MCMP – Logic

Peter Verdée (Université catholique de Louvain) gives a talk at the MCMP Colloquium (16 April, 2015) titled "On Relevance of Entailment in a Classical Logic Context". Abstract: In this talk I present a logic that aims to determine whether implications in a classical logic context express a relevant connection between antecedent and consequent. Just like in the relevance logic tradition, the connective '->' is added to the language of classical logic as a formalisation of the relevant entailment relation in the object language (without nesting restrictions). The recursively enumerable set of theorems of the logic is defined by a set of axioms closed under a number of rules and the principle of uniform substitution. Unlike the logics in the relevance logic literature, the presented logic leaves classical negation intact in the sense that the law of non-contradiction can be used to obtain relevant implications, as long as there is a connection between antecedent and consequent. In order to realise this strengthening of traditional relevance logics, the traditional requirement that a theory of relevance should also define a new standard of deduction is given up. I present and argue for a list of requirements such a logical theory of relevant entailment in classical contexts needs to meet. I then formulate the system by presenting its axiomatisation and an algebraic semantics and demonstrate that it respects each of these requirements. This system is conceived by means of a translation into the relevant logic R and by presupposing an asymmetric treatment of symbols that occur in antecedents and symbols that occur in consequents. I moreover will give a philosophical motivation for our non-standard relevant implication and the asymmetric interpretation of antecedents and consequents. Finally I argue that there is no a priori reason to require that a successful formal theory of relevant entailment should also be a successful formal theory of (the standard of) deduction.

Catarina Dutilh Novaes (Groningen) gives a talk at the MCMP Colloquium (22 January, 2015) titled "A dialogical analysis of structural rules". Abstract: Recently, a number of substructural approaches to the semantic paradoxes have been proposed, which identify one or another structural rule as the ‘culprit’ giving rise to paradox. This is often taken to be sufficient motivation to restrict or altogether ban the rule in question. However, it seems that independent, i.e. not paradox-related, motivations would be required for such restrictions to be truly convincing. In other words, we need independent arguments on what is wrong with this or that structural rule, and to this end, we need to discuss the (putative) reasons why the rule was thought to be plausible in the first place. In this talk, I adopt a dialogical conception of deductive proofs in order to discuss the rationale behind weakening, contraction, and reflexivity. I argue that, while they may seem prima facie plausible dialogical principles, there are also purely dialogical reasons to restrict all three.

Joao Marcos (UFRN) gives a talk at the MCMP Colloquium (5 June, 2014) titled "Logical consequence explicated in terms of cognitive attitudes". Abstract: The received notions of logical consequence, either introduced by semantical means or by way of some convenient proof formalism, or even studied in their own right as abstract relations/operations between sentences or collections of sentences, are often explicated in terms of standard judgments such as assertion and refutation/denial. As a matter of fact, from the semantical viewpoint such judgments are often confused with truth-values. For a fresh view on the matter, we propose substituting judgments by a richer collection of primitive cognitive attitudes concerning acceptance or rejection, by an agent, of a given piece of information, and organize such attitudes into an opposition structure from which we show how to extract a generous bidimensional notion of entailment, henceforth called B-entailment, that generalizes the well-known approaches by Tarski, by Scott, and by

Shoesmith & Smiley ([7]). We study and prove a general characterization result about the underlying abstract consequence relations in terms of a bilattice-based structure of truth-values, show that it extends earlier results by G. Malinowski and S. Frankowski ([6,4]), and show how this connects to older and newer research on the structure of truth-values or of the space of valuations ([2,8,5]). Finally, we also prove a normal form result that shows how the B-entailment formalism is expressive enough so as to define any 4-valued (partial) nondeterministic matrix ([1,3]). This reports on joint work with Carolina Blasio and Carlos Caleiro.

Catarina Dutilh Novaes (Groningen) gives a talk at the MCMP Colloquium (22 January, 2015) titled "A dialogical analysis of structural rules". Abstract: Recently, a number of substructural approaches to the semantic paradoxes have been proposed, which identify one or another structural rule as the ‘culprit’ giving rise to paradox. This is often taken to be sufficient motivation to restrict or altogether ban the rule in question. However, it seems that independent, i.e. not paradox-related, motivations would be required for such restrictions to be truly convincing. In other words, we need independent arguments on what is wrong with this or that structural rule, and to this end, we need to discuss the (putative) reasons why the rule was thought to be plausible in the first place. In this talk, I adopt a dialogical conception of deductive proofs in order to discuss the rationale behind weakening, contraction, and reflexivity. I argue that, while they may seem prima facie plausible dialogical principles, there are also purely dialogical reasons to restrict all three.

Michael De (Konstanz) gives a talk at the MCMP Colloquium (18 December, 2014) titled "Negation as modality". Abstract: In a recent paper, Francesco Berto (forthcoming) defends an account of negation as a modality. According to that account, the negation of a sentence is true at a world x just in case the sentence is not true in any world compatible with x. Alternatively, the negation of a sentence is true at x just in case all the worlds at which the sentence is true are incompatible with x. Compatibility is taken to be the key notion in the account, and what minimal properties a negation has comes down to which minimal conditions compatibility satisfies. In this talk, we first point out what we take to be serious problems for this modal account. Second, we propose an alternative, non-modal, understanding of negation.

Joao Marcos (UFRN) gives a talk at the MCMP Colloquium (5 June, 2014) titled "Logical consequence explicated in terms of cognitive attitudes". Abstract: The received notions of logical consequence, either introduced by semantical means or by way of some convenient proof formalism, or even studied in their own right as abstract relations/operations between sentences or collections of sentences, are often explicated in terms of standard judgments such as assertion and refutation/denial. As a matter of fact, from the semantical viewpoint such judgments are often confused with truth-values. For a fresh view on the matter, we propose substituting judgments by a richer collection of primitive cognitive attitudes concerning acceptance or rejection, by an agent, of a given piece of information, and organize such attitudes into an opposition structure from which we show how to extract a generous bidimensional notion of entailment, henceforth called B-entailment, that generalizes the well-known approaches by Tarski, by Scott, and by

Shoesmith & Smiley ([7]). We study and prove a general characterization result about the underlying abstract consequence relations in terms of a bilattice-based structure of truth-values, show that it extends earlier results by G. Malinowski and S. Frankowski ([6,4]), and show how this connects to older and newer research on the structure of truth-values or of the space of valuations ([2,8,5]). Finally, we also prove a normal form result that shows how the B-entailment formalism is expressive enough so as to define any 4-valued (partial) nondeterministic matrix ([1,3]). This reports on joint work with Carolina Blasio and Carlos Caleiro.

Maarten McKubre-Jordens (Canterbury) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "Doing mathematics paraconsistently. A manifesto". Abstract: In this talk, we outline several motivations for conducting mathematics–in the style of the working mathematician–without dependence on assumptions of non-contradiction. The story involves a short analysis of theorem and counterexample, what it is to reason paraconsistently within mathematics, and takes note of some non-traditional obstacles and attempts to resolve them. In part, this will provide motivation to the mathematician to think outside the box when approaching surprising conclusions within the usual framework. Then, as we delve into the mathematics, we survey some recent results in elementary analysis when performed paraconsistently, and outline some conjectures for future research. This talk is of interest both to provide reasons and techniques for paraconsistent mathematics, and to show how rich a picture can be painted without recourse to assumptions of non-contradiction.

Bryson Brown (Lethbridge) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "On the Preservation of Reliability". Abstract: …all models are wrong, but some are useful. (G E.P. Box and N. R. Draper, 1987). C.S. Peirce examined several broad methods for arriving at beliefs in “On the Fixation of Belief”; the central theme of his essay is the importance of having a method that leads to stable agreement amongst the members of a society. Peirce argues that the ‘scientific method meets this standard, generalizing Hobbes’ observation that Harvey’s hypothesis of the circulation of the blood is an important example of a once-controversial view that came to be accepted even by those who initially rejected it, and a demonstration of the special epistemic success of science, in contrast with other forms of inquiry. But stable agreements are sometimes overturned. The brilliant, wide-ranging successes of classical physics made the basic principles of Newton’s theory laws of nature in the eyes of physicists, philosophers and the educated public. Yet they have been superseded by new principles. This pattern of success followed by failure and replacement is the key premise of Laudan’s pessimistic induction argument against scientific realism. Yet we have confidence, often justified confidence, in many scientific inferences, including inferences that begin with models and theoretical principles known to be false. Though the equations used and descriptions of actual systems our models are applied to are false, the results of the calculations (even when only approximate) are considered reliable for many purposes, and with good reason. Thus models provided by orbital mechanics are used to place probes in desired orbits and even land on other planets, while atmospheric GCMs are used to estimate large-scale climate changes likely to occur given various scenarios for human GHG emissions. Neither the equations used nor the descriptions of the systems such models include are true; the results of the calculations (even assuming the calculations are exact) cannot reasonably be taken to be true, either. Yet we do rely on them, and with good reason. Many measurable physical quantities that models allow us to calculate values for turn out to be very close to the results of actual observations under a wide range of specifiable conditions. This paper develops a pragmatic view of theories, models and the inferences we use our models and theories to make. The preservation of reliability allows for the use of incompatible theories and principles in our inferences, along with contextually-determined levels of acceptable approximation, and conceptual shifts in how measurement results are interpreted in the light of the different principles relied on at different points. But it is also compatible with a modest scientific realism: the criteria by which we decide when a theory can be relied on generally include measurable parameters whose values, even though our understanding of them remains imperfect, reliably indicate when and to what extent a given theory is reliable and constitutes reliably settled science in a sense that Peirce might have found satisfying even without an accepted background theory in which we can explain both that reliability and its limits.

Otávio Bueno (Miami) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "Inconsistent scientific Theories: A Framework". Abstract: Four important issues need to be considered when inconsistent scientific theories are under discussion: (1) To begin with, are there–and can there be–such things as inconsistent scientific theories? On standard conceptions of the structure of scientific theories, such as the semantic and the syntactic approaches (Suppe [1989], and van Fraassen [1980]), there is simply no room for such theories, given the classical underpinnings of these views. In fact, both the syntactic and the semantic approaches assume that the underlying logic is classical, and as is well known, in classical logic everything follows from an inconsistent theory. Despite this fact, it seems undeniable that inconsistent scientific theories have been entertained–or, at least, stumbled upon–throughout the history of science. So, it looks as though we need to make room for them. (2) But once some room is made for inconsistent scientific theories, how exactly should they be accommodated? In particular, it seems crucial that we are able to understand the styles of reasoning that involve inconsistencies; that is, the various ways in which scientists and mathematicians reason from inconsistent assumptions without deriving everything from them. It is tempting, of course, to adopt a paraconsistent logic to model some of the reasoning styles in question (see da Costa and French [2003], da Costa, Krause, and Bueno [2007], and da Costa, Bueno, and French [1998]). This is certainly a possibility. However, actual scientific practice is not typically done using paraconsistent logic. And if our goal is to understand that practice in its own terms, rather than to produce a parallel discourse about that practice that somehow justifies the adequacy of the latter by invoking tools that are foreign to it, an entirely different strategy is called for. (3) What are the sources of the inconsistencies in scientific theories? Do such inconsistencies emerge from empirical reasons, from conceptual reasons, from both, or by sheer mistake? By identifying the various sources in question, we can handle and assess the significance of the inconsistencies in a better way. Perhaps some inconsistencies are more important, troublesome, or heuristically fruitful than others—and this should be part of their assessment. (4) Several scientific theories become inconsistent due to the mathematical framework they assume. For example, the theories may refer to infinitesimals, as the latter were originally formulated in the early versions of the calculus (see Robinson [1974] and Bell [2005]), the theories may invoke Dirac’s delta function (Dirac [1958]), or some other arguably inconsistent mathematical framework. The issue then arises as to how we should deal with inconsistent applied mathematical theories. What is the status of these theories? Which commitments do they bring? Are we committed to the existence of inconsistent objects if we use such theories in explaining the phenomena? Can an inconsistent scientific theory ever be indispensable? Questions of this sort need to be answered so that we can make sense of the role of inconsistent theories in applications. (For an insightful discussion, see Colyvan [2009].) In this paper, I examine these four issues, and develop a framework–in terms of partial mappings (Bueno, French and Ladyman [2002], and Bueno [2006]), and the inferential conception of the application of mathematics (Bueno and Colyvan [forthcoming])–to represent and interpret inconsistent theories in science. Along the way, I illustrate how the framework can be used to make sense of various allegedly inconsistent theories, from the early formulations of the calculus through Dirac’s delta function and Bohr’s atomic model (Bohr [1913]).

Holger Andreas (MCMP/LMU) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "A Paraconsistent Generalization of Carnap's Logic of Theoretical Terms".