Truth and Falsehood: An Inquiry into Generalized Logical Values: 36 (Trends in Logic)

Philosophical Analysis in the Twentieth Century, Volume 2. Consistency, Contradiction and Negation. Logical and Computational Perspectives. Real Analysis via Sequences and Series. Selected Readings on Transformational Theory. Martingales in Banach Spaces. Mind, Language, and Metaphilosophy. Larisa Maksimova on Implication, Interpolation, and Definability. Logic, Language, and Probability.

Stochastic Integration in Banach Spaces. Interpreted Languages and Compositionality.

A Graduate Course in Probability. Rational Points and Arithmetic of Fundamental Groups. One Hundred Prisoners and a Light Bulb. Procedural Semantics for Hyperintensional Logic. Readings in Formal Epistemology. Between Logic and Reality. Hybrid Logic and its Proof-Theory. Proof, Computation and Agency. Objectivity, Realism, and Proof. The Argument of Mathematics.

New Directions in Paraconsistent Logic. Conditionals and Modularity in General Logics. Introduction to Stochastic Calculus. Rohit Parikh on Logic, Language and Society. Twelve Landmarks of Twentieth-Century Analysis. Michael Dunn on Information Based Logics. Philosophy of Mathematics Today. Belief Revision in Non-Classical Logics. The Logic of Time. Model Theory in Algebra, Analysis and Arithmetic. Lou van den Dries. Advances in Natural Deduction. The Logic of Infinity.

Current Trends in Asia. Logical Thinking in the Pyramidal Schema of Concepts: The Logical and Mathematical Elements. Structural Analysis of Non-Classical Logics. The Life and Work of Leon Henkin. Handbook of Philosophical Logic. Logical Theory and Semantic Analysis. Intuitionistic Proof Versus Classical Truth. Concentration Inequalities for Sums and Martingales. A New Perspective on Nonmonotonic Logics. A Course in Probability Theory.

And so must C3 and C4, because the number, such that Sir Walter Scott is the man who wrote that many Waverley Novels altogether is the same as the number of counties in Utah, namely If this is indeed the case, then C1 and C4 must have the same denotation designation as well. But it seems that the only semantically relevant thing these sentences have in common is that both are true. Thus, taken that there must be something what the sentences designate, one concludes that it is just their truth value.

As Church remarks, a parallel example involving false sentences can be constructed in the same way by considering, e. Stated generally, the pattern of the argument goes as follows cf. One starts with a certain sentence, and then moves, step by step, to a completely different sentence. Every two sentences in any step designate presumably one and the same thing. Hence, the starting and the concluding sentences of the argument must have the same designation as well. But the only semantically significant thing they have in common seems to be their truth value.

Thus, what any sentence designates is just its truth value. Quine , too, presents a variant of the slingshot using class abstraction, see also Shramko and Wansing It is worth noticing that the formal versions of the slingshot show how to move—using steps that ultimately preserve reference—from any true false sentence to any other such sentence.

In view of this result, it is hard to avoid the conclusion that what the sentences refer to are just truth values. The slingshot argument has been analyzed in detail by many authors see especially the comprehensive study by Stephen Neale Neale and references therein and has caused much controversy notably on the part of fact-theorists, i. Also see the supplement on the slingshot argument. Truth values evidently have something to do with a general concept of truth.

Therefore it may seem rather tempting to try to incorporate considerations on truth values into the broader context of traditional truth-theories, such as correspondence, coherence, anti-realistic, or pragmatist conceptions of truth. Yet, it is unlikely that such attempts can give rise to any considerable success. It does not commit one to any specific metaphysical doctrine of truth.

In one significant respect, however, the idea of truth values contravenes traditional approaches to truth by bringing to the forefront the problem of its categorial classification. In most of the established conceptions, truth is usually treated as a property. By contrast with this apparently quite natural attitude, the suggestion to interpret truth as an object may seem very confusing, to say the least.

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Nevertheless this suggestion is also equipped with a profound and strong motivation demonstrating that it is far from being just an oddity and has to be taken seriously cf. First, it should be noted that the view of truth as a property is not as natural as it appears on the face of it. In this case a superficial grammatical analogy is misleading. This idea gave an impetus to the deflationary conception of truth advocated by Ramsey, Ayer, Quine, Horwich, and others, see the entry on the deflationary theory of truth.

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However, even admitting the redundancy of truth as a property, Frege emphasizes its importance and indispensable role in some other respect. Namely, truth, accompanying every act of judgment as its ultimate goal, secures an objective value of cognition by arranging for every assertive sentence a transition from the level of sense the thought expressed by a sentence to the level of denotation its truth value.

This circumstance specifies the significance of taking truth as a particular object. As Tyler Burge explains:. The object, in the sense of the point or objective , of sentence use was truth. It is illuminating therefore to see truth as an object. As it has been observed repeatedly in the literature cf. He considered philosophical statements to be not mere judgements but rather assessments , dealing with some fundamental values, the value of truth being one of the most important among them.

This latter value is to be studied by logic as a special philosophical discipline. Thus, from a value-theoretical standpoint, the main task of philosophy, taken generally, is to establish the principles of logical, ethical and aesthetical assessments, and Windelband accordingly highlighted the triad of basic values: Later this triad was taken up by Frege in when he defined the subject-matter of logic see below. Gabriel points out The decisive move made by Frege was to bring together a philosophical and a mathematical understanding of values on the basis of a generalization of the notion of a function on numbers.

If predicates are construed as a kind of functional expressions which, being applied to singular terms as arguments, produce sentences, then the values of the corresponding functions must be references of sentences. Taking into account that the range of any function typically consists of objects, it is natural to conclude that references of sentences must be objects as well.

And if one now just takes it that sentences refer to truth values the True and the False , then it turns out that truth values are indeed objects, and it seems quite reasonable to generally explicate truth and falsity as objects and not as properties. A statement contains no empty place, and therefore we must take its Bedeutung as an object.

But this Bedeutung is a truth-value. Thus the two truth-values are objects. But even Dummett If truth values are accepted and taken seriously as a special kind of objects, the obvious question as to the nature of these entities arises. The above characterization of truth values as objects is far too general and requires further specification.

1. Truth Values (Stanford Encyclopedia of Philosophy);
2. 1. Truth values as objects and referents of sentences;
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One way of such specification is to qualify truth values as abstract objects. Among the other logical objects Frege pays particular attention to are sets and numbers, emphasizing thus their logical nature in accordance with his logicist view. Since then it is customary to label truth values as abstract objects, thus allocating them into the same category of entities as mathematical objects numbers, classes, geometrical figures and propositions.

2. Truth values as logical values

One may pose here an interesting question about the correlation between Fregean logical objects and abstract objects in the modern sense see the entry on abstract objects. Obviously, the universe of abstract objects is much broader than the universe of logical objects as Frege conceives them. Generally, the class of abstracta includes a wide diversity of platonic universals such as redness, youngness, or geometrical forms and not only those of them which are logically necessary.

Nevertheless, it may safely be said that logical objects can be considered as paradigmatic cases of abstract entities, or abstract objects in their purest form. It should be noted that finding an adequate definition of abstract objects is a matter of considerable controversy. According to a common view, abstract entities lack spatio-temporal properties and relations, as opposed to concrete objects which exist in space and time Lowe In this respect truth values obviously are abstract as they clearly have nothing to do with physical spacetime.

In a similar fashion truth values fulfill another requirement often imposed upon abstract objects, namely the one of a causal inefficacy see, e. Here again, truth values are very much like numbers and geometrical figures: Finally, it is of interest to consider how truth values can be introduced by applying so-called abstraction principles , which are used for supplying abstract objects with criteria of identity. The idea of this method of characterizing abstract objects is also largely due to Frege, who wrote:. If the symbol a is to designate an object for us, then we must have a criterion that decides in all cases whether b is the same as a , even if it is not always in our power to apply this criterion.

More precisely, one obtains a new object by abstracting it from some given kind of entities, in virtue of certain criteria of identity for this new abstract object. This abstraction is performed in terms of an equivalence relation defined on the given entities see Wrigley The celebrated slogan by Quine For truth values such a criterion has been suggested in Anderson and Zalta This idea can be formally explicated following the style of presentation in Lowe Namely, he points out a strong analogy between extensions of predicators and truth values of sentences.

And then, Carnap remarks, it seems quite natural to take truth values as extensions for sentences. Note that this criterion employs a functional dependency between an introduced abstract object in this case a truth value and some other objects sentences. The criterion of identity for truth values is formulated then through the logical relation of equivalence holding between these other objects—sentences, propositions, or the like with an explicit quantification over them. It should also be remarked that the properties of the object language biconditional depend on the logical system in which the biconditional is employed.

Biconditionals of different logics may have different logical properties, and it surely matters what kind of the equivalence connective is used for defining truth values. This means that the concept of a truth value introduced by means of the identity criterion that involves a biconditional between sentences is also logic-relative. Taking into account the role truth values play in logic, such an outcome seems to be not at all unnatural. Anderson and Zalta The notion of a truth value in general is then defined as an object which is the truth value of some proposition:.

The latter result is expected, if one bears in mind that what the definitions above actually introduce are the classical truth values as the underlying logic is classical. That is, the connective of material equivalence divides sentences into two distinct collections. Due to the law of excluded middle these collections are exhaustive, and by virtue of the law of non-contradiction they are exclusive.

Thus, we get exactly two equivalence classes of sentences, each being a hypostatized representative of one of two classical truth values. By doing so, logic is interested in truth as such, understood objectively, and not in what is merely taken to be true. Now, if one admits that truth is a specific abstract object the corresponding truth value , then logic in the first place has to explore the features of this object and its interrelations to other entities of various other kinds. As he paradigmatically put it:. All true propositions denote one and the same object, namely truth, and all false propositions denote one and the same object, namely falsehood.

I consider truth and falsehood to be singular objects in the same sense as the number 2 or 4 is. The objects denoted by propositions are called logical values. Truth is the positive, and falsehood is the negative logical value. This definition may seem rather unconventional, for logic is usually treated as the science of correct reasoning and valid inference.

The latter understanding, however, calls for further justification. This becomes evident, as soon as one asks, on what grounds one should qualify this or that pattern of reasoning as correct or incorrect. In answering this question, one has to take into account that any valid inference should be based on logical rules which, according to a commonly accepted view, should at least guarantee that in a valid inference the conclusion s is are true if all the premises are true.

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Translating this demand into the Fregean terminology, it would mean that in the course of a correct inference the possession of the truth value The True should be preserved from the premises to the conclusion s. Thus, granting the realistic treatment of truth values adopted by Frege, the understanding of logic as the science of truth values in fact provides logical rules with an ontological justification placing the roots of logic in a certain kind of ideal entities see Shramko Popper and also Burge Among the subdomains of this third realm one finds, e.

The set of truth values may be regarded as forming another such subdomain, namely the one of logical values , and logic as a branch of science rests essentially on this logical domain and on exploring its features and regularities. According to Frege, there are exactly two truth values, the True and the False. This opinion appears to be rather restrictive, and one may ask whether it is really indispensable for the concept of a truth value. One should observe that in elaborating this conception, Frege assumed specific requirements of his system of the Begriffsschrift , especially the principle of bivalence taken as a metatheoretical principle, viz.

On the object-language level this principle finds its expression in the famous classical laws of excluded middle and non-contradiction. The further development of modern logic, however, has clearly demonstrated that classical logic is only one particular theory although maybe a very distinctive one among the vast variety of logical systems. In fact, the Fregean ontological interpretation of truth values depicts logical principles as a kind of ontological postulations, and as such they may well be modified or even abandoned.

For example, by giving up the principle of bivalence, one is naturally led to the idea of postulating many truth values. By generalizing this idea and also adopting the above understanding of the subject-matter of logic, one naturally arrives at the representation of particular logical systems as a certain kind of valuation systems see, e. In later publications, Gottwald has changed his terminology and states that.

For example, the set of tautologies logical laws is directly specified by the given set of designated truth values: Thus, if a valuation system is said to determine a logic, the valuation system by itself is, properly speaking, not a logic, but only serves as a semantic basis for some logical system. Valuation systems are often referred to as logical matrices. With a view on semantically defining a many-valued logic, these restrictions are very natural and have been taken up in Shramko and Wansing and elsewhere.

In this way Fregean, i. A Gentzen-style formulation can be found in Font The syntactic notion of a single-conclusion consequence relation has been extensively studied by representatives of the Polish school of logic, most notably by Alfred Tarski, who in fact initiated this line of research see Tarski a,b; cf. If additionally a consequence relation has the substitution property 7 , then it is called structural.

Generally speaking, the framework of valuation systems not only perfectly suits the conception of logic as the science of truth values, but also turns out to be an effective technical tool for resolving various sophisticated and important problems in modern logic, such as soundness, completeness, independence of axioms, etc. The idea of truth as a graded notion has been applied to model vague predicates and to obtain a solution to the Sorites Paradox, the Paradox of the Heap see the entry on the Sorites Paradox.

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However, the success of applying many-valued logic to the problem of vagueness is highly controversial. In any case, the vagueness of concepts has been much debated in philosophy see the entry on vagueness and it was one of the major motivations for the development of fuzzy logic see the entry on fuzzy logic. In the s, Lotfi Zadeh introduced the notion of a fuzzy set. A fuzzy set has a membership function ranging over the real interval [0,1]. The application of continuum-valued logics to the Sorites Paradox has been suggested by Joseph Goguen The Sorites Paradox in its so-called conditional form is obtained by repeatedly applying modus ponens in arguments such as:.

Whereas it seems that all premises are acceptable, because the first premise is true and one grain does not make a difference to a collection of grains being a heap or not, the conclusion is, of course, unacceptable. To overcome the problem of assigning precise values to predications of vague concepts, Zadeh introduced fuzzy truth values as distinct from the numerical truth values in [0, 1], the former being fuzzy subsets of the set [0, 1], understood as true , very true , not very true , etc.

The interpretation of continuum-valued logics in terms of fuzzy set theory has for some time be seen as defining the field of mathematical fuzzy logic. Haack emphasizes that her criticisms of fuzzy logic do not apply to the base logics. Moreover, it should be pointed out that mathematical fuzzy logics are nowadays studied not in the first place as continuum-valued logics, but as many-valued logics related to residuated lattices see Hajek ; Cignoli et al.

A fundamental concern about the semantical treatment of vague predicates is whether an adequate semantics should be truth-functional, that is, whether the truth value of a complex formula should depend functionally on the truth values of its subformulas. Whereas mathematical fuzzy logic is truth-functional, Williamson According to Williamson, the degree of truth of a conjunction, a disjunction, or a conditional just fails to be a function of the degrees of truth of vague component sentences.

One way of in a certain sense non-truthfunctionally reasoning about vagueness is supervaluationism. The method of supervaluations has been developed by Henryk Mehlberg and Bas van Fraassen and has later been applied to vagueness by Kit Fine , Rosanna Keefe and others. Even if one grants atomic sentences containing non-denoting singular terms and that some attributions of vague predicates are neither true nor false, it nevertheless seems natural not to preclude that compound sentences of a certain shape containing non-denoting terms or vague predications are either true or false, e.

Supervaluational semantics provides a solution to this problem. The property of being superfalse is defined analogously. Supervaluationism is, however, not truth-functional with respect to supervalues. The supervalue of a disjunction, for example, does not depend on the supervalue of the disjuncts.

An argument against the charge that supervaluationism requires a non-truth-functional semantics of the connectives can be found in MacFarlane cf. One might, perhaps, think that the mere existence of many-valued logics shows that there exist infinitely, in fact, uncountably many truth values.

However, this is not at all clear recall the more cautious use of terminology advocated by Gottwald. Whereas the algebraic values are elements of an algebraic structure and referents of formulas, the logical value true is used to define valid consequence: If every premise is true, then so is at least one of the conclusion s.

The other logical value, false , is preserved in the opposite direction: If the every conclusion is false, then so is at least one of the premises. The logical values are thus represented by a bi-partition of the set of algebraic values into a set of designated values truth and its complement falsity. Essentially the same idea has been taken up earlier by Dummett in his influential paper, where he asks.