scholarly journals Meaning-Preserving Translations of Non-classical Logics into Classical Logic: Between Pluralism and Monism

2021 ◽  
Author(s):  
Gerhard Schurz
Keyword(s):  
Classical Logic ◽  
Epistemic Value ◽  
Logical System ◽  
Ceteris Paribus ◽  
Quantum Logics ◽  
Logical Rules

AbstractIn order to prove the validity of logical rules, one has to assume these rules in the metalogic. However, rule-circular ‘justifications’ are demonstrably without epistemic value (sec. 1). Is a non-circular justification of a logical system possible? This question attains particular importance in view of lasting controversies about classical versus non-classical logics. In this paper the question is answered positively, based on meaning-preserving translations between logical systems. It is demonstrated that major systems of non-classical logic, including multi-valued, paraconsistent, intuitionistic and quantum logics, can be translated into classical logic by introducing additional intensional operators into the language (sec. 2–5). Based on this result it is argued that classical logic is representationally optimal. In sec. 6 it is investigated whether non-classical logics can be likewise representationally optimal. The answer is predominantly negative but partially positive. Nevertheless the situation is not symmetric, because classical logic has important ceteris paribus advantages as a unifying metalogic.

scholarly journals Why classical logic is privileged: justification of logics based on translatability

Synthese ◽  
2021 ◽  
Author(s):  
Gerhard Schurz
Keyword(s):  
Classical Logic ◽  
A Priori ◽  
The Other ◽  
Ceteris Paribus ◽  
Quantum Logics ◽  
Other Hand

AbstractIn Sect. 1 it is argued that systems of logic are exceptional, but not a priori necessary. Logics are exceptional because they can neither be demonstrated as valid nor be confirmed by observation without entering a circle, and their motivation based on intuition is unreliable. On the other hand, logics do not express a priori necessities of thinking because alternative non-classical logics have been developed. Section 2 reflects the controversies about four major kinds of non-classical logics—multi-valued, intuitionistic, paraconsistent and quantum logics. Its purpose is to show that there is no particular domain or reason that demands the use of a non-classical logic; the particular reasons given for the non-classical logic can also be handled within classical logic. The result of Sect. 2 is substantiated in Sect. 3, where it is shown (referring to other work) that all four kinds of non-classical logics can be translated into classical logic in a meaning-preserving way. Based on this fact a justification of classical logic is developed in Sect. 4 that is based on its representational optimality. It is pointed out that not many but a few non-classical logics can be likewise representationally optimal. However, the situation is not symmetric: classical logic has ceteris paribus advantages as a unifying metalogic, while non-classical logics can have local simplicity advantages.

scholarly journals A Certain Kind of Formal Theories

1965 ◽  
Vol 25 ◽  
pp. 59-86 ◽  
Author(s):  
Katuzi Ono
Keyword(s):  
Classical Logic ◽  
Axiom System ◽  
Logical System ◽  
First Order ◽  
Formal Theories

A common feature of formal theories is that each theory has its own system of axioms described in terms of some symbols for its primitive notions together with logical symbols. Each of these theories is developed by deduction from its axiom system in a certain logical system which is usually the classical logic of the first order.

scholarly journals CONSERVATIVELY EXTENDING CLASSICAL LOGIC WITH TRANSPARENT TRUTH

2012 ◽  
Vol 5 (2) ◽  
pp. 354-378 ◽  
Author(s):  
DAVID RIPLEY
Keyword(s):  
Classical Logic ◽  
The Other ◽  
Proof System ◽  
Truth Predicate ◽  
Logical System ◽  
Proof Systems ◽  
The Face

This paper shows how to conservatively extend a classical logic with a transparent truth predicate, in the face of the paradoxes that arise as a consequence. All classical inferences are preserved, and indeed extended to the full (truth-involving) vocabulary. However, not all classical metainferences are preserved; in particular, the resulting logical system is nontransitive. Some limits on this nontransitivity are adumbrated, and two proof systems are presented and shown to be sound and complete. (One proof system features admissible Cut, but the other does not.)

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.

scholarly journals On an Application of Intermediate Logics

1960 ◽  
Vol 16 ◽  
pp. 119-133
Author(s):  
Toshio Umezawa
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Logical System ◽  
Intermediate Logics ◽  
Individual Variables ◽  
Classical Predicate

In [1] I investigated some logics intermediate between intuitionistic and classical predicate logics. The purpose of this paper is to show the possibility of applying some intermediate logics to mathematics namely, to show that some mathematical theorems which are provable in the classical logic but not provable in the intuitionistic logic are provable in some intermediate logics. Let LZ be the logical system obtained from LJ′ a variant of Gentzen’s LJ [2], by adding as axioms all those sequents which can be obtained from a sequent scheme Z by substitution for propositional, predicate, or individual variables.

scholarly journals LOGIC IN THE TRACTATUS

2017 ◽  
Vol 10 (1) ◽  
pp. 1-50 ◽  
Author(s):  
MAX WEISS
Keyword(s):  
Set Theory ◽  
Classical Logic ◽  
Logical System ◽  
Image Position ◽  
Countably Infinite ◽  
The Universe

AbstractI present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named.There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is $\Pi _1^1$-complete. But third, it is only granted the assumption of countability that the class of tautologies is ${\Sigma _1}$-definable in set theory.Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects.

scholarly journals LOGIC FOR THE THEORY OF CONCEPTS

ARHE ◽  
2021 ◽  
Vol 27 (34) ◽  
pp. 85-102
Author(s):  
JOVANA KOSTIĆ
Keyword(s):  
Classical Logic ◽  
Predicate Calculus ◽  
Logical System ◽  
Logical Basis ◽  
Logical Foundation ◽  
Starting Point ◽  
Partial Predicates ◽  
Classical Predicate

In this paper, we follow Gödel’s remarks on an envisioned theory of concepts to determine which properties should a logical basis of such a theory have. The discussion is organized around the question of suitability of the classical predicate calculus for this role. Some reasons to think that classical logic is not an appropriate basis for the theory of concepts, will be presented. We consider, based on these reasons, which alternative logical system could fare better as a logical foundation of, in Gödel’s opinion, the most important theory in logic yet to be developed. This paper should, in particular, motivate the study of partial predicates in a certain system of three-valued logic, as a promising starting point for the foundation of the theory of concepts.

Completeness of global intuitionistic set theory

1997 ◽  
Vol 62 (2) ◽  
pp. 506-528 ◽  
Author(s):  
Satoko Titani
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Classical Logic ◽  
Mathematical Object ◽  
The Other ◽  
Logical System ◽  
Logical Operators ◽  
Logical Symbol ◽  
Image Position ◽  
Intuitionistic Set Theory

Gentzen's sequential system LJ of intuitionistic logic has two symbols of implication. One is the logical symbol → and the other is the metalogical symbol ⇒ in sequentsConsidering the logical system LJ as a mathematical object, we understand that the logical symbols ∧, ∨, →, ¬, ∀, ∃ are operators on formulas, and ⇒ is a relation. That is, φ ⇒ Ψ is a metalogical sentence which is true or false, on the understanding that our metalogic is a classical logic. In other words, we discuss the logical system LJ in the classical set theory ZFC, in which φ ⇒ Ψ is a sentence.The aim of this paper is to formulate an intuitionistic set theory together with its metatheory. In Takeuti and Titani [6], we formulated an intuitionistic set theory together with its metatheory based on intuitionistic logic. In this paper we postulate that the metatheory is based on classical logic.Let Ω be a cHa. Ω can be a truth value set of a model of LJ. Then the logical symbols ∧, ∨, →, ¬, ∀x, ∃x are interpreted as operators on Ω, and the sentence φ ⇒ Ψ is interpreted as 1 (true) or 0 (false). This means that the metalogical symbol ⇒ also can be expressed as a logical operators such that φ ⇒ Ψ is interpreted as 1 or 0.

scholarly journals Prawda i fałsz wypowiedzi literackiej

2020 ◽  
Vol 15 (6) ◽  
pp. 72-91
Author(s):  
Roman Magryś ◽  
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Specific Model ◽  
Logical System ◽  
Theoretical Assumptions ◽  
Point Of Reference

The article “Truth and Falsity of Literary Statements” investigates the issue of logical values, and consequently communication related status of sentences in indicative mood occurring in works of literature. The problem is discussed with reference to theoretical assumptions of classical logic, intuitionistic logic, and other possible intensional logics as well as phenomenological concepts proposed by Roman Ingarden. In this context it is suggested that intensional systems, mainly intuitionistic logic be adequately applied to identify logical value of literary sentences. As a result, it is assumed that the logical value of literary sentences depends on the specific logical system selected; according to the standards of intuitionistic logic, literary statements are true, likely or false. In this context it seems necessary to revise Roman Ingarden’s phenomenological assumption that sentences in indicative mood in a work of fiction do not have objective point of reference. It is suggested that such sentences be recognised as false, and therefore indicative of the group of their intentional meanings as a specific model of reality which can be deemed true or false.

Ceteris paribus

Quadrature ◽  
2010 ◽  
pp. 45-48
Author(s):  
Jean-Philippe Uzan
Keyword(s):  
Ceteris Paribus
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