scholarly journals Metainferences from a Proof-Theoretic Perspective, and a Hierarchy of Validity Predicates

2021 ◽  
Author(s):  
Rea Golan
Keyword(s):  
Paraconsistent Logics ◽  
Philosophical Logic ◽  
Soundness And Completeness ◽  
The One

AbstractI explore, from a proof-theoretic perspective, the hierarchy of classical and paraconsistent logics introduced by Barrio, Pailos and Szmuc in (Journal o f Philosophical Logic, 49, 93-120, 2021). First, I provide sequent rules and axioms for all the logics in the hierarchy, for all inferential levels, and establish soundness and completeness results. Second, I show how to extend those systems with a corresponding hierarchy of validity predicates, each one of which is meant to capture “validity” at a different inferential level. Then, I point out two potential philosophical implications of these results. (i) Since the logics in the hierarchy differ from one another on the rules, I argue that each such logic maintains its own distinct identity (contrary to arguments like the one given by Dicher and Paoli in 2019). (ii) Each validity predicate need not capture “validity” at more than one metainferential level. Hence, there are reasons to deny the thesis (put forward in Barrio, E., Rosenblatt, L. & Tajer, D. (Synthese, 2016)) that the validity predicate introduced in by Beall and Murzi in (Journal o f Philosophy, 110(3), 143–165, 2013) has to express facts not only about what follows from what, but also about the metarules, etc.

Logic and Philosophical Methodology

2016 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Classical Logic ◽  
Proof Theory ◽  
Recursion Theory ◽  
Theory Model ◽  
Philosophical Methodology ◽  
Paraconsistent Logics ◽  
Philosophical Logic

This article explores the role of logic in philosophical methodology, as well as its application in philosophy. The discussion gives a roughly equal coverage to the seven branches of logic: elementary logic, set theory, model theory, recursion theory, proof theory, extraclassical logics, and anticlassical logics. Mathematical logic comprises set theory, model theory, recursion theory, and proof theory. Philosophical logic in the relevant sense is divided into the study of extensions of classical logic, such as modal or temporal or deontic or conditional logics, and the study of alternatives to classical logic, such as intuitionistic or quantum or partial or paraconsistent logics. The nonclassical consists of the extraclassical and the anticlassical, although the distinction is not clearcut.

New Light on Medieval Philosophy: The Sophismata of Richard Kilvington

Dialogue ◽  
1992 ◽  
Vol 31 (3) ◽  
pp. 517-522 ◽  
Author(s):  
E. J. Ashworth
Keyword(s):  
Philosophy Of Language ◽  
Modern Science ◽  
Philosophical Logic ◽  
Medieval Philosophy ◽  
Medieval Logic ◽  
Before And After ◽  
The One ◽  
Domain I ◽  
Mathematics And Physics ◽  
Analysis Of Motion

The fourteenth-century English philosopher and theologian Richard Kilvington (1302/5–61) presents a useful correction to popular views of medieval philosophy in two ways. On the one hand, he reminds us that to think of medieval philosophy in terms of Aquinas, Duns Scotus and Ockham, or to think of medieval logic in terms of Aristotelian syllogistic, is to overlook vast areas of intellectual endeavour. Kilvington, like many before and after him, was deeply concerned with problems that would now be assigned to philosophy of language; philosophical logic and philosophy of science. He discussed topics in epistemic logic, semantic paradoxes, problems of reference, particularly those connected with the interplay between quantifiers and modal or temporal operators, and problems arising from the use of infinite series in the analysis of motion and change. On the other hand, this very account of his work raises the important issue of conceptual domain. I have spoken as if Kilvington's work can be neatly classified in terms of contemporary interests; and the temptation to read medieval philosophy in modern terms is only strengthened when one recognizes Kilvington as the first member of the group of Oxford calculatores, men such as William Heytesbury and Richard Swineshead, whose discussions of mathematics and physics have caused them to be hailed as forerunners of modern science.

scholarly journals De Finettian Logics of Indicative Conditionals Part II: Proof Theory and Algebraic Semantics

2021 ◽  
Author(s):  
Paul Égré ◽  
Lorenzo Rossi ◽  
Jan Sprenger
Keyword(s):  
Proof Theory ◽  
Future Research ◽  
Algebraic Semantics ◽  
Sequent Calculi ◽  
Truth Conditions ◽  
Indicative Conditionals ◽  
Tarski Algebra ◽  
Soundness And Completeness ◽  
Tableau Calculi ◽  
The One

AbstractIn Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti (1936) and Reichenbach (1935, 1944) on the one hand, and by Cooper (Inquiry, 11, 295–320, 1968) and Cantwell (Notre Dame Journal of Formal Logic, 49, 245–260, 2008) on the other. Here we provide the proof theory for the resulting logics and , using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: allows for algebraic completeness, but not for the construction of a canonical model, while fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects.

scholarly journals The Many and the One

2021 ◽  
Author(s):  
Salvatore Florio ◽  
Øystein Linnebo
Keyword(s):  
Expressive Power ◽  
Philosophical Logic ◽  
Plural Logic ◽  
Theoretical Frameworks ◽  
Higher Order Logic ◽  
Systematic Analysis ◽  
Novel Approach ◽  
The Many ◽  
The One ◽  
And Linguistics

Plural logic has become a well-established subject, especially in philosophical logic. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between this logic and other theoretical frameworks such as set theory, mereology, higher-order logic, and modal logic. The applications of plural logic rely on two assumptions, namely that this logic is ontologically innocent and has great expressive power. These assumptions are shown to be problematic. The result is a more nuanced picture of plural logic’s applications than has been given so far. Questions about the correct logic of plurals play a central role in the last part of the book, where traditional plural logic is rejected in favor of a “critical” alternative. The most striking feature of this alternative is that there is no universal plurality. This leads to a novel approach to the relation between the many and the one. In particular, critical plural logic paves the way for an account of sets capable of solving the set-theoretic paradoxes.

Enthymematic classical recapture 1

2018 ◽  
Vol 28 (5) ◽  
pp. 817-831
Author(s):  
Henrique Antunes
Keyword(s):  
Classical Logic ◽  
Paraconsistent Logics ◽  
Logical Theory ◽  
Oxford University ◽  
The One ◽  
Oxford University Press

AbstractPriest (2006, Ch.8, 2nd edn. Oxford University Press), argues that classical reasoning can be made compatible with his preferred (paraconsistent) logical theory by proposing a methodological maxim authorizing the use of classical logic in consistent situations. Although Priest has abandoned this proposal in favour of the one in G. Priest (1991, Stud. Log., 50, 321–331), I shall argue that due to the fact that the derivability adjustment theorem holds for several logics of formal (in)consistency (cf. W. A. Carnielli and M. E. Coniglio, 2016, Springer), these paraconsistent logics are particularly well suited to accommodate classical reasoning by means of a version of that maxim, yielding thus an enthymematic account of classical recapture.

scholarly journals Philosophy and Model Theory

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Philosophy Of Language ◽  
Philosophy Of Mathematics ◽  
Philosophical Logic ◽  
Philosophical Discussion ◽  
Mathematical Structures ◽  
Philosophical Significance ◽  
Basic Model ◽  
The One

Model theory is used in every theoretical branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant results are found only in mathematics textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a serious background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are scattered across disconnected pockets of papers. The first aim of this book, then, is to consider the philosophical uses of model theory, focusing on the central topics of reference, realism, and doxology. Its second aim is to address important questions in the philosophy of model theory, such as: sameness of theories and structure, the boundaries of logic, and the classification of mathematical structures. Philosophy and Model Theory will be accessible to anyone who has completed an introductory logic course. It does not assume that readers have encountered model theory before, but starts right at the beginning, discussing philosophical issues that arise even with conceptually basic model theory. Moreover, the book is largely self-contained: model-theoretic notions are defined as and when they are needed for the philosophical discussion, and many of the most philosophically significant results are given accessible proofs.

scholarly journals Sequent Calculi for the Propositional Logic of HYPE

Studia Logica ◽  
2021 ◽  
Author(s):  
Martin Fischer
Keyword(s):  
Mathematical Logic ◽  
Propositional Logic ◽  
The Other ◽  
Cut Elimination ◽  
Philosophical Logic ◽  
Admissible Rule ◽  
Sequent Calculi ◽  
Complete Proof ◽  
Elimination Procedure ◽  
The One

AbstractIn this paper we discuss sequent calculi for the propositional fragment of the logic of HYPE. The logic of HYPE was recently suggested by Leitgeb (Journal of Philosophical Logic 48:305–405, 2019) as a logic for hyperintensional contexts. On the one hand we introduce a simple $$\mathbf{G1}$$ G 1 -system employing rules of contraposition. On the other hand we present a $$\mathbf{G3}$$ G 3 -system with an admissible rule of contraposition. Both systems are equivalent as well as sound and complete proof-system of HYPE. In order to provide a cut-elimination procedure, we expand the calculus by connections as introduced in Kashima and Shimura (Mathematical Logic Quarterly 40:153–172, 1994).

scholarly journals Propositional Linear Temporal Logic with Initial Validity Semantics

2015 ◽  
Vol 23 (4) ◽  
pp. 379-386
Author(s):  
Mariusz Giero
Keyword(s):  
Temporal Logic ◽  
Formal System ◽  
Computer Programs ◽  
Linear Temporal Logic ◽  
Deduction Theorem ◽  
Temporal Logics ◽  
Initial State ◽  
Formal Systems ◽  
Soundness And Completeness ◽  
The One

Summary In the article [10] a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of “until” operator in a very strict version. The very strict “until” operator enables to express all other temporal operators. In this article we construct a formal system for LTLB with the initial semantics [12]. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article [10] and the one introduced in this article. Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in [10] and in this article can be used to carry out such verifications in Mizar [4].

scholarly journals Seguir una regla: tres interpretaciones

1986 ◽  
Vol 18 (53) ◽  
pp. 3-32
Author(s):  
Álvaro Rodríguez Tirado
Keyword(s):  
Philosophy Of Mind ◽  
Philosophy Of Mathematics ◽  
Recent Book ◽  
Philosophical Logic ◽  
Private Language ◽  
John Mcdowell ◽  
First Order ◽  
The One ◽  
The Right ◽  
Naming And Necessity

It is my contention that the profundity of Wittgenstein’s discussion of the problem of following a rule has not yet been fully appreciated in our philosophical environment. This is, to say the least, rather surprising, given its multitudinous connections with many other philosophical problems of the first order, especially, in the philosophy of mathematics, the philosophy of mind and philosophical logic. Saul Kripke’s latest contribution to philosophy has been a book whose title, Wittgenstein: On Rules and Private Language, deals precisely with these issues. Kripke’s discussion, brilliant and lucid as it was to be expected from the author of Naming and Necessity, has exerted an amazing influence on analytical philosophers dealing with problems as diverse as realism in semantics, the notion of ‘proof’ in mathematics, the possibility of a private language, the theory of meaning for a natural language, behaviourism in the philosophy of mind, the notion of objectivity, and many others. Notwithstanding the immense amount of resources which Kripke brought to hear in his discussion, I believe that if we follow him all the way, we end up with a feeling that the point we have reached is very different from the one Wittgenstein wanted and, indeed, argued for. If, then, my reading of Wittgenstein’s texts is on anything like the right lines, one should be a bit skeptical about Kripke’s exegesis. Two years after the publication of Kripke’s book, John McDowell wrote a splendid essay entitled ‘Wittgenstein on Following a Rule’ in which he challenges Kripke’s interpretation and, to my mind, some of McDowell’s arguments prove to be devastating of the position endorsed by Kripke. But McDowell considers it to be absolutely essential, for his own arguments to go through, to assume what I call ‘the community view’ on the practice of following a rule and this, I think, is a mistake. In a recent book, Colin McGinn has endorsed this conclusion, and I’ve tried to make it more appealing by exploring the possibility of bringing into play a causal theory of understanding.

Note on Selection of Coordinate Systems for Geodynamics

1975 ◽  
Vol 26 ◽  
pp. 395-407
Author(s):  
S. Henriksen
Keyword(s):  
Sea Level ◽  
The Other ◽  
Coordinate Systems ◽  
Mean Sea Level ◽  
The Earth ◽  
The One ◽  
Static Systems ◽  
Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

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