On Słupecki T-functions

1958 ◽  
Vol 23 (3) ◽  
pp. 267-270 ◽  
Author(s):  
Trevor Evans ◽  
P. B. Schwartz
Keyword(s):  
Propositional Logic ◽  
Value Function ◽  
Truth Values ◽  
Truth Value ◽  
The Many

In [2] E. L. Post defined a many-valued propositional logic to be functionally complete if, for every function on the set of truth-values, there exists a formula of the logic having that function as its associated truth-value function. He proved that the logic with truth-values 1, 2, …, m and (i) a unary connective ∼ such that ∼p has truth-value i+1 (mod m) when p has truth-value i, (ii) a binary connective ∨ such that p ∨ q has truth-value min(i, j) when p, q have truth-values i, j respectively, is functionally complete.The many-valued logics described by Łukasiewicz and Tarski [1] are not functionally complete. These logics have truth-values 1, 2, …, m and (i) a unary connective ~ such that ~p has truth-value m−i+1 when p has truth-value i, (ii) a binary connective → such that if p, q have truth-values i, j respectively, then p → q has truth-value 1 for i ≧ j, and truth-value 1 for i ≧ j. The functional incompleteness of these logics is immediate, since there exists no formula in p having truth-value i (≠ 1 or m) when p has truth-value 1.In [4] Słupecki showed that if a new unary connective T, such that T(p) has truth-value 2 for all truth-values assigned to p, is added to the 3-valued Łukasiewicz-Tarski logic, then the resulting logic is functionally complete. In [3] Rosser and Turquette proved this result for the m-valued (m ≧ 3) logic.

ENRICHED INTERVAL BILATTICES AND PARTIAL MANY-VALUED LOGICS: AN APPROACH TO DEAL WITH GRADED TRUTH AND IMPRECISION

1994 ◽  
Vol 02 (01) ◽  
pp. 37-54 ◽  
Author(s):  
FRANCESC ESTEVA ◽  
PERE GARCIA-CALVÉS ◽  
LLUÍS GODO
Keyword(s):  
Approximate Reasoning ◽  
Truth Values ◽  
Partial Models ◽  
Truth Value ◽  
The Many ◽  
Available Information ◽  
States Of Knowledge

Within the many-valued approach for approximate reasoning, the aim of this paper is two-fold. First, to extend truth-values lattices to cope with the imprecision due to possible incompleteness of the available information. This is done by considering two bilattices of truth-value intervals corresponding to the so-called weak and strong truth orderings. Based on the use of interval bilattices, the second aim is to introduce what we call partial many-valued logics. The (partial) models of such logics may assign intervals of truth-values to formulas, and so they stand for representations of incomplete states of knowledge. Finally, the relation between partial and complete semantical entailment is studied, and it is provedtheir equivalence for a family of formulas, including the so-called free well formed formulas.

Truth and Bivalence

2017 ◽  
Author(s):  
Jody Azzouni
Keyword(s):  
General Point ◽  
Coherence Theory ◽  
Truth Values ◽  
Perception Model ◽  
Truth Conditional ◽  
Truth Value ◽  
The Many

Some of the many ways that sentences with non-referring terms, such as “witch,” “Frodo,” and “casts spells,” are induced to have truth values are sketched. Three models are the axiomatic model, the fiction model, and the perception model. The general point is that the methods that we use to discover the truth values of sentences with referring terms can be generalized to sentences with non-referring terms. Even though truth-value inducing, in general, does not force a truth value on every sentence in a discourse, a commitment to bivalence is preserved by the use of expressions of ignorance. It’s also argued that traditional truth-conditional semantics should not be required to describe language-world relations. How adopting the coherence theory of truth for certain classes of sentences with non-referring terms avoids traditional objections to coherence views of truth is described.

Propositional logic

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Fuzzy Logic ◽  
Propositional Logic ◽  
Building Blocks ◽  
English Word ◽  
The Other ◽  
American Women ◽  
Truth Values ◽  
Other Hand ◽  
Truth Value ◽  
The Relationship

In propositional logic, atomic formulas are propositions. Any assertion will do. For example, . . . A = “Aristotle is dead,” B = “Barcelona is on the Seine,” and C = “Courtney Love is tall” . . . are atomic formulas. Atomic formulas are the building blocks used to construct sentences. In any logic, a sentence is regarded as a particular type of formula. In propositional logic, there is no distinction between these two terms. We use “formula” and “sentence” interchangeably. In propositional logic, as with all logics we study, each sentence is either true or false. A truth value of 1 or 0 is assigned to the sentence accordingly. In the above example, we may assign truth value 1 to formula A and truth value 0 to formula B. If we take proposition C literally, then its truth is debatable. Perhaps it would make more sense to allow truth values between 0 and 1. We could assign 0.75 to statement C if Miss Love is taller than 75% of American women. Fuzzy logic allows such truth values, but the classical logics we study do not. In fact, the content of the propositions is not relevant to propositional logic. Henceforth, atomic formulas are denoted only by the capital letters A, B, C,. . . (possibly with subscripts) without referring to what these propositions actually say. The veracity of these formulas does not concern us. Propositional logic is not the study of truth, but of the relationship between the truth of one statement and that of another. The language of propositional logic contains words for “not,” “and,” “or,” “implies,” and “if and only if.” These words are represented by symbols: . . . ￢ for “not,” ∧ for “and,” ∨ for “or,” → for “implies,” and ↔ for “if and only if.” . . . As is always the case when translating one language into another, this correspondence is not exact. Unlike their English counterparts, these symbols represent concepts that are precise and invariable. The meaning of an English word, on the other hand, always depends on the context.

Contextualism and the Many Senses of Knowledge

2005 ◽  
Vol 69 (1) ◽  
pp. 147-164 ◽  
Author(s):  
René van Woudenberg
Keyword(s):  
The Core ◽  
Knowledge Ascriptions ◽  
Truth Value ◽  
The Many ◽  
Peter Unger

Contextualists explain certain intuitions regarding knowledge ascriptions by means of the thesis that 'knowledge' behaves like an indexical. This explanation denies what Peter Unger has called invariantism, i.e., the idea that knowledge ascriptions have truth value independent of the context in which they are issued. This paper aims to provide an invariantist explanation of the contextualist's intuitions, the core of which is that 'knowledge' has many different senses.

The cylindric algebras of three-valued logic

1998 ◽  
Vol 63 (4) ◽  
pp. 1201-1217
Author(s):  
Norman Feldman
Keyword(s):  
Effective Procedure ◽  
Cylindric Algebras ◽  
Truth Values ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Truth Value ◽  
The One ◽  
Definition Of

In this paper we consider the three-valued logic used by Kleene [6] in the theory of partial recursive functions. This logic has three truth values: true (T), false (F), and undefined (U). One interpretation of U is as follows: Suppose we have two partially recursive predicates P(x) and Q(x) and we want to know the truth value of P(x) ∧ Q(x) for a particular x0. If x0 is in the domain of definition of both P and Q, then P(x0) ∧ Q(x0) is true if both P(x0) and Q(x0) are true, and false otherwise. But what if x0 is not in the domain of definition of P, but is in the domain of definition of Q? There are several choices, but the one chosen by Kleene is that if Q(X0) is false, then P(x0) ∧ Q(x0) is also false and if Q(X0) is true, then P(x0) ∧ Q(X0) is undefined.What arises is the question about knowledge of whether or not x0 is in the domain of definition of P. Is there an effective procedure to determine this? If not, then we can interpret U as being unknown. If there is an effective procedure, then our decision for the truth value for P(x) ∧ Q(x) is based on the knowledge that is not in the domain of definition of P. In this case, U can be interpreted as undefined. In either case, we base our truth value of P(x) ∧ Q(x) on the truth value of Q(X0).

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.

Parmenides on Ascertainment of the Real

1975 ◽  
Vol 4 (4) ◽  
pp. 623-633 ◽  
Author(s):  
T. M. Robinson
Keyword(s):  
Propositional Logic ◽  
Theory Of Knowledge ◽  
The Real ◽  
The Many

In this paper I want to suggest that, while the argued philosophical distinction between logic, epistemolgoy and ontology is one of the many achievements of Aristotle, his predecessor Parmenides was in fact already operating with a theory of knowledge and an elementary propositional logic that are of abiding philosophical interest. As part of the thesis I shall be obliged to reject a number of interpretations of particular passages in his poem, including one or two currently fashionable ones. Since so much turns on points of translation, I note for purposes of comparison what seem to be significant alternatives to my own in any particular instance. The line numbers are those of the DK text.

Contextualism, minimalism, and situationalism

2007 ◽  
Vol 15 (1) ◽  
pp. 115-137 ◽  
Author(s):  
Eros Corazza
Keyword(s):  
Semantic Content ◽  
Truth Values ◽  
Context Dependent ◽  
Truth Value ◽  
True False

After discussing some difficulties that contextualism and minimalism face, this paper presents a new account of the linguistic exploitation of context, situationalism. Unlike the former accounts, situationalism captures the idea that the main intuitions underlying the debate concern not the identity of propositions expressed but rather how truth-values are situation-dependent. The truth-value of an utterance depends on the situation in which the proposition expressed is evaluated. Hence, like in minimalism, the proposition expressed can be truth-evaluable without being enriched or expanded. Along with contextualism, it is argued that an utterance’s truth-value is context dependent. But, unlike contextualism and minimalism, situationalism embraces a form of relativism in so far as it maintains that semantic content must be evaluated vis-à-vis a given situation and, therefore, that a proposition cannot be said to be true/false eternally.

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.

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