scholarly journals Many-valued logics—implications and semantic consequences

2013 ◽  
Vol 5 (2) ◽  
pp. 145-166
Author(s):  
Katalin Pásztor Varga ◽  
Gábor Alagi
Keyword(s):  
Classical Logic ◽  
Propositional Logic ◽  
Matrix Method ◽  
Decision Problem ◽  
Modus Ponens ◽  
Deduction Theorem ◽  
Classical Propositional Logic ◽  
Consequence Operation ◽  
Semantic Consequence ◽  
Logical Calculi

Abstract In this paper an application of the well-known matrix method to an extension of the classical logic to many-valued logic is discussed: we consider an n-valued propositional logic as a propositional logic language with a logical matrix over n truth-values. The algebra of the logical matrix has operations expanding the operations of the classical propositional logic. Therefore we look over the Łukasiewicz, Post, Heyting and Rosser style expansions of the operations negation, conjunction, disjunction and with a special emphasis on implication. In the frame of consequence operation, some notions of semantic consequence are examined. Then we continue with the decision problem and the logical calculi. We show that the cause of difficulties with the notions of semantic consequence is the weakness of the reviewed expansions of negation and implication. Finally, we introduce an approach to finding implications that preserve both the modus ponens and the deduction theorem with respect to our definitions of consequence.

scholarly journals An Holistic Extension for Classical Logic via Quantum Fredkin Gate

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1178
Author(s):  
Hector Freytes ◽  
Giuseppe Sergioli
Keyword(s):  
Quantum Computation ◽  
Classical Logic ◽  
Propositional Logic ◽  
Classical Propositional Logic ◽  
Mixed States ◽  
Fredkin Gate ◽  
Bipartite States ◽  
Extended Notion

An holistic extension for classical propositional logic is introduced in the framework of quantum computation with mixed states. The mentioned extension is obtained by applying the quantum Fredkin gate to non-factorizable bipartite states. In particular, an extended notion of classical contradiction is studied in this holistic framework.

Implication with possible exceptions

2001 ◽  
Vol 66 (2) ◽  
pp. 517-535
Author(s):  
Herman Jurjus ◽  
Harrie de Swart
Keyword(s):  
Propositional Logic ◽  
Modus Ponens ◽  
The Other ◽  
Classical Propositional Logic ◽  
Consequence Relation ◽  
Other Hand

AbstractWe introduce an implication-with-possible-exceptions and define validity of rules-with-possible-exceptions by means of the topological notion of a full subset. Our implication-with-possible-exceptions characterises the preferential consequence relation as axiomatized by Kraus, Lehmann and Magidor [Kraus, Lehmann, and Magidor, 1990]. The resulting inference relation is non-monotonic. On the other hand, modus ponens and the rule of monotony, as well as all other laws of classical propositional logic, are valid-up-to-possible exceptions. As a consequence, the rules of classical propositional logic do not determine the meaning of deducibility and inference as implication-without-exceptions.

scholarly journals Holistic Type Extension for Classical Logic via Toffoli Quantum Gate

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 636 ◽  
Author(s):  
Hector Freytes ◽  
Roberto Giuntini ◽  
Giuseppe Sergioli
Keyword(s):  
Quantum Computation ◽  
Classical Logic ◽  
Propositional Logic ◽  
Quantum States ◽  
Quantum Gate ◽  
Logical System ◽  
Classical Propositional Logic ◽  
Mixed States ◽  
Werner States ◽  
To Receive

A holistic extension of classical propositional logic is introduced via Toffoli quantum gate. This extension is based on the framework of the so-called “quantum computation with mixed states”, where also irreversible transformations are taken into account. Formal aspects of this new logical system are detailed: in particular, the concepts of tautology and contradiction are investigated in this extension. These concepts turn out to receive substantial changes due to the non-separability of some quantum states; as an example, Werner states emerge as particular cases of “holistic" contradiction.

FRACTIONAL SEMANTICS FOR CLASSICAL LOGIC

2019 ◽  
Vol 13 (4) ◽  
pp. 810-828
Author(s):  
MARIO PIAZZA ◽  
GABRIELE PULCINI
Keyword(s):  
Classical Logic ◽  
Propositional Logic ◽  
Logical Formula ◽  
Classical Propositional Logic ◽  
Multivalued Semantics

AbstractThis article presents a new (multivalued) semantics for classical propositional logic. We begin by maximally extending the space of sequent proofs so as to admit proofs for any logical formula; then, we extract the new semantics by focusing on the axiomatic structure of proofs. In particular, the interpretation of a formula is given by the ratio between the number of identity axioms out of the total number of axioms occurring in any of its proofs. The outcome is an informational refinement of traditional Boolean semantics, obtained by breaking the symmetry between tautologies and contradictions.

The Development of Modus Ponens in Antiquity : From Aristotle to the 2nd Century AD

Phronesis ◽  
2002 ◽  
Vol 47 (4) ◽  
pp. 359-394 ◽  
Author(s):  
Susanne Bobzien
Keyword(s):  
20Th Century ◽  
Propositional Logic ◽  
Late Antiquity ◽  
Modus Ponens ◽  
Classical Propositional Logic ◽  
Aristotelian Logic ◽  
Gradual Development ◽  
Stoic Logic ◽  
Short Presentation

Abstract'Aristotelian logic', as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classified as 'hypothetical syllogisms'. However, Aristotle did not discuss such arguments, nor did he call any arguments 'hypothetical syllogisms'. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them 'hypothetical syllogisms'; nor did they describe them as ponendo ponens, etc. The tradition of the four argument forms and the classification of the arguments as hypothetical syllogisms hence need some explaining. In this paper, I offer some explanations by tracing the development of certain elements of Aristotle's logic via the early Peripatetics to the logic of later antiquity. I consider the questions: How did the four argument forms arise? Why were there four of them? Why were arguments of these forms called 'hypothetical syllogisms'? On what grounds were they considered valid? I argue that such arguments were neither part of Aristotle's dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle's logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some 20th century theories.

PEIRCE’S CALCULI FOR CLASSICAL PROPOSITIONAL LOGIC

2018 ◽  
Vol 13 (3) ◽  
pp. 509-540 ◽  
Author(s):  
MINGHUI MA ◽  
AHTI-VEIKKO PIETARINEN
Keyword(s):  
Propositional Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Theoretic Analysis ◽  
Classical Propositional Logic ◽  
Deduction System ◽  
Natural Deduction System ◽  
Logical Algebra ◽  
Graphical System ◽  
Logical Calculi

AbstractThis article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted byPC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to presentPCas a sequent calculus. The law of distributivity, which Peirce states in 1880, is proved using Peirce’s Rule, which is a residuation, inPC. The transitional systems of the algebra of the copula that Peirce develops since 1880 paved the way to the 1896 graphical system of the alpha graphs. It is shown how the rules of the alpha system reinterpret Boolean algebras, answering Peirce’s statement that logical graphs supply a new system of fundamental assumptions to logical algebra. A proof-theoretic analysis is given for the connection betweenPCand the alpha system.

An Essay on Complex Valued Propositional Logic

2013 ◽  
Vol 11 (1) ◽  
pp. 2-7 ◽  
Author(s):  
V. Sgurev
Keyword(s):  
Boolean Algebra ◽  
Propositional Logic ◽  
Modus Ponens ◽  
Complex Variables ◽  
Inference Rules ◽  
Modus Tollens ◽  
Classical Propositional Logic ◽  
Admissible Solutions ◽  
Complex Valued ◽  
Truth Tables

Abstract In decision making logic it is often necessary to solve logical equations for which, due to the features of disjunction and conjunction, no admissible solutions exist. In this paper an approach is suggested, in which by the introduction of Imaginary Logical Variables (ILV), the classical propositional logic is extended to a complex one. This provides a possibility to solve a large class of logical equations.The real and imaginary variables each satisfy the axioms of Boolean algebra and of the lattice. It is shown that the Complex Logical Variables (CLV) observe the requirements of Boolean algebra and the lattice axioms. Suitable definitions are found for these variables for the operations of disjunction, conjunction, and negation. A series of results are obtained, including also the truth tables of the operations disjunction, conjunction, negation, implication, and equivalence for complex variables. Inference rules are deduced for them analogous to Modus Ponens and Modus Tollens in the classical propositional logic. Values of the complex variables are obtained, corresponding to TRUE (T) and FALSE (F) in the classic propositional logic. A conclusion may be made from the initial assumptions and the results achieved, that the imaginary logical variable i introduced hereby is “truer” than condition “T” of the classic propositional logic and i - “falser” than condition “F”, respectively. Possibilities for further investigations of this class of complex logical structures are pointed out

scholarly journals Control effects as a modality

2009 ◽  
Vol 19 (1) ◽  
pp. 17-26 ◽  
Author(s):  
HAYO THIELECKE
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Propositional Logic ◽  
The Other ◽  
Classical Propositional Logic ◽  
Double Negation ◽  
Other Hand

AbstractWe combine ideas from types for continuations, effect systems and monads in a very simple setting by defining a version of classical propositional logic in which double-negation elimination is combined with a modality. The modality corresponds to control effects, and it includes a form of effect masking. Erasing the modality from formulas gives classical logic. On the other hand, the logic is conservative over intuitionistic logic.

scholarly journals Natural three-valued logics and classical logic

2013 ◽  
Vol 19 ◽  
pp. 344-352 ◽  
Author(s):  
Н.Е. Томова
Keyword(s):  
Classical Logic ◽  
Propositional Logic ◽  
Classical Propositional Logic

In this paper implicative fragments of natural three- valued logic are investigated. It is proved that some fragments are equivalent by set of tautologies to implicative fragment of classical logic. It is also shown that some natural three-valued logics verify all tautologies of classical propositional logic.

Hilbert-style proof Calculus for Propositional Logic in ABC notation

2019 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Propositional Logic ◽  
Inference Rule ◽  
Modus Ponens ◽  
Axiomatic System ◽  
First Contact

All nine axioms and a single inference rule of logic (Modus Ponens) within the Hilbert axiomatic system are presented using capital letters (ABC) in order to familiarize the beginner student in hers/his first contact with the topic.

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