This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko’s Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some (...) recent works studying these in the particular cases of Łukasiewicz’s many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties. (shrink)

In VAGUENESS AND DEGREES OF TRUTH, Nicholas Smith develops a new theory of vagueness: fuzzy plurivaluationism. -/- A predicate is said to be VAGUE if there is no sharply defined boundary between the things to which it applies and the things to which it does not apply. For example, 'heavy' is vague in a way that 'weighs over 20 kilograms' is not. A great many predicates -- both in everyday talk, and in a wide array of theoretical vocabularies, (...) from law to psychology to engineering -- are vague. -/- Smith argues, based on a detailed account of the defining features of vagueness, that an accurate theory of vagueness must involve the idea that truth comes in degrees. The core idea of degrees of truth is that while some sentences are true and some are false, others possess intermediate truth values: they are truer than the false sentences, but not as true as the true ones. Degree-theoretic treatments of vagueness have been proposed in the past, but all have encountered significant objections. In light of these, Smith develops a new type of degree theory. Its innovations include a definition of logical consequence that allows the derivation of a classical consequence relation from the degree-theoretic semantics, a unified account of degrees of belief and their relationships with degrees of truth and subjective probabilities, and the incorporation of semantic indeterminacy -- the view that vague statements need not have unique meanings -- into the degree-theoretic framework. -/- As well as being essential reading for those working on vagueness, Smith's book provides an excellent entry-point for newcomers to the area -- both from elsewhere in philosophy, and from computer science, logic and engineering. It contains a thorough introduction to existing theories of vagueness and to the requisite logical background. (shrink)

The purpose of this article is to compare the theory that there are degrees of truth with putnam's intuitionist theory as rival solutions to the sorites paradox. I argue that intuitionist logic lacks explanatory support and is self-Defeating. The degree theory on the other hand offers an illuminating explanation of the sorites fallacy.

The paper introduces a concept of logic applied to a formalization of the so-called inferences preserving degrees of truth. Semantical and syntactical characterizations of three kinds of logics preserving degrees of truth are provided. The other approach than in [3] and [9] to the problem of expressing that a sentence is less true than a sentence is presented.

Many theorists think nowadays that vagueness is a widespread phenomenon that affects and infects almost all terms and concepts of our thought and language, and for some philosophers degree of truth theories are the best way to cope with vagueness and sorites susceptible concepts. In this paper I argue that many of the allegedly vague concepts are not vague in the last analysis the philosopher or scientist could offer if compelled to, and that much of the vagueness of the (...) properly vague ones comes from its contextual dependence alone. I also argue that degree of truth approaches -- particularly the infinitist ones -- and fuzzy logics do not solve practically any of the puzzles brought about by vagueness and sorites arguments, and conversely they have many additional problems of their own. Concerning recalcitrant cases of vagueness, I would tentatively commend the epistemic theory of vagueness, from an inference to the best explanation. (shrink)

Putnam's intuitionist proposal for a logic of vague terms is defended. It is argued that both classical logic and the degrees of truth approach are committed to treating vague terms as having hidden precise borderlines. This is a crucial failing in a logic of vagueness. Intuitionism, because of the nature of intuitionist negation, avoids this failing.

Nicholas Smith argues that an adequate account of vagueness must involve\ndegrees of truth. The basic idea of degrees of truth is that while\nsome sentences are true and some are false, others possess intermediate\ntruth values: they are truer than the false sentences, but not as\ntrue as the true ones. This idea is immediately appealing in the\ncontext of vagueness--yet it has fallen on hard times in the philosophical\nliterature, with existing degree-theoretic treatments of vagueness\nfacing apparently insuperable objections. Smith seeks to (...) turn the\ntide in favor of a degree-theoretic treatment of vagueness, by motivating\nand defending the basic idea that truth can come in degrees, by arguing\nthat no theory of vagueness that does not countenance degrees of\ntruth can be correct, and by developing a new degree-theoretic treatment\nof vagueness--fuzzy plurivaluationism--that solves the problems plaguing\nearlier degree theories. (shrink)

A number of authors have noted that vagueness engenders degrees of belief, but that these degrees of belief do not behave like subjective probabilities. So should we countenance two different kinds of degree of belief: the kind arising from vagueness, and the familiar kind arising from uncertainty, which obey the laws of probability? I argue that we cannot coherently countenance two different kinds of degree of belief. Instead, I present a framework in which there is a single notion (...) of degree of belief, which in certain circumstances behaves like a subjective probability assignment and in other circumstances does not. The core idea is that one’s degree of belief in a proposition P is one’s expectation of P’s degree of truth. (shrink)

Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully (...) adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens. (shrink)

Involutive Stone algebras were introduced by R. Cignoli and M. Sagastume in connection to the theory of $n$-valued Łukasiewicz–Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras named Six. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds of (...) truth values, and hence not only preserving the value $1$. Among other things, we prove that Six is a many-valued logic that can be determined by a finite number of matrices. Besides, we show that Six is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of Six providing a Gentzen calculus for it, which is sound and complete with respect to the logic. (shrink)

(No abstract is available for this citation).

When considering m-sequents, it is always possible to obtain an m-sequent calculus VL for every m-valued logic (defined from an arbitrary finite algebra L of cardinality m) following for instance the works of the Vienna Group for Multiple-valued Logics. The Gentzen relations associated with the calculi VL are always finitely equivalential but might not be algebraizable. In this paper we associate an algebraizable 2-Gentzen relation with every sequent calculus VL in a uniform way, provided the original algebra L has a (...) reduct that is a distributive lattice or a pseudocomplemented distributive lattice. We also show that the sentential logic naturally associated with the provable sequents of this algebraizable Gentzen relation is the logic that preserves degrees of truth with respect to the original algebra (in contrast with the more common logic that merely preserves truth). Finally, for some particular logics we obtain 2-sequent calculi that axiomatize the algebraizable Gentzen relations obtained so far. (shrink)

What is the nature of truth? This question has been answered by philosophers in quite different ways, while F. H. Bradley asserts that truths have degrees and that no proposition can be stated to be simply true or false. In this paper, I briefly illustrate what he calls the doctrine of degrees of truth and try to address the problems it entails. I first explain what he means by truth and error/falsehood (he does not make (...) a clear distinction between the two terms); then, I concentrate on his criticisms of three theories of truth, followed by a discussion of his own identity theory of truth. I will be focusing on his doctrine of the degrees of truth and highlight its difficulties. I show that his theory faces some insurmountable difficulties, and it should be motivated by a particular form of monism that he insisted on, saying existence monism. (shrink)

In this essay, I argue that F.H. Bradley’s controversial theory of “degrees of truth and reality” is the logical development of Hegel’s own theory of truth when it is placed within the metaphysical system of the Science of Logic. Despite Bradley’s own claim that with regards to the theory of degrees of truth and reality he is indebted even more than anywhere else to Hegel, this connection has been little examined in the secondary literature. Through (...) a careful examination of both Bradley’s works and the structure of Hegel’s logic, it will become clear that Bradley’s development of the theory is the only logical conclusion that the consistent Hegelian can make. This essay has clear ramifications for our understanding of Bradley’s philosophy and, through uncovering the logical connections that led Bradley to develop the theory, I reveal an important implication of Hegel’s thought that has been entirely overlooked. (shrink)

Formalization of reasoning which accepts rules of inference leading to conclusions whose logical values are not smaller than the logical value of the “weakest” premise leads to the concept of consequence operation preserving degrees of truth. Several examples of such consequence operation have already been considered . In the present paper we give a general notion of the consequence operation preserving degrees of truth and its characterization in terms of projective generation and selfextensionality.

Vagueness is one of the most persistent and challenging topics in the intersection of philosophy and logic. At least five other noteworthy books on vagueness have been written by philosophers since 1991 [2, 6, 11, 12, 15]. A (necessarily incomplete) bibliography that has been compiled for the Arché project Vagueness: its Nature and Logic (2004-2006) of the University of St Andrews lists more than 350 articles and books on vagueness until 2005.1 Many new and interesting contributions have appeared since. The (...) book under review is much more than yet another addition to this prolific discourse. Nicholas Smith manages to tackle two different tasks that are potentially in tension. On the one hand, he provides a comprehensive, systematic and well written account of various approaches to vagueness that have been debated so far. On the other hand, Smith carefully explains and defends his own theory of vagueness, called fuzzy plurivaluationism. Given the complex and almost unsurmountably large amount of relevant literature and the fact that theories of vagueness based on fuzzy logic have almost universally been rejected by philosophers so far this is no simple feat. In the comments below, I will largely follow the structure of book. If along the way I cannot resist to make side remarks or even take issue with some of.. (shrink)

In his book _Paradoxes, Mark Sainsbury suggests that degrees of truth can be justified and explained by analogy with degrees of belief. Considerations of vagueness place theoretical limitations on degrees of belief which require degrees of truth. This paper argues that considerations of vagueness and degrees of belief do nothing to illuminate degrees of truth. An account of vagueness need not postulate degrees of truth.

We study the weak truth table degree spectra of first-order relational structures. We prove a dichotomy among the possible wtt degree spectra along the lines of Knight’s upward-closure theorem for Turing degree spectra. We prove new results contrasting the wtt degree spectra of finite- and infinite-signature structures. We show that, as a method of defining classes of reals, the wtt degree spectrum is, except for some trivial cases, strictly more expressive than the Turing degree spectrum.

A framework of degrees of belief, or credences, is often advocated to model our uncertainty about how things are or will turn out. It has also been employed in relation to the kind of uncertainty or indefiniteness that arises due to vagueness, such as when we consider “a is F” in a case where a is borderline F. How should we understand degrees of belief when we take into account both these phenomena? Can the right kind of theory (...) of the semantics of vagueness help us answer this? Nicholas J.J. Smith defends a unified account, according to which “degree of belief is expected truth-value”; this builds on his Degree Theory of vagueness that offers an account of the semantics and logic of vagueness in terms of degrees of truth. I argue that his account fails. Degree theories of vagueness do not help us understand degrees of belief and, I argue, we shouldn’t expect a theory of vagueness to yield a detailed uniform story about this. The route from the semantics to psychological states needn’t be straightforward or uniform even before we attempt to combine vagueness with probabilistic uncertainty. (shrink)

We often evaluate belief-forming processes, agents, or entire belief states for reliability. This is normally done with the assumption that beliefs are all-or-nothing. How does such evaluation go when we’re considering beliefs that come in degrees? I consider a natural answer to this question that focuses on the degree of truth-possession had by a set of beliefs. I argue that this natural proposal is inadequate, but for an interesting reason. When we are dealing with all-or-nothing belief, high reliability (...) leads to high levels of truth-possession. However, when it comes to degrees of belief, reliability and truth-possession part ways. The natural answer thus fails to be a good way to evaluate degrees of belief for reliability. I propose and develop an alternative method based on the notion of calibration, suggested by Frank Ramsey, which does not have this problem and consider why we should care about such assessments of reliability even if they are not tied directly to truth-possession. (shrink)

If we think, as Ramsey did, that a degree of belief that P is a stronger or weaker tendency to act as if P, then it is clear that not only uncertainty, but also vagueness, gives rise to degrees of belief. If I like hot coffee and do not know whether the coffee is hot or cold, I will have some tendency to reach for a cup; if I like hot coffee and know that the coffee is borderline hot, (...) I will have some tendency to reach for a cup. Suppose that we take degrees of belief arising from uncertainty to obey the laws of probability and that we model vagueness using degrees of truth. We then encounter a problem: it does not look as though degrees of belief arising from vagueness should obey the laws of probability. One response would be to countenance two different sorts of degrees of belief: degrees of belief arising from uncertainty, which obey the laws of probability; and degrees of belief arising from vagueness, which obey a different set of laws. I argue, however, that if a degree of belief that P is a stronger or weaker tendency to act as if P, then this option is not open. Instead, I propose an account of the behaviour of degrees of belief that integrates subjective probabilities and degrees of truth. On this account, degrees of belief are expectations of degrees of truth. The account explains why degrees of belief behave in accordance with the laws of probability in cases involving only uncertainty, while also allowing degrees of belief to behave differently in cases involving only vagueness, and in mixed cases involving both uncertainty and vagueness. Justifications of the account are given both via Dutch books and in terms of epistemic accuracy. (shrink)

Degrees of belief are familiar to all of us. Our confidence in the truth of some propositions is higher than our confidence in the truth of other propositions. We are pretty confident that our computers will boot when we push their power button, but we are much more confident that the sun will rise tomorrow. Degrees of belief formally represent the strength with which we believe the truth of various propositions. The higher an agent’s degree (...) of belief for a particular proposition, the higher her confidence in the truth of that proposition. For instance, Sophia’s degree of belief that it will be sunny in Vienna tomorrow might be .52, whereas her degree of belief that the train will leave on time might be .23. The precise meaning of these statements depends, of course, on the underlying theory of degrees of belief. These theories offer a formal tool to measure degrees of belief, to investigate the relations between various degrees of belief in different propositions, and to normatively evaluate degrees of belief. (shrink)

What has been learned about logic by means of "uninterpreted" logistic systems can be supplemented by comparing the latter with systems which are more uninterpreted, as well as with others which are less uninterpreted than the well-known logistic systems. By somewhat extending the meaning of 'uninterpreted', I hope to establish certain claims about the nature of logistic systems and also to cast some light on the nature of "logic itself." My procedure involves looking at three major "degrees" of interpretation: (...) first, systems uninterpreted both semantically and syntactically, second, systems uninterpreted semantically but not syntactically, and third, systems uninterpreted neither semantically nor syntactically. We shall be forced to limit ourselves to the truth-functional part of logic in this brief study. What are usually called uninterpreted systems can be seen on a continuum of "degrees" of interpretation, from ordinary reasoning at one extreme to a "thoroughly uninterpreted system" at the other. "Logic itself" apparently lies nearer to the interpreted, deformalized end of the spectrum than to the uninterpreted, formalized end. Logic is not identical with any particular logistic system, but is that which the particular logistic systems aim to formalize or model or capture. I propose that it is that minimum set of logical, rather than syntactical, primitive terms, definitions, and rules which is needed to generate logically, rather than syntactically, the principles of ordinary reasoning as it is used by logicians in their metalinguistic discussions and informal proofs of metatheorems. This minimum set is somewhat larger than the primitive bases of most logistic systems; its truth-functional part is presented in the "Deformalized Logic" at degree twelve. (shrink)

We can classify the (truth-theoretic) paradoxes according to their degrees of paradoxicality. Roughly speaking, two paradoxes have the same degrees of paradoxicality, if they lead to a contradiction under the same conditions, and one paradox has a (non-strictly) lower degree of paradoxicality than another, if whenever the former leads to a contradiction under a condition, the latter does so under the same condition. In this paper, we outline some results and questions around the degrees of paradoxicality (...) and summarize recent progress. (shrink)

This paper argues that the doctrines of determinism and supervenience, while logically independent, are importantly linked in physical mechanics—and quite interestingly so. For it is possible to formulate classical mechanics in such a way as to take advantage of the existence of mathematical devices that represent the advance of time—and which are such as to inspire confidence in the truth of determinism—in order to prevent violation of supervenience. It is also possible to formulate classical mechanics-and to do so in (...) an observationally equivalent, and thus equally empirically respectable, way—such that violations of supervenience are (on the one hand) routine, and (on the other hand) necessary for achieving complete descriptions of the motions of mechanical systems—necessary, therefore, for achieving a deterministic mechanical theory. Two such formulations—only one of which preserves supervenience universally—will conceive of mechanical law in quite different ways. What's more, they will not admit of being extended to treat thermodynamical questions in the same way. Thus we will find that supervenience is a contingent matter, in the following rather surprising and philosophically interesting way: we cannot in mechanics separate our decisions to conceive of physical law in certain ways from our decisions to treat macroscopic quantities in certain ways. (shrink)

Scientific realism is the position that success of a scientific theory licenses an inference to its approximate truth. The argument from pessimistic meta-induction maintains that this inference is undermined due to the existence of theories from the history of science that were successful, but false. I aim to counter pessimistic meta-induction and defend scientific realism. To do this, I adopt a notion of success that admits of degrees, and show that our current best theories enjoy far higher (...) class='Hi'>degrees of success than any of the successful, but refuted theories of the past. (shrink)

I identify the special sort of stability (invariance, resilience, etc.) that distinguishes laws from accidental truths. Although an accident can have a certain invariance under counterfactual suppositions, there is no continuum between laws and accidents here; a law's invariance is different in kind, not in degree, from an accident's. (In particular, a law's range of invariance is not "broader"--at least in the most straightforward sense.) The stability distinctive of the laws is used to explicate what it would mean for there (...) to be multiple grades (or degrees) of physical necessity. Whether there are is for science to discover. (shrink)

Still, it is but fair for me to point out that several of the mainstays of the present proposal owe very little to the influence of the philosophers whose epistemological views have attracted me most — or for that matter to that of other analytical philosophers. I am referring to my acknowledging degrees of truth and existence and, consequently, degrees of knowledge, too.