MCMP – Logic

Itala M. Loffredo D'Ottaviano (Campinas) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "Can a paraconsistent differential calculus extend the classical differential calculus?". Abstract:In 2000, da Costa proposes the construction of a paraconsistent differential calculus, whose language is the language L of his known paraconsistent logic C1, extended to the language of his paraconsistent set theory CHU1, introduced in 1986. We have studied and improved the calculus proposed by da Costa, having obtained extensions of several fundamental theorems of the classical differential calculus. From the introduction of the concept of paraconsistent super-structure X over a set X of atoms of CHU1 and of the concept of monomorphism between paraconsistent super-structures, we will present a Transference Theorem that “translates” the classical differential calculus into da Costa’s paraconsistent calculus.

Graham Priest (CUNY and St Andrews) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "Models of Paraconsistent Set Theory". Abstract: Any adequate paraconsistent set theory must be able to validate at least a major part of the standard results of orthodox set theory. One way to achieve this is to take the universe or universes of sets to be such as to validate not only the naïve principles, but also all the theorems of Zermelo Fraenkel set theory. In this talk I will discuss various constructions of models of set theory which do just this.

Andreas Kapsner (MCMP/LMU) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "Why designate gluts?". Abstract: In this talk, I want to explore the following idea: Truth value gluts should be allowed in the semantics of logical systems, as they are in many non-classical systems. However, unlike what is standard in such systems, these gluts should be treated as undesignated values. I shall give my reasons for taking this to be a view worth exploring and discuss its effects on such topics as dialetheism, paraconsistency and relevance. On the whole, it will turn out to be a surprisingly attractive view that deals well with epistemic inconsistencies and semantic paradoxes. Some of the greatest difficulties arise in the attempt to account for interesting inconsistent scientific and mathematical theories; this, then, will be the touchstone for the proposed view.

Diderik Batens (Ghent) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "Transitory and Permanent Applications of Paraconsistency". Abstract: The advent of paraconsistency offers an excellent opportunity to unveil past prejudices. These do not only concern the truth or sensibility of inconsistencies, but many aspects of the nature of logic(s). My aim will be to raise questions (and possibly arrive at insights) on such topics as the following: methodological versus descriptive application contexts of logics, logical pluralism (in those application contexts), arguments for considering a specific application as suitable, truth-functionality of logical operators, sensibility of (certain uses of) classical negation, fallibilism in logic.

Francesco Berto (Amsterdam) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "Inconsistent Thinking, Fast and Slow". Abstract: This plays on Kahneman’s Thinking Fast and Slow. We implement two reasoning systems: our Slow system is logical-rule-based. Our Fast system is associative, context-sensitive, and integrates what we conceive via background information Slow inconsistent thinking may rely on paraconsistent logical rules, but I focus on Fast inconsistent thinking. I approach our Fast-conceiving inconsistencies in terms of ceteris paribus intentional operators: variably restricted quantifiers on possible and impossible worlds. The explicit content of an inconsistent conception is similar to a ceteris paribus relevant conditional antecedent. I discuss how such operators invalidate logical closure for conceivability, and how similarity works when impossible worlds are around.

Marie Duzi (Technical University Ostrava) gives a talk at the MCMP Colloquium (15 May, 2014) titled "A plea for beta-reduction by value". Abstract: This paper solves, in a logically rigorous manner, a problem discussed in a 2004 paper by Stephen Neale and originally advanced as a counterexample to Chomsky’s theory of binding. The example I will focus on is the following. John loves his wife. So does Peter. Therefore, John and Peter share a property. Only which one? There are two options. (1) Loving John’s wife. Then John and Peter love the same woman (and there is trouble on the horizon). (2) Loving one’s own wife. Then, unless they are married to the same woman, John loves one woman and Peter loves another woman (and both are exemplary husbands). On the strict reading of “John loves his wife, and so does Peter” property (1) is the one they share. On the sloppy reading, property (2) is the one they share. The dialectics of this contribution is to move from linguistics through logic to semantics. An issue originally bearing on binding in linguistics is used to make a point about -conversion in the typed ß-calculus. Since the properties loving John’s wife and loving one’s own wife as attributed to John are distinct, there is room for oscillation between the sloppy and the strict reading. But once we feed the formal renditions of attribution of these two properties to John into the widespread ß-calculus for logical analysis, a logical problem arises. The problem is this. Their respective ß-redexes are distinct, for sure, but they share the same ß-contractum. This contractum corresponds to the strict reading. So ß-conversion predicts, erroneously, that two properties applied to John ß-reduce to one. The result is that the sloppy reading gets squeezed out. ß-reduction blots out the anaphoric character of ‘his wife’, while the resulting contractum is itself ß-expandable back into both the strict and the sloppy reading. Information is lost in transformation. The information lost when performing ß-reduction on the formal counterparts of “John loves his wife” is whether the property that was applied was (1) or (2), since both can be reconstructed from the contractum, though neither in particular. The sentence “John loves his wife, and so does Peter” ostensibly shows that the ß-calculus is too crude an analytical tool for at least one kind of perfectly natural use of indexicals. The problematic reduction and its solution will both be discussed within the framework of Tichý’s Transparent Intensional Logic. Tichý’s TIL was developed simultaneously with Montague’s Intensional Logic. The technical tools of the two disambiguations of the analysandum will be familiar from Montague’s intensional logic, with two important exceptions. One is that we ß-bind separate variables w1,…,wn ranging over possible worlds and t1,…,tn ranging over times. This dual binding is tantamount to explicit intensionalization and temporalization. The other exception is that functional application is the logic both of extensionalization of intensions (functions from possible worlds) and of predication. I will demonstrates that, and how, the ß-calculus is up for the challenge, provided a rule of ß-conversion by value is adopted. The logical contribution of the paper is a generally valid form of ß-reduction by value rather than by name. The philosophical application of ß-reduction by value to a context containing anaphora is another contribution of this paper. The standard approach to VP ellipsis based on ß-abstracts and variable binding can, thus, be safely upheld. Our solution has the following features. First, unambiguous terms and expressions with a pragmatically incomplete meaning, like ‘his wife’ or “So does Peter”, are analyzed in all contexts as expressing an open construction containing at least one free variable with a fixed domain of quantification. Second, the solution uses ß-conversion by value, rather than conversion by name. The generally valid rule of ß-conversion by value exploits our substitution method and I show that the application of this rule does not yield a loss of analytic information. Third, the substitution method is applied to sentences containing anaphora, like ‘so does’ and ‘his’, in order to pre-process the meaning of the incomplete clause. Our declarative procedural semantics also makes it straightforward to infer that there is a property that John and Peter share.

Sander Beckers (Leuven) gives a talk at the MCMP Colloquium (6 November, 2013) titled "Actual causation using CP-logic". Abstract: Since Pearl’s seminal work on providing a formal language for causality, the subject has garnered a lot of interest among philosophers and researchers in artificial intelligence alike. One of the most debated topics in this context regards the notion of actual causation, or token causation, which concerns itself with specific - as opposed to general - causal claims. The search for a proper formal definition of actual causation has evolved into a controversial debate, that is pervaded with ambiguities and confusion. One of the goals of our research is to provide a clear way for comparing competing definitions, by translating the main existing approaches into one formal paradigm, namely CP-logic (Causal Probabilistic logic). Another goal we have is to relate the abstract philosophical discussion to practical applications of causal reasoning. In this talk we shall do so by focusing on the definition(s) offered by Ned Hall (2004, 2007). We will show how his ideas can be expressed in the CP-logic framework, and how this can be seen as an improvement. Further, we shall contrast his definition with our own.

Florian Steinberger (MCMP/LMU) gives a talk at the Workshop on Meaning: Models and Proofs (11-12 Oct, 2013, co-organized by the MCMP and the Arché Research Centre, St Andrews) titled "Understanding and inference".

Alex Yates (Arché) gives a talk at the Workshop on Meaning: Models and Proofs (11-12 Oct, 2013, co-organized by the MCMP and the Arché Research Centre, St Andrews) titled "Truth-conditional Theories of Meaning within Fregean Limits".

Ole T. Hjortland (MCMP/LMU) gives a talk at the Workshop on Meaning: Models and Proofs (11-12 Oct, 2013, co-organized by the MCMP and the Arché Research Centre, St Andrews) titled "Proof Theoretic Harmony in Substructural Logics".