Indeterminate Truth Research Articles

The Modal Future

<i>The Modal Future</i>

A few little steps beyond Knuth’s Boolean Logic Table with Neutrosophic Logic: A Paradigm Shift in Uncertain Computation

The present article delves into the extension of Knuth’s fundamental Boolean logic table to accommodate the complexities of indeterminate truth values through the integration of neutrosophic logic (Smarandache Christianto, 2008). Neutrosophic logic, rooted in Florentin Smarandache’s groundbreaking work on Neutrosophic Logic (cf. Smarandache, 2005, and his other works), introduces an additional truth value, ‘indeterminate,’ enabling a more comprehensive framework to analyze uncertainties inherent in computational systems. By bridging the gap between traditional boolean operations and the indeterminacy present in various real-world scenarios, this extension redefines logic tables, introducing neutrosophic operators that capture nuances beyond the binary realm. Through a thorough exploration of neutrosophic logic's principles and its implications in computational paradigms, this study proposes a novel approach to logic design that accommodates uncertain, imprecise, and incomplete information. This paradigm shift in logic tables not only broadens the spectrum of computing methodologies but also holds promise in fields such as decision-making systems and data analytics. This article amalgamates insights from over twelve key references encompassing seminal works in boolean logic, neutrosophic logic, and their applications in diverse scientific and computational domains, aiming to pave the way for a more robust and adaptable logic framework in computation.

Lógica difusa y el riesgo financiero. Una propuesta de clasificación de riesgo financiero al sector cooperativo

Apply fuzzy logic in financial indicators is a proposal not widespread in the accounting field. This methodology allows us to observe the results of financial ratios with a broader perspective, showing not completely true nor completely false results, because they can take an indeterminate truth value within a set of values, applying the theory of fuzzy logic. The aim of this paper is to present the reader with the application of fuzzy logic in indicators of financial risk, using ratios of the cooperatives segment each Ecuador, and thus validate the level of relevance that has this indicator to compared with the standard meta model and CAMEL ratings. To apply these theory linguistic variables were used, which were valued at scales ranges from 0 to 1. It is determined that the diffuse methodologist applied to financial risks presented a higher level of membership to good credit rating by ensuring a level of low risk and very good solvency. However, in periods of low economic activity it would stagnate at this level for the increased risk.

Making sense of (in)determinate truth: the semantics of free variables

It is argued that truth value of a sentence containing free variables in a context of use (or the truth value of the proposition it expresses in a context of use), just as the reference of the free variables concerned, depends on the assumptions and posits given by the context. However, context may under-determine the reference of a free variable and the truth value of sentences in which it occurs. It is argued that in such cases a free variable has indeterminate reference and a sentence in which it occurs may have indeterminate truth value. On letting, say, x be such that x^2=4, the sentence ‘Either x=2 or x=-2’ is true but the sentence ‘x=2’ has an indeterminate truth value: it is determinate that the variable x refers to either 2 or -2, but it is indeterminate which of the two it refers to, as a result ‘x=2’ has a truth value but its truth value is indeterminate. The semantic indeterminacy is analysed in a ‘radically’ supervaluational (or plurivaluational) semantic framework closely analogous to the treatment of vagueness in McGee and McLaughlin (South J Philos 33:203–251, 1994, Linguist Philos 27:123–136, 2004) and Smith (Vagueness and degrees of truth, Oxford University Press, Oxford, 2008), which saves bivalence, the T-schema and the truth-functional analysis of the boolean connectives. It is shown that on such an analysis the modality ‘determinately’ is quite clearly not an epistemic modality, avoiding a potential objection raised by Williamson (Vagueness, Routledge, London, 1994) against such ‘radically’ supervaluational treatments of vagueness, and that determinate truth (rather than truth simpliciter) is the semantic value preserved in classically valid arguments. The analysis is contrasted with the epistemicist proposal of Breckenridge and Magidor (Philos Stud 158:377–400, 2012) which implies that (in the given context) ‘x=2’ has a determinate but unknowable truth value.

Art and fiction are signals with indeterminate truth values.

Menninghaus et al. distinguish art from fiction, but no current arguments or data suggest that the concept of art can be meaningfully circumscribed. This is a problem for aesthetic psychology. I sketch a solution by rejecting the distinction: Unlike most animal communication, in which signals are either true or false, art and fiction consist of signals without determinate truth values.

SUPER LIARS

Kripke’s theory of truth offered a trivalent semantics for a language which, like English, contains a truth predicate and means of self-reference; but it did so by severely restricting the expressive power of the logic. In Kripke’s analysis, the Liar (e.g.,This very sentence is not true) receives the indeterminate truth value, but this fact cannot be expressed in the language; by contrast, it is straightforward to say in English thatthe Liar is something other than true. Kripke’s theory also fails to handle the Strengthened Liar, which can be expressed in English as:This very sentence is something other than true. We develop a theory which seeks to overcome these difficulties, and is based on a detailed analysis of some of the linguistic means by which the Strengthened Liar can be expressed in English. In particular, we propose to takeliterallythe quantificational form of the negative expressionsomething other than true.Like other quantifiers, it may have different implicit domain restrictions, which give rise to a variety of negations of different strengths (e.g.,something other than true among the values{0, 1}, oramong{0, 1, 2}, etc). This analysis naturally leads to a logic with as many truth values as there are ordinals—a conclusion reached independently by Cook (2008a). We develop the theory within a generalization of the Strong Kleene Logic, augmented with negations that each have a nonmonotonic semantics. We show that fixed points can be constructed for our logic, and that it enjoys a limited form of ‘expressive completeness’. Finally, we discuss the relation between our theory and various alternatives, including one in which the wordtrue(rather than negation) is semantically ambiguous, and gives rise to a hierarchy of truth predicates of increasing strength.

Tacit verification of determinate and indeterminate text ideas.

Singer presented evidence that reading time reflects, in part, processes of continuously verifying current clauses with reference to their text antecedents. This study extended that analysis to text ideas of indeterminate truth in their discourse context. For example, having read simply that Norm's house was destroyed, one might later read that Norm's cousin believed that Norm's house was destroyed by a fire. Research concerning people's reports of not knowing something suggested that the reading of clauses of indeterminate truth would be faster than for false or inconsistent clauses. In two experiments, target sentences varied in their truth, verb entailments, and use of negation with reference to their antecedents. Reading time for indeterminate targets was sometimes distinctly shorter and was never longer than for corresponding false targets. The effects were modulated by the entailments of factive and nonfactive main verbs and by negative expression. All of the conditions that were identical to ones of Singer (2006) replicated the former findings. It was concluded that encountering text ideas of indeterminate truth does not initiate protracted memory searches for comparable concepts.

Multiple Universes of Sets and Indeterminate Truth Values

Multiple Universes of Sets and Indeterminate Truth Values

A modal logic with temporal variables.

There appear thus far to have been five general methods suggested in the literature for formalizing indeterministic tense-modal logics. The first consists in the introduction of a third truth-value, but this has the absurb consequence that the law of non-contradiction must be denied since the conjunction of any two propositions having the indeterminate truth-value must be indeterminate rather than false. (The absurdity is apparent from the fact that in this context “having an indeterminate truth- value” means the same as, “possibly being true”). A second alternative called by Prior a “Peircean tense logic” has the very awkward result that true, contingent propositions about the future cannot be -tated at all. The third method which Prior calls “Ockhamist” is the one considered below. (Cahn’s and Thomason’s systems I have tried to investigate elsewhere.)1

The effect of jury size and judge’s instructions on memory for pragmatic implications from courtroom testimony

Subjects working in groups of one, two, and three heard an excerpt of courtroom testimony and were later asked to rate statements about material in the testimony as true, false, or of indeterminate truth value. Half of the subjects heard any given piece of information (Mr. X rang the burglar alarm) directly asserted (I rang the burglar alarm), while the other half heard it only implied (I ran up to the burglar alarm). Half of the subjects initially heard specific instructions about the pitfalls of interpreting implied information as if it were asserted fact, while the other half heard no such instructions. Subjects in both groups remembered a majority of implications as true. The implication instructions did not significantly reduce the number of implied statements remembered as true, and there were no differences as a function of group size.

Memory for pragmatic implications from courtroom testimony

Subjects heard an excerpt of mock courtroom testimony and were later asked to rate statements about material in the testimony as “true,” “false,” or “of indeterminate truth value.” Half of the subjects heard a given piece of information (Mr. X rang the burglar alarm) directly asserted (I rang the burglar alarm), while the other half heard it only pragmatically implied or suggested (I ran up to the burglar alarm). Half of the subjects initially heard specific instructions about the pitfalls of interpreting pragmatically implied information as if it were asserted fact, while the other half had no such instructions. Half of the subjects performed the response task immediately after hearing the testimony, while the other half did so 2 days later. Subjects generally remembered both implications and assertions as definite fact, even when specifically warned not to do so.

Memory and comprehension of implications and inferences of complex sentences

Subjects heard a list of complex sentences, all containing complement sentences as objects of the main verbs ( Miss America said that she played the tuba ). They were asked, either at the end of the entire list (memory group) or after each sentence (comprehension group), to judge the complement sentence ( Miss America played the tuba ) as true, false, or of indeterminate truth value. On sentences where the complement had no logically necessary truth value, comprehension subjects typically judged the truth value indeterminate, while memory subjects typically judged it according to its invited inference. Results were interpreted in the context of an interactionist view of the higher mental processes, where distinctions between different types of stored semantic information are obscured in memory.