures for non-psychiatric populations and their containing PsychEval Personality Profiles (PPPs) for psychiatric populations, and, on the other hand, Szondi's personality profiles (SPPs), a less well-known but, as we show, finer personality measure for psychiatric as well as non-psychiatric populations (conceived as a unification of the depth psychology of S. Freud, C.G. Jung, and A. Adler). The practical significance of our result is that our Galois-connection provides a pair of computable, interpreting translations between the two personality spaces of PPPs (containing the 16PFs) and SPPs: one concrete from PPP-space to SPP-space (because SPPs are finer than PPPs) and one abstract from SPP-space to PPP-space (because PPPs are coarser than SPPs). Thus Cattell's and Szondi's personality-test results are mutually interpretable and inter-translatable, even automatically by computers.Comment: closely related to arXiv:1403.2000 as explained in the first paragrap

A Galois-Connection between Myers-Briggs' Type Indicators and Szondi's Personality Profiles

We propose a computable Galois-connection between Myers-Briggs' Type Indicators (MBTIs), the most widely-used personality measure for non-psychiatric populations (based on C.G. Jung's personality types), and Szondi's personality profiles (SPPs), a less well-known but, as we show, finer personality measure for psychiatric as well as non-psychiatric populations (conceived as a unification of the depth psychology of S. Freud, C.G. Jung, and A. Adler). The practical significance of our result is that our Galois-connection provides a pair of computable, interpreting translations between the two personality spaces of MBTIs and SPPs: one concrete from MBTI-space to SPP-space (because SPPs are finer) and one abstract from SPP-space to MBTI-space (because MBTIs are coarser). Thus Myers-Briggs' and Szondi's personality-test results are mutually interpretable and inter-translatable, even automatically by computers

Computer-Aided Discovery and Categorisation of Personality Axioms

We propose a computer-algebraic, order-theoretic framework based on intuitionistic logic for the computer-aided discovery of personality axioms from personality-test data and their mathematical categorisation into formal personality theories in the spirit of F.~Klein's Erlanger Programm for geometrical theories. As a result, formal personality theories can be automatically generated, diagrammatically visualised, and mathematically characterised in terms of categories of invariant-preserving transformations in the sense of Klein and category theory. Our personality theories and categories are induced by implicational invariants that are ground instances of intuitionistic implication, which we postulate as axioms. In our mindset, the essence of personality, and thus mental health and illness, is its invariance. The truth of these axioms is algorithmically extracted from histories of partially-ordered, symbolic data of observed behaviour. The personality-test data and the personality theories are related by a Galois-connection in our framework. As data format, we adopt the format of the symbolic values generated by the Szondi-test, a personality test based on L.~Szondi's unifying, depth-psychological theory of fate analysis.Comment: related to arXiv:1403.200

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

Parametric Constructive Kripke-Semantics for Standard Multi-Agent Belief and Knowledge (Knowledge As Unbiased Belief)

We propose parametric constructive Kripke-semantics for multi-agent KD45-belief and S5-knowledge in terms of elementary set-theoretic constructions of two basic functional building blocks, namely bias (or viewpoint) and visibility, functioning also as the parameters of the doxastic and epistemic accessibility relation. The doxastic accessibility relates two possible worlds whenever the application of the composition of bias with visibility to the first world is equal to the application of visibility to the second world. The epistemic accessibility is the transitive closure of the union of our doxastic accessibility and its converse. Therefrom, accessibility relations for common and distributed belief and knowledge can be constructed in a standard way. As a result, we obtain a general definition of knowledge in terms of belief that enables us to view S5-knowledge as accurate (unbiased and thus true) KD45-belief, negation-complete belief and knowledge as exact KD45-belief and S5-knowledge, respectively, and perfect S5-knowledge as precise (exact and accurate) KD45-belief, and all this generically for arbitrary functions of bias and visibility. Our results can be seen as a semantic complement to previous foundational results by Halpern et al. about the (un)definability and (non-)reducibility of knowledge in terms of and to belief, respectively

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

Asymptotic cones of finitely presented groups

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that R-rank(S) is at least 2, and let $\Gamma$ be a uniform lattice in G. (a) If $CH$ holds, then $\Gamma$ has a unique asymptotic cone up to homeomorphism. (b) If $CH$ fails, then $\Gamma$ has $2^{2^{\omega}}$ asymptotic cones up to homeomorphism.Comment: To appear in Advances in Mathematic

Pulsar Spin--Velocity Alignment: Further Results and Discussion

The reported alignment between the projected spin-axes and proper motion directions of pulsars is revisited in the light of new data from Jodrell Bank and Effelsberg. The present investigation uses 54 pulsars, the largest to date sample of pulsars with proper-motion and absolute polarisation, to study this effect. Our study has found strong evidence for pulsar spin-velocity alignment, excluding that those two vectors are completely uncorrelated, with >99% confidence. Although we cannot exclude the possibility of orthogonal spin-velocity configurations, comparison of the data with simulations shows that the scenario of aligned vectors is more likely than that of the orthogonal case. Moreover, we have determined the spread of velocities that a spin-aligned and spin-orthogonal distribution of kicks must have to produce the observed distribution of spin-velocity angle offsets. If the observed distribution of spin-velocity offset angles is the result of spin-aligned kicks, then we find that the distribution of kick-velocity directions must be broad with {\sigma}_v~30\degree if the orthogonal-kick scenario is assumed, then the velocity distribution is much narrower with {\sigma}_v<10\degree. Finally, in contrast to previous studies, we have performed robustness tests on our data, in order to determine whether our conclusions are the result of a statistical and/or systematic bias. The conclusion of a correlation between the spin and velocity vectors is independent of a bias introduced by subsets in the total sample. Moreover, we estimate that the observed alignment is robust to within 10% systematic uncertainties on the determination of the spin-axis direction from polarisation data.Comment: 20 pages, 7 figures, 1 Table, accepted in MNRA