Syntactical investigations intoBI logic andBB′I logic

In this note, we will study four implicational logicsB, BI, BB and BBI. In [5], Martin and Meyer proved that a formula is provable inBB if and only if is provable inBBI and is not of the form of » . Though it gave a positive solution to theP - W problem, their method was semantical and not easy to grasp. We shall give a syntactical proof of the syntactical relation betweenBB andBBI logics. It also includes a syntactical proof of Powers and Dwyer's theorem that is proved semantically in [5]. Moreover, we shall establish the same relation betweenB andBI logics asBB andBBI logics. This relation seems to say thatB logic is meaningful, and so we think thatB logic is the weakest among meaningful logics. Therefore, by Theorem 1.1, our Gentzentype system forBI logic may be regarded as the most basic among all meaningful logics. It should be mentioned here that the first syntactical proof ofP - W problem is given by Misao Nagayama [6].Presented byHiroakira Ono     

Learning Łukasiewicz logic

The integration between connectionist learning and logic-based reasoning is a longstanding foundational question in artificial intelligence, cognitive systems, and computer science in general. Research into neural-symbolic integration aims to tackle this challenge, developing approaches bridging the gap between sub-symbolic and symbolic representation and computation. In this line of work the core method has been suggested as a way of translating logic programs into a multilayer perceptron computing least models of the programs. In particular, a variant of the core method for three valued Łukasiewicz logic has proven to be applicable to cognitive modelling among others in the context of Byrne’s suppression task. Building on the underlying formal results and the corresponding computational framework, the present article provides a modified core method suitable for the supervised learning of Łukasiewicz logic (and of a closely-related variant thereof), implements and executes the corresponding supervised learning with the backpropagation algorithm and, finally, constructs a rule extraction method in order to close the neural-symbolic cycle. The resulting system is then evaluated in several empirical test cases, and recommendations for future developments are derived.     

On the logic of event-causation Jaśkowski-style systems of causal logic

Causality is a concept which is sometimes claimed to be easy to illustrate, but hard to explain. It is not quite clear whether the former part of this claim is as obvious as the latter one. I will not present any specific theory of causation. Our aim is much less ambitious; to investigate the formal counterparts of causal relations between events, i.e. to propose a formal framework which enables us to construct metamathematical counterparts of causal relations between singular events. This should be a good starting point to define formal counterparts for concepts like causal law, causal explanation and so on.Children in their simplicity keep asking why. The person of understanding has given this up; every why, he has long found out, is merely the end of a thread that vanishes into the thick snare of infinity, which no one can truly unravel, let him tug and worry at it as much as he likes.W. Busch,The Butterfly, translated by W. ArndtThe work on this paper was supported by grants from the Humboldt-Foundation, the Center for Philosophy of Science at the University of Pittsburgh and from the Fulbright-Foundation.This paper concerns a larger project on causal logic, Uwe Scheffler from Humboldt-University and I have been working on for several months. This collaboration explains the usage of plural forms wherever they occur in section 1 and 6. However, all results of this paper, if not stated explicitely otherwise, are due to its (single) author.Presented byJan Zygmunt;     

What is quantum logic?

The paper describes in detail the procedure of identification of the inner language and an inner logico of a physical theory. The procedure is a generalization of the original ideas of J. von Neuman and G. Birkhoff about quantum logic.     

Revisiting Reichenbach’s logic

Synthese - In this paper we show that, when analyzed with contemporary tools in logic—such as Dunn-style semantics, Reichenbach’s three-valued logic exhibits many interesting features,...     

An extension of the Łukasiewicz logic to the modal logic of quantum mechanics

An attempt is made to include the axioms of Mackey for probabilities of experiments in quantum mechanics into the calculus _x0 of ukasiewicz. The obtained calculusQ contains an additional modal signQ and four modal rules of inference. The propositionQx is read x is confirmed. The most specific rule of inference may be read: for comparable observations implication is equivalent to confirmation of material implication.The semantic truth ofQ is established by the interpretation with the help of physical objects obeying to the rules of quantum mechanics. The embedding of the usual quantum propositional logic inQ is accomplished.Allatum est die 9 Junii 1976     

Hegel’s logic of finitude

In ??Violence and Metaphysics?? Jacques Derrida suggests that ??the only effective position to take in order not to be enveloped by Hegel would seem to be??to consider false-infinity??irreducible.?? Inversely, refuting the charge of logocentrism associated with Hegelian true infinity (wahrhafte Unendlichkeit) would involve showing that Hegel??s speculative logic does not establish the infinity of being exempt from the negativity of the finite. This paper takes up Derrida??s challenge, and argues that true infinity is crucial to Hegel??s understanding of ideality as a question of normative authority, which does not fall prey to logocentrism. Through an exposition of the dialectic of the finite and the infinite in Hegel??s Science of Logic, I argue that true infinity is not an ontological category that eliminates division, but rather refers to the metalogical standpoint involved in a philosophical account of determinacy. Although fully achieved at the end of the Logic, the metalogical standpoint that Hegel elaborates in the Seinslogik under the banner of the true infinite already clarifies that determinacy is a product of normative authority that is itself precarious.     

Field’s logic of truth

T. M. Scanlon’s buck-passing account of value (BPA) has been subjected to a barrage of criticisms. Recently, to be helpful to BPA, Roger Crisp has suggested that a number of these criticisms can be met if one makes some revisions to BPA. In this paper, I argue that if advocates of the buck-passing account accepted these revisions, they would effectively be giving up the buck-passing account as it is typically understood, that is, as an account concerned with the conceptual priority of reasons or the right vis-à-vis value or the good. I conclude by addressing some of the broader implications of my arguments for the current debate about the buck-passing account of value.     

Hempel’s logic of confirmation

This paper presents a new analysis of C.G. Hempel’s conditions of adequacy for any relation of confirmation [Hempel C. G. (1945). Aspects of scientific explanation and other essays in the philosophy of science. New York: The Free Press, pp. 3–51.], differing from the one Carnap gave in §87 of his [1962. Logical foundations of probability (2nd ed.). Chicago: University of Chicago Press.]. Hempel, it is argued, felt the need for two concepts of confirmation: one aiming at true hypotheses and another aiming at informative hypotheses. However, he also realized that these two concepts are conflicting, and he gave up the concept of confirmation aiming at informative hypotheses. I then show that one can have Hempel’s cake and eat it too. There is a logic that takes into account both of these two conflicting aspects. According to this logic, a sentence H is an acceptable hypothesis for evidence E if and only if H is both sufficiently plausible given E and sufficiently informative about E. Finally, the logic sheds new light on Carnap’s analysis.     

Intuitionist logic — Subsystem of,extension of,or rival to,classical logic?

Strictly speaking, intuitionistic logic is not a modal logic. There are, after all, no modal operators in the language. It is a subsystem of classical logic, not [like modal logic] an extension of it. But... (thus Fitting, p. 437, trying to justify inclusion of a large chapter on intuitionist logic in a book that is largely about modal logics).     

Averaging the truth-value in Łukasiewicz logic

Chang's MV algebras are the algebras of the infinite-valued sentential calculus of ukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of average degree of truth of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.     

Axiomatization of ‘Peircean’ branching-time logic

The branching-time logic called Peircean by Arthur Prior is considered and given an infinite axiomatization. The axiomatization uses only the standard deduction rules for tense logic.     

Is there a logic of incoherence?

What should we do when we discover that our assessment of probabilities is incoherent? I explore the hypothesis that there is a logic of incoherence—a set of universally valid rules that specify how incoherent probability assessments are to be repaired. I examine a pair of candidate‐rules of incoherence logic that have been employed in philosophical reconstructions of scientific arguments. Despite their intuitive plausibility, both rules turn out to be invalid. There are presently no viable candidate‐rules for an incoherence logic on the table. Other ways of dealing with incoherence are surveyed, and found either to be unsatisfactory or to rely on a logic of incoherence in the end. The resolution of these antagonistic conclusions is left to future researchers.     

The logic of Simpson’s paradox

There are three distinct questions associated with Simpson’s paradox. (i) Why or in what sense is Simpson’s paradox a paradox? (ii) What is the proper analysis of the paradox? (iii) How one should proceed when confronted with a typical case of the paradox? We propose a “formal” answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson’s paradox. Our account contrasts sharply with Pearl’s causal (and questionable) account of the first two questions. We argue that the “how to proceed question?” does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth’s account of the paradox. Our research on the paradox suggests that the “how to proceed question” needs to be divorced from what makes Simpson’s paradox “paradoxical.”