scholarly journals The Propositional Logic of Frege’s Grundgesetze: Semantics and Expressiveness

2017 ◽  
Vol 5 (6) ◽  
Author(s):  
Eric D. Berg ◽  
Roy T. Cook
Keyword(s):  
Propositional Logic ◽  
Truth Values ◽  
Logical Operators ◽  
The Way

In this paper we compare the propositional logic of Frege’s Grundgesetze der Arithmetik to modern propositional systems, and show that Frege does not have a separable propositional logic, definable in terms of primitives of Grundgesetze, that corresponds to modern formulations of the logic of “not”, “and”, “or”, and “if…then…”. Along the way we prove a number of novel results about the system of propositional logic found in Grundgesetze, and the broader system obtained by including identity. In particular, we show that the propositional connectives that are definable in terms of Frege’s horizontal, negation, and conditional are exactly the connectives that fuse with the horizontal, and we show that the logical operators that are definable in terms of the horizontal, negation, the conditional, and identity are exactly the operators that are invariant with respect to permutations on the domain that leave the truth-values fixed. We conclude with some general observations regarding how Frege understood his logic, and how this understanding differs from modern views.

Inferentialism, Logicism, Harmony, and a Counterpoint

2020 ◽  
pp. 223-248
Author(s):  
Neil Tennant
Keyword(s):  
Special Reference ◽  
Natural Logic ◽  
Logical Operators ◽  
Crispin Wright ◽  
Truth Conditional ◽  
The Way

Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truth-conditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings. The logically reformist inferentialism of Dummett and Prawitz is contrasted with the more recent quietist inferentialism of Brandom. Various other issues are highlighted for inferentialism in general, by reference to which different kinds of inferentialism can be characterized. Inferentialism for the logical operators is explained, with special reference to the Principle of Harmony. The statement of that principle in the author’s book Natural Logic is fine-tuned here in the way obviously required in order to bar an interesting would-be counterexample furnished by Crispin Wright, and to stave off any more of the same.

A Mathematical Setting for Fuzzy Logics

1997 ◽  
Vol 05 (03) ◽  
pp. 223-238 ◽  
Author(s):  
Mai Gehrke ◽  
Carol Walker ◽  
Elbert Walker
Keyword(s):  
Fuzzy Logic ◽  
Propositional Logic ◽  
Normal Forms ◽  
Unit Interval ◽  
Fuzzy Logics ◽  
Real Numbers ◽  
Truth Values ◽  
Finite Algebras ◽  
Finite Algorithms ◽  
Interval Valued

The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and -x=1-x for conjunction, disjunction and negation, respectively, is the standard propositional fuzzy logic. This is shown to be the same as three-valued logic. The propositional logic obtained when the algebra of truth values is the set {(a, b)|a≤ b and a,b∈[0,1]} of subintervals of the unit interval with component-wise operations, is propositional interval-valued fuzzy logic. This is shown to be the same as the logic given by a certain four element lattice of truth values. Since both of these logics are equivalent to ones given by finite algebras, it follows that there are finite algorithms for determining when two statements are logically equivalent within either of these logics. On this topic, normal forms are discussed for both of these logics.

scholarly journals Tableau-based Decision Procedure for Non-Fregean Logic of Sentential Identity

2021 ◽  
pp. 41-57
Author(s):  
Joanna Golińska-Pilarek ◽  
Taneli Huuskonen ◽  
Michał Zawidzki
Keyword(s):  
Propositional Logic ◽  
Decision Procedure ◽  
Classical Propositional Logic ◽  
Complexity Bound ◽  
Sentential Calculus ◽  
Truth Values

AbstractSentential Calculus with Identity ($$\mathsf {SCI}$$ SCI ) is an extension of classical propositional logic, featuring a new connective of identity between formulas. In $$\mathsf {SCI}$$ SCI two formulas are said to be identical if they share the same denotation. In the semantics of the logic, truth values are distinguished from denotations, hence the identity connective is strictly stronger than classical equivalence. In this paper we present a sound, complete, and terminating algorithm deciding the satisfiability of $$\mathsf {SCI}$$ SCI -formulas, based on labelled tableaux. To the best of our knowledge, it is the first implemented decision procedure for $$\mathsf {SCI}$$ SCI which runs in NP, i.e., is complexity-optimal. The obtained complexity bound is a result of dividing derivation rules in the algorithm into two sets: decomposition and equality rules, whose interplay yields derivation trees with branches of polynomial length with respect to the size of the investigated formula. We describe an implementation of the procedure and compare its performance with implementations of other calculi for $$\mathsf {SCI}$$ SCI (for which, however, the termination results were not established). We show possible refinements of our algorithm and discuss the possibility of extending it to other non-Fregean logics.

scholarly journals The natural logic of language and cognition

Pragmatics ◽  
2006 ◽  
Vol 16 (1) ◽  
pp. 103-138 ◽  
Author(s):  
Pieter A.M. Seuren
Keyword(s):  
Set Theory ◽  
Propositional Logic ◽  
Predicate Logic ◽  
Human Cognition ◽  
Natural Logic ◽  
Analogous Argument ◽  
Standard Set ◽  
Language And Cognition ◽  
The Way

This paper aims at an explanation of the discrepancies between natural intuitions and standard logic in terms of a distinction between NATURAL and CONSTRUCTED levels of cognition, applied to the way human cognition deals with sets. NATURAL SET THEORY (NST) restricts standard set theory cutting it down to naturalness. The restrictions are then translated into a theory of natural logic. The predicate logic resulting from these restrictions turns out to be that proposed in Hamilton (1860) and Jespersen (1917). Since, in this logic, NO is a quantifier in its own right, different from NOT-SOME, and given the assumption that natural lexicalization processes occur at the level of basic naturalness, single-morpheme lexicalizations for NOT-ALL should not occur, just as there is no single-morpheme lexicalization for NOT-SOME at that level. An analogous argument is developed for the systematic absence of lexicalizations for NOT-AND in propositional logic.

scholarly journals Recovery operators, paraconsistency and duality

2019 ◽  
Vol 28 (5) ◽  
pp. 624-656 ◽  
Author(s):  
Walter Carnielli ◽  
Marcelo E Coniglio ◽  
Abilio Rodrigues
Keyword(s):  
Common Core ◽  
Classical Logic ◽  
Propositional Logic ◽  
Inference Rules ◽  
Object Language ◽  
Logical Operators ◽  
Formal Systems ◽  
Excluded Middle

Abstract There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express metalogical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the logics of formal inconsistency (LFIs) and by the logics of formal undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core—in the case studied here, this common core is classical positive propositional logic. The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of LFI and undeterminedness, namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion one at a time. The last sections offer an algebraic account for such logics by adapting the swap structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite nondeterministic matrices.

Thinking at the Edges

2020 ◽  
Vol 36 (1) ◽  
pp. 95-116
Author(s):  
Nathan Houser ◽  
Keyword(s):  
Way Of Life ◽  
Social Unrest ◽  
Truth Values ◽  
Mental Operations ◽  
Charles Peirce ◽  
The Way

The field of semiotic studies requires borders to function as a discipline but as a living science it is essential that those borders be unheeded. When Charles Peirce opened the modern field of semiotic studies he understood that he was an intellectual pioneer preparing the way for future semioticians. Peirce’s decision to equate semiotics with logic would likely seem bizarre to most professional logicians today yet his decision followed naturally from his view that all mental operations are sign actions and that semiosis is inferential. Peirce’s life-long study of sign types eventually led to a detailed, though provisional, classification of sixty-six distinct varieties of semiosis, many of which generate emotions or reactions rather than thoughts. Only twenty-one classes of signs yield interpretants that carry truth values or purport to be truth-preserving; the sign actions associated with these signs constitute the sphere of intellectual semiosis. The remaining forty-five non-intellectual sign classes drive perception and dominate the often unconscious mental operations that support and enrich day-to-day life. But this is also the realm of semiosis where memes flourish, where emoji function, and where propaganda first strikes a chord. This is the semiotic sphere where communal feeling can be engendered, but it is also the sphere of mob psychology. We are in troubled times during which signs are being used strategically to create dissension and social unrest and to generate disrespect for the very institutions that maintain the intelligence and practices that are fundamental for the survival of our way of life. It is time for semioticians to join forces against the weaponization of signs and I believe an investigation of the more primitive non-intellectual sign classes that Peirce identified will help lay the groundwork for the coming battle.

scholarly journals Djinn, Monotonic

10.29007/33k5 ◽  
2018 ◽  
Author(s):  
Conor McBride
Keyword(s):  
Propositional Logic ◽  
Program Synthesis ◽  
Intuitionistic Propositional Logic ◽  
Proof Search ◽  
Compositional Approach ◽  
Termination Proof ◽  
Simple Program ◽  
Synthesis Tool ◽  
The Way

Dyckhoff's algorithm for contraction-free proof search in intuitionistic propositional logic (popularized by Augustsson as the type-directed program synthesis tool, Djinn) is a simple program with a rather tricky termination proof. In this talk, I describe my efforts to reduce this program to a steady structural descent. On the way, I shall present an attempt at a compositional approach to explaining termination, via a uniform presentation of memoization.

scholarly journals Constructing illoyal algebra-valued models of set theory

2021 ◽  
Vol 82 (3) ◽  
Author(s):  
Benedikt Löwe ◽  
Robert Paßmann ◽  
Sourav Tarafder
Keyword(s):  
Set Theory ◽  
Propositional Logic ◽  
Truth Values ◽  
Models Of Set Theory

AbstractAn algebra-valued model of set theory is called loyal to its algebra if the model and its algebra have the same propositional logic; it is called faithful if all elements of the algebra are truth values of a sentence of the language of set theory in the model. We observe that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of illoyal models: tail stretches, transposition twists, and maximal twists.

scholarly journals Truth Tables Without Truth Values: On 4.27 and 4.42 of Wittgenstein’s Tractatus

2021 ◽  
pp. 212-220
Author(s):  
Tabea Rohr
Keyword(s):  
Truth Values ◽  
Logical Connectives ◽  
Truth Tables ◽  
The Way

AbstractIn 4.27 and 4.42 of his Tractatus Wittgenstein introduces quite complicated formulas, which are equivalent to $$2^n$$ 2 n and $$2^{2^{n}}$$ 2 2 n . This paper shows, however, that the formulas Wittgenstein presents fit particularly well with the way he thinks about truth values, logical connectives, tautologies, and contradictions. Furthermore, it will be shown how Wittgenstein could have avoided truth values even more radically. In this way it is demonstrated that the reference to truth values can indeed be substituted by talking of existing and non-existing facts.

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