Axiomatization of Some Basic and Modal Boolean Connexive Logics

Boolean connexive logic is an extension of Boolean logic that is closed under Modus Ponens and contains Aristotle’s and Boethius’ theses. According to these theses (i) a sentence cannot imply its negation and the negation of a sentence cannot imply the sentence; and (ii) if the antecedent implies the consequent, then the antecedent cannot imply the negation of the consequent and if the antecedent implies the negation of the consequent, then the antecedent cannot imply the consequent. Such a logic was first introduced by Jarmużek and Malinowski, by means of so-called relating semantics and tableau systems. Subsequently its modal extension was determined by means of the combination of possible-worlds semantics and relating semantics. In the following article we present axiomatic systems of some basic and modal Boolean connexive logics. Proofs of completeness will be carried out using canonical models defined with respect to maximal consistent sets.

Logic and the Basis of Theology: an Investigation into the Help Logic Can Give to the Understanding of Different Theologies, Following the Contributions to Both Disciplines of Arthur Prior, 1914-1969

<p>My thesis is that modem symbolic mathematical logics have an important contribution to make to theologies. I demonstrate this firstly in a 'theoretical section' (i) by showing what logics are and why they can be trusted; (ii) by showing how all theologies may be correctly treated as axiomatic systems; (iii) by outlining some modern logics which can assist theological thinking, including a logic I construct for this purpose called the Theologic. I demonstrate this, secondly, in an 'applied logic' section, by looking at (iv) the theology of one current branch of Christianity in detail, outlining its logical problems and the consequences of trying to avoid them; (v) 'post-modern' Christian theologies, firstly those that suggest that the word 'God' is a symbol rather than a name, and secondly at three feminist theologies two of which are logically quite radical; (vi) pantheism, in particular at Spinoza's ideas and Lovelock's Gaia; (vii) two religions, Buddhism and Confucianism, which, in their basic religious thinking, can be said to have no gods. I find that all religions I have studied - and they are representative of religions actual, proposed and imagined - have serious logical flaws, some known of old, others brought to light by the modern logics. The consequences of making the religions more logically sound are generally unacceptable to the members of the faiths. The suggestion that the gods use a different sort of logic to us is generally logically unacceptable. This does not leave abandoning religion as the only other possibility: the work of theologians in future, assisted by mathematical logic, may be (a) to bring about changes in basic beliefs, and (b) to assist in the birth of new, logically sound, religions. These investigations are carried out in the spirit of A N Prior, who came to logic through a Christian upbringing which gave him an interest in theology, a desire to make that theology more consistent, and as Professor of Philosophy at Canterbury College (as it then was) taught me. My upbringing was similar. We both, in the end, found conventional Christianity too illogical to believe. Time having past, I have been able to examine the logic of other, and newer, theologies.</p>

Logic and the Basis of Theology: an Investigation into the Help Logic Can Give to the Understanding of Different Theologies, Following the Contributions to Both Disciplines of Arthur Prior, 1914-1969

<p>My thesis is that modem symbolic mathematical logics have an important contribution to make to theologies. I demonstrate this firstly in a 'theoretical section' (i) by showing what logics are and why they can be trusted; (ii) by showing how all theologies may be correctly treated as axiomatic systems; (iii) by outlining some modern logics which can assist theological thinking, including a logic I construct for this purpose called the Theologic. I demonstrate this, secondly, in an 'applied logic' section, by looking at (iv) the theology of one current branch of Christianity in detail, outlining its logical problems and the consequences of trying to avoid them; (v) 'post-modern' Christian theologies, firstly those that suggest that the word 'God' is a symbol rather than a name, and secondly at three feminist theologies two of which are logically quite radical; (vi) pantheism, in particular at Spinoza's ideas and Lovelock's Gaia; (vii) two religions, Buddhism and Confucianism, which, in their basic religious thinking, can be said to have no gods. I find that all religions I have studied - and they are representative of religions actual, proposed and imagined - have serious logical flaws, some known of old, others brought to light by the modern logics. The consequences of making the religions more logically sound are generally unacceptable to the members of the faiths. The suggestion that the gods use a different sort of logic to us is generally logically unacceptable. This does not leave abandoning religion as the only other possibility: the work of theologians in future, assisted by mathematical logic, may be (a) to bring about changes in basic beliefs, and (b) to assist in the birth of new, logically sound, religions. These investigations are carried out in the spirit of A N Prior, who came to logic through a Christian upbringing which gave him an interest in theology, a desire to make that theology more consistent, and as Professor of Philosophy at Canterbury College (as it then was) taught me. My upbringing was similar. We both, in the end, found conventional Christianity too illogical to believe. Time having past, I have been able to examine the logic of other, and newer, theologies.</p>

Axiomatizing Hybrid XPath with Data

In this paper we introduce sound and strongly complete axiomatizations for XPath with data constraints extended with hybrid operators. First, we present HXPath=, a multi-modal version of XPath with data, extended with nominals and the hybrid operator @. Then, we introduce an axiomatic system for HXPath=, and we prove it is strongly complete with respect to the class of abstract data models, i.e., data models in which data values are abstracted as equivalence relations. We prove a general completeness result similar to the one presented in, e.g., [BtC06], that ensures that certain extensions of the axiomatic system we introduce are also complete. The axiomatic systems that can be obtained in this way cover a large family of hybrid XPath languages over different classes of frames, for which we present concrete examples. In addition, we investigate axiomatizations over the class of tree models, structures widely used in practice. We show that a strongly complete, finitary, first-order axiomatization of hybrid XPath over trees does not exist, and we propose two alternatives to deal with this issue. We finally introduce filtrations to investigate the status of decidability of the satisfiability problem for these languages.

To Know Them, Remove Them: An Outer Methodological Approach to Biophysics and Humanities

Set theory faces two difficulties: formal definitions of sets/subsets are incapable of assessing biophysical issues; formal axiomatic systems are complete/inconsistent or incomplete/consistent. To overtake these problems reminiscent of the old-fashioned principle of individuation, we provide formal treatment/validation/operationalization of a methodological weapon termed &ldquo;outer approach&rdquo; (OA). The observer&rsquo;s attention shifts from the system under evaluation to its surroundings, so that objects are investigated from outside. Subsets become just &ldquo;holes&rdquo; devoid of information inside larger sets. Sets are no longer passive containers, rather active structures enabling their content&rsquo;s examination. Consequences/applications of OA include: a) operationalization of paraconsistent logics, anticipated by unexpected forerunners, in terms of advanced truth theories of natural language, anthropic principle and quantum dynamics; b) assessment of embryonic craniocaudal migration in terms of Turing&rsquo;s spots; c) evaluation of hominids&rsquo; social behaviors in terms of evolutionary modifications of facial expression&rsquo;s musculature; d) treatment of cortical action potentials in terms of collective movements of extracellular currents, leaving apart what happens inside the neurons; e) a critique of Shannon&rsquo;s information in terms of the Arabic thinkers&rsquo; active/potential intellects. Also, OA provides an outer view of a) humanistic issues such as the enigmatic Celestino of Verona&rsquo;s letter, Dante Alighieri&rsquo;s &ldquo;Hell&rdquo; and the puzzling Voynich manuscript; b) historical issues such as Aldo Moro&rsquo;s death and the Liston/Clay boxing fight. Summarizing, the safest methodology to quantify phenomena is to remove them from our observation and tackle an outer view, since mathematical/logical issues such as selective information deletion and set complement rescue incompleteness/inconsistency of biophysical systems.

Exploring the Jungle of Intuitionistic Temporal Logics

The importance of intuitionistic temporal logics in Computer Science and Artificial Intelligence has become increasingly clear in the last few years. From the proof-theory point of view, intuitionistic temporal logics have made it possible to extend functional programming languages with new features via type theory, while from the semantics perspective, several logics for reasoning about dynamical systems and several semantics for logic programming have their roots in this framework. We consider several axiomatic systems for intuitionistic linear temporal logic and show that each of these systems is sound for a class of structures based either on Kripke frames or on dynamic topological systems. We provide two distinct interpretations of “henceforth”, both of which are natural intuitionistic variants of the classical one. We completely establish the order relation between the semantically defined logics based on both interpretations of “henceforth” and, using our soundness results, show that the axiomatically defined logics enjoy the same order relations.

Two systems of epistemic logic

Drawing on the results of chapters 2 to 4, two non-normal, multimodal axiomatic systems for both knowledge (k) and being in a position to know (K) are introduced—an idealized system and a weaker, more realistic system. Both share important theorems governing the complex operators ‘¬K¬K’ and ‘¬K¬K’, whose availability will be of crucial importance in later chapters. Unlike the realistic system, the idealized system requires subjects to be logically omniscient and must therefore ultimately be rejected in favour of the realistic system. A semantic characterization of the idealized system is devised that shows it to be sound and allows us to invalidate principles we previously found unacceptable for independent reasons. Since the realistic system is weaker, this result implies that it too has these features. Both systems imply that each of ⌜¬K¬Kφ‎⌝, ⌜K¬Kφ‎⌝, and ⌜¬K¬Kφ‎⌝ encodes a luminous condition. The scenario of the unmarked clock presents a prima facie case against this implication. It is shown that the relevant anti-luminosity argument presupposes the principle that being in a position to know (K) distributes across provable conditionals—a principle that has been shown to be deeply problematic and that the realistic system is designed to flout.

«GeOrigaMetry» An Approach to the Accessibility of Geometry for Blind People

Making geometry accessible for blind people, apart from the formal aspects, can pose some difficulties, especially in terms of accessibility to figures. To deal with this problem this article focuses on paper folding where both Euclidean and origami axiomatic systems are used simultaneously. In the first case, with a ruler and compass, we can solve quadratic problems in a plane. In addition, the axioms of origami allow us to address unanswered questions with classical geometry methods, which involve cubic equations, such as the trisection of an angle. An experiment with INJA (National Institute for Blind Youth, Paris) students and other blind people will take place so that we can see the possibilities offered by this method, which brings a ludic, but rigorous approach to these complex and frequently off-putting issues. We believe that this dynamic pedagogical approach can increase interest and motivation, encourage tactile stimulation and facilitate the development of specific structures of brain plasticity. The article is written in a linear way, accessible to blind people; figures are provided to facilitate understanding for "visually impaired" people, who are not used to following a geometric concept without pictures. Finally, it should be noted that the method is particularly suitable in an inclusive education context.