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LINGUISTIX&LOGIK, Tony Marmo's blog
Thursday, 23 September 2004


Combining possibility and knowledge

By Alexandre Costa-Leite
Source: CLE

This paper is an attempt to define a new modality with philosophical interest by combining the basic modal ingredients of possibility and knowledge. This combination is realized via product of modal frames so as to construct a knowability modality, which is a bidimensional constructor of arity one defined in a two-dimensional modal frame. A semantical interpretation for the operator is proposed, as well as an axiomatic system able to account for inferences related to this new modality. The resulting logic for knowability LK is shown to be sound and complete with respect to its class of modal-epistemic product models.


Posted by Tony Marmo at 01:01 BST
Updated: Thursday, 23 September 2004 05:12 BST
Wednesday, 22 September 2004


Inconsistency without Contradiction

by Achille C. Varzi
Source: Notre Dame J. Formal Logic

Lewis has argued that impossible worlds are nonsense: if there were such worlds, one would have to distinguish between the truths about their contradictory goings-on and contradictory falsehoods about them; and this--Lewis argues--is preposterous. In this paper I examine a way of resisting this argument by giving up the assumption that `in so-and-so world' is a restricting modifier which passes through the truth-functional connectives. The outcome is a sort of subvaluational semantics which makes a contradiction 'A and not-A' false even when both 'A' and 'not-A' are true, just as supervaluational semantics makes a tautology 'A and not-A' true even when neither 'A' and 'not-A' are.


Posted by Tony Marmo at 01:01 BST
Updated: Wednesday, 22 September 2004 03:44 BST
Saturday, 11 September 2004


An Intensional Schr?dinger Logic

By Newton C. A. da Costa and D?cio Krause
Source: Notre Dame J. Formal Logic 38 (1997), no. 2, 179-194

We investigate the higher-order modal logic SwI, which is a variant of the system Sw presented in our previous work. A semantics for that system, founded on the theory of quasi sets, is outlined. We show how such a semantics, motivated by the very intuitive base of Schr?dinger logics, provides an alternative way to formalize some intensional concepts and features which have been used in recent discussions on the logical foundations of quantum mechanics; for example, that some terms like 'electron' have no precise reference and that 'identical' particles cannot be named unambiguously. In the last section, we sketch a classical semantics for quasi set theory.


General Logic
See also on Kai von Fintel's blog:
Klement on Logical Grammar in Frege, Russell, and Wittgenstein.

Posted by Tony Marmo at 01:01 BST
Updated: Saturday, 11 September 2004 20:12 BST
Thursday, 9 September 2004




da Costa & Krause
(excerpt, page 5)

It should be remarked that in the `classical world', which at first glance can be described by using standard logic and mathematics, if alpha and beta are both theses or theorems of a theory (founded on classical logic), then alpha & beta is also a thesis of that theory. This is what we intuitively mean when we say that on the grounds of classical logic, a true proposition cannot `exclude' another true proposition.

In classical logic, if from some group Delta1 of axioms of a theory T we deduce gamma, and if from another group Delta2 we deduce non-gamma, then gamma & non-gamma is also deductible in T.

Normally, our group Delta of axioms of T is finite, so that we may talk of the conjunction of its sentences instead of Delta itself. Then, if alpha and beta are respectively the conjunctions associated to Delta1 and Delta2, as above, we are looking for a theory T such that in T we may have alpha |- gamma and beta |- non-gamma, but in which gamma & non-gamma is not a theorem of T.

Therefore, our goal is to describe a way to formally avoid that Delta1 U Delta2 (or alpha&beta) entails a contradiction, since we do not intend to rule out `complementary situations'.

Posted by Tony Marmo at 01:53 BST
Updated: Thursday, 9 September 2004 01:57 BST


The Logic of Confusion

by Jean-Yves Beziau
The logic of confusion is a way to handle together incompatible viewpoints.
These viewpoints can be information data, physical experiments, sets of opinions or believes. Logics of confusion are obtained by generalizing Jaskowski-type semantics and combining it with many-valued semantics.

Keywords: paraconsistent logic, discussive logic, many-valued logic, quantum physics


Posted by Tony Marmo at 01:01 BST
Updated: Tuesday, 7 September 2004 01:15 BST
Tuesday, 7 September 2004



by Newton C. A. da Costa & D?cio Krause

Bohr's Principle of Complementarity is controversial and there has been much dispute over its precise meaning. Here, without trying to provide a detailed exegesis of Bohr's ideas, we take a very plausible interpretation of what may be understood by a theory which encompasses complementarity in a definite sense, which we term C-theories. The underlying logic of such theories is a kind of logic which has been termed `paraclassical', obtained from classical logic by a suitable modification of the notion of deduction. Roughly speaking, C-theories are non-trivial theories which may have `physically' incompatible theorems (and, in particular, contradictory theorems). So, their underlying logic is a kind of paraconsistent logic.

Keywords: Complementarity, Paraconsistency, Paraclassical Logic.


Posted by Tony Marmo at 01:01 BST
Updated: Thursday, 9 September 2004 02:00 BST


The Dialogical Dynamics of Adaptive Paraconsistency

Shahid Rahman & Jean Paul van Bendegem

The dialogical approach to paraconsistency as developed by Rahman and Carnielli ([1]), Rahman and Roetti ([2]) and Rahman ([3], [4] and [5]) suggests a way of studying the dynamic process of arguing with inconsistencies.
In his paper on Paraconsistency and Dialogue Logic ([6]) Van Bendegem suggests that an adaptive version of paraconsistency is the natural way of capturing the inherent dynamics of dialogues. The aim of this paper is to develop a formulation of dialogical paraconsistent logic in the spirit of an adaptive approach and which explores the possibility of eliminating inconsistencies by means of logical preference strategies.

One way to formulate paraconsistent logic within the dialogical approach as developed in Rahman and Carnielli ([1]), Rahman and Roetti ([2]) and Rahman ([4]) can be achieved in the following way. Assume that to the structural rules of the standard dialogical logic [i] we add the following:

Negative Literal Rule:
The Proponent is allowed to attack the negation of an atomic (propositional) statement (the so called negative literal) if and only if the Opponent has already attacked the same statement before.

This structural rule can be considered in analogy to the formal rule for positive literals. The idea behind this rule is the following: An inconsistency of the Opponent may be tolerated by using a type of charity principle. The inconsistency might involve different semantic contexts in which, say, aand ?ahave been asserted. Now, if the Opponent attacks ?awith ahe concedes thereby that there is some common context between ?aand awhich makes an attack on ?apossible. This allows the Proponent to attack the corresponding negation of the Opponent.

In Rahman and Carnielli ([1]) the logics produced by this rule were called Literal Dialogues , or shorter: L-D. In order to distinguish between the intuitionistic and the classical version Rahman and Carnielli wrote L-D i(for the intuitionistic version) and L-D c(for the classical version). To be precise we should call these logical systems literal dialogues with classical structural rules and literal dialogues with intuitionistic structural rules respectively.
The Proponent loses because he is not allowed to attack the move (5) (see negative literal rule). In other words the Opponent may have contradicted himself, but the semantic context of the negative literal is not available to the Proponent until the Opponent starts an attack on the same negative literal ?an attack which in this case will not take place.


Posted by Tony Marmo at 01:01 BST
Updated: Tuesday, 7 September 2004 16:52 BST
Monday, 6 September 2004



Roman Tuziak

Paraconsistent logic was introduced in order to provide the framework for inconsistent but nontrivial theories. It was initiated by J. Lukasiewicz (1910) in Poland and, independently, by N. A. Vasilev (1911-13) in Russia, but only in 1948 the first paraconsistent formal system was designed. Since then thousands of papers have been published in this field. Paraconsistency became one of the fastest growing branches of logic, due to its fruitful applications to computer science, information theory, and artificial intelligence. K. R. Popper touched on the problem in his paper ,,What is Dialectic?" (1940). Although only mentioned, his basic idea of the possibility of a formal system of such a logic was fresh and original. Another attempt of exploring the logic of contradiction, this time as a dual to intuitionistic logic, was made by Popper in his paper ,,On the Theory of Deduction I and II" (1948). The same idea was formalized by N. D. Goodman (1981) and developed by D. Miller (1993) under a label ,,Logic for Falsificationists".
Popper`s contribution to the subject of paraconsistent logic has not been properly recognized so far. Since Lukasiewicz`s and Vasilev`s works were still not translated into any West European languages in the 1940s, he should be undoubtedly regarded as an independent forerunner of paraconsistency. On the other hand, it seems tempting to adapt some of Popper`s other ideas for the theory of paraconsistent logic (the way it was done with Vasilev`s very general concepts) and, especially, for the theory of artificial intelligence.


Posted by Tony Marmo at 14:46 BST
Sunday, 5 September 2004


Paraconsistent Logics and Paraconsistency:
Technical and Philosophical Developments

by Newton C. A. da Costa, D?cio Krause & Ot?vio Bueno

Saying in a few words, paraconsistent logics (PL) are the logics that can be the logics of inconsistent but non-trivial theories. A deductive theory is paraconsistent if its underlying logic is paraconsistent. A theory is inconsistent if there is a formula (a grammatically well formed expression of its language) such that the formula and its negation are both theorems of the theory; otherwise, the theory is called consistent. A theory is trivial if all formulas of its language are theorems. Roughly speaking, in a trivial theory 'everything' (expressed in its language) can be proved. If the underlying logic of a theory is classical logic, or even any of the standard logical systems like intuitionistic logic, inconsistency entails triviality, and conversely. So, how can we speak of inconsistent but non trivial theories? Of course by exchanging the underlying logic to one which may admit inconsistency without making the system trivial. Paraconsistent logics do the job.
Let us remark that our use of terms like 'consistency', 'inconsistency', 'contradictory' and similar ones is syntactical, what is in accord with the original metamathematical terminology of Hilbert and his school. On the other hand, in order to treat such terms from the semantical point of view, in the field of paraconsistency, one must be able to build, for instance, a paraconsistent set theory beforehand. This is possible, as we shall see, although most semantics for paraconsistent logics are classical, i.e., constructed inside classical set theories. So, the best, to begin with, is to employ the above terms syntactically.

Source: CLE


See also:

Science and Partial Truth - A Unitary Approach to Models and Scientific Reasoning

by Newton C. A. da Costa & Steven French

Da Costa and French explore the consequences of adopting a 'pragmatic' notion of truth in the philosophy of science. Their framework sheds new light on issues to do with belief, theory acceptance, and the realism-antirealism debate, as well as the nature of scientific models and their heuristic development.

Check it

Posted by Tony Marmo at 01:01 BST
Updated: Sunday, 5 September 2004 13:56 BST
Saturday, 4 September 2004


Towards a Logic for Pragmatics. Assertions and Conjectures

by Gianluigi Bellin and Corrado Biasi

The logic for pragmatics extends classical logic in order to characterize the logical properties of the operators of illocutionary force such as that of assertion and obligation and of the pragmatic connectives which are given an intuitionistic interpretation. Here we consider the cases of assertions and conjectures : the assertion that a mathematical proposition _ is true is justified by the capacity to present an actual proof of _, while the conjecture that _ is true is justified by the absence of a refutation of _. We give sequent calculi of type G3i and G3i m inspired by Girard's LU , with subsystems characterizing intuitionistic reasoning and some forms of classical reasoning with such operators. Extending Godel, McKinsey, Tarski and Kripke's translations of intuitionistic logic into S4 , we show that our sequent calculi are sound and complete with respect to Kripke's semantics for S4 .


See also
G. Bellin and Corrado Biasi.
Cut-Elimination for the logic of pragmatics
(This is work in progress, which summarizes dr Corrado Biasi's Tesi di Laurea. It substantially refines and develops the ideas and the direction of research presented at the IMLA-FLOC'02 conference.)


Posted by Tony Marmo at 01:01 BST
Updated: Saturday, 4 September 2004 12:03 BST
Friday, 27 August 2004

Here is a new website for paraconsistency:

The home page for the philosophy of paraconsistency

Aim of this site

This site results, among others, from discussions between Jean-Yves Beziau and Alessio Moretti, about the philosophical relevance of paraconsistent logic. It's aim is to offer a frame to all searchers interested in exploring the philosophical implications of the many logical systems and approaches going under the general banner "paraconsistency". The analysis is, more or less, the following :
1) paraconsistent logic is by now a quite well established branch of mathematics, exploring mainly the properties of logical "negation". One has to note that the many technical results about paraconsistency are still quite unknown to the non-paraconsistent working mathematician, not to speak about the philosopher not used to mathematical formalisms.
2) Although paraconsistency was, at his beginning, motivated by deep philosophical reasons, it seems, by some kind of paradox, that now that the technical machinery has reached a satisfactory level of maturity, the philosophical interpretations of this new logical landscape are very few and still seem to lack of philosophical depth, at least in the sense of not (yet) reaching the scope of possible applications.
3) This unsatisfactory situation of the philosophical side of the enterprise is not due to some intrinsic reason of the subject, say "paraconsistency is not philosophically interesting". On the contrary, paraconsistency is highly interesting, probably crucial, to the philosopher (i.e. to all kinds of philosophers), and the relative poverty of the reflection thereupon is more a matter of coordination of the researchers and of accessibility, for the professional philosopher, of the technical discoveries, these being most of the time rather unintelligible to that respectable majority of philosophers not that much used to mathematical sophistication.
4) Which gives us the policy to follow for our present project : <...>


Posted by Tony Marmo at 01:01 BST
Updated: Monday, 30 August 2004 02:02 BST
Monday, 16 August 2004


A Paraconsistent Higher Order Logic

by J?rgen Villadsen

Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.

Posted by Tony Marmo at 01:01 BST
Saturday, 14 August 2004


What does Paraconsistency do?

The case of belief revision

by Koji Tanaka

In this talk, I apply a paraconsistent logic to the Grove's sphere semantics, that is a model for the AGM theory of belief revision. Firstly, I examine the soundness of the paraconsistent sphere semantics with respect to the AGM postulates. Secondly, I discuss some differences between classical (AGM) and a paraconsistent approach. I then argue that the theory of belief revision that is based on paraconsistent logic is simple and elegant, and of universal use.
Download link

Posted by Tony Marmo at 06:44 BST
Updated: Saturday, 14 August 2004 06:50 BST


Assertion and Denial, Commitment and Entitlement, and Incompatibility

by Greg Restall

In this short paper, I compare and contrast the kind of symmetricalist treatment of negation favoured in different ways by Huw Price (in "Why `Not'?") and by me (in "Multiple Conclusions") with Robert Brandom's analysis of scorekeeping in terms of commitment, entitlement and incompatibility.

Both kinds of account provide a way to distinguish the inferential significance of " A" and "A is warranted" in terms of a subtler analysis of our practices: on the one hand, we assert as well as deny; on the other, by distingushing downstream commitments from upstream entitlements and the incompatibility definable in terms of these. In this note I will examine the connections between these different approaches.


Posted by Tony Marmo at 01:01 BST
Wednesday, 11 August 2004


Paraconsistency Everywhere

By Greg Restall

Paraconsistent logics are, by definition, inconsistency tolerant:
In a paraconsistent logic, inconsistencies need not entail everything. However, there is more than one way a body of information can be inconsistent. In this paper I distinguish contradictions from other inconsistencies, and I show that several different logics are, in an important sense, 'paraconsistent' in virtue of being inconsistency tolerant without thereby being contradiction tolerant. For example, even though no inconsistencies are tolerated by intuitionistic propositional logic, some inconsistencies are tolerated by intuitionistic predicate logic. In this way, intuitionistic predicate logic is, in a mild sense, paraconsistent. So too are orthologic and quantum propositional logic and other formal systems. Given this fact, a widespread view that traditional paraconsistent logics are especially repugnant because they countenance inconsistencies is undercut. Many well-understood nonclassical logics countenance inconsistencies as well.

Download link

Posted by Tony Marmo at 01:01 BST

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