A paradox is generally a puzzling conclusion we seem to be driven towards by our reasoning, but which is highly counterintuitive, nevertheless. There are, amongst these, a large variety of paradoxes of a logical nature which have teased even professional logicians, in some cases for several millennia. But what are now sometimes isolated as 'the logical paradoxes' are a much less heterogeneous collection: they are a group of antinomies centered on the notion of self-reference, some of which (...) were known in Classical times, but most of which became particularly prominent in the early decades of last century. Quine distinguished amongst paradoxes such antinomies. He did so by first isolating the 'veridical' and 'falsidical' paradoxes, which, although puzzling riddles, turned out to be plainly true, or plainly false, after some inspection. In addition, however, there were paradoxes which 'produce a self-contradiction by accepted ways of reasoning', and which, Quine thought, established 'that some tacit and trusted pattern of reasoning must be made explicit, and henceforward be avoided or revised' (Quine 1966, p7). We will first look, more broadly, and historically, at several of the main conundrums of a logical nature which have proved difficult, some since antiquity, before concentrating later on the more recent troubles with paradoxes of self-reference. They will all be called 'logical paradoxes'. (shrink)

The author considers following question: is a consistent semantically closed language possible? The negative answer is the orthodox answer in the logic of the 20th century. It was presented in Russell's theory of types and Tarski's semantic theory of metalanguages. Nevertheless, contemporary logicians and philosophers of language return to this problem time and again, pointing to its relevance in various aspects. In particular, it is asserted that semantically closed language is a very important tool for expressing logical and philosophical (...) ideas. In logic of the 20th century, the problem of semantically closed language was discussed in connection with the problem of logical paradoxes. Russell and Tarski saw a fundamental cause of paradoxes in the phenomenon of self-reference that arises in semantically closed language. Accordingly, a solution for paradoxes was seen in eliminating the cause, that is, in prohibiting semantically closed language by means of a hierarchy of logical types of classes (Russell) or a hierarchy of metalanguages (Tarski). However, some contemporary logicians criticize the hierarchical approach, whose main argument consists in asserting that the approach was wrong in its diagnosis of the cause of paradoxes. This author does not try to correct the diagnostics of the hierarchical approach by identifying another cause of paradoxes. Instead, the author recognizes that a general solution of the problem of paradoxes cannot be given, a priori, by eliminating what fundamentally generates them. In this article, a new “ad hoc solution" of the problem is offered that rests upon an empirical method of identifying and eliminating paradoxes. A specific characteristic of the method is admitting the existence of consistent semantically closed language. (shrink)

The logic B+ is Routley and Meyer’s basic positive logic. Wedefine the logics BK+ and BK′+ by adding to B+ the K rule and to BK+the characteristic S4 axiom, respectively. These logics are endowed witha relatively strong non-constructive negation. We prove that all the logicsdefined lack the K axiom and the standard paradoxes of consistency.

This chapter contains sections titled: Introduction Semantic Paradoxes: Some Proposals Sets and Extensions Three Paradoxes A Contextual Approach A Singularity Proposal Universality.

In this contribution I will examine Cotnoir’s (2013) solution to the problems that alethic pluralism faces when it comes to logical paradoxes. I will argue that his proposal fails to be a viable option and I will put forward an alternative approach, more Tarskian in spirit, but with the same “pluralist” trait of Cotnoir’s solution. Such an alternative approach is based on the idea that each truth predicate can be associated with an index that fully describes its relation with (...) other truth predicates.I will show how such indexes must be constrained in order to avoid logical paradoxes. (shrink)

This chapter contains sections titled: Paradoxes Stimulate Theory Development An Analogy with Perceptual Illusions Do Logical Paradoxes Exist? Imagination Overflows Logical Possibility Paradoxes Evoke Logical Analogies An Implication about the Nature of Paradox.

According to Hume’s classical definition of causal relations, a cause must fulfill two distinct conditions: a) be a sufficient condition for the occurrence of its effect; b) be temporally prior to it. However, a careful logical analysis shows that the combination of sufficiency and temporality is impossible. This is because if a complete cause is a sufficient condition for its effect to occur-then the effect is a necessary condition for the occurrence of its own complete cause! Thus, there can (...) be no complete cause for anything to occur, which means that causal determination is logically impossible. The paper goes on to examine some possible responses to this argument, and then concludes with a nontemporal version of the paradox that shows that even the first of Hume’s above conditions cannot be fulfilled on its own. This is because if something is a sufficient condition of something else, then it cannot be a sufficient condition of itself. (shrink)

The paper investigates from a proof-theoretic perspective various non-contractive logical systems, which circumvent logical and semantic paradoxes. Until recently, such systems only displayed additive quantifiers (Grišin and Cantini). Systems with multiplicative quantifiers were proposed in the 2010s (Zardini), but they turned out to be inconsistent with the naive rules for truth or comprehension. We start by presenting a first-order system for disquotational truth with additive quantifiers and compare it with Grišin set theory. We then analyze the reasons behind (...) the inconsistency phenomenon affecting multiplicative quantifiers. After interpreting the exponentials in affine logic as vacuous quantifiers, we show how such a logic can be simulated within a truth-free fragment of a system with multiplicative quantifiers. Finally, we establish that the logic for these multiplicative quantifiers (but without disquotational truth) is consistent, by proving that an infinitary version of the cut rule can be eliminated. This paves the way to a syntactic approach to the proof theory of infinitary logic with infinite sequents. (shrink)

Wonder Woman wields a number of magical Amazonian devices: her bulletproof bracelets, her invisible plane, and most importantly for this chapter, her golden lasso of truth. The first thing to notice about the golden lasso is that evildoers bound by it are not only compelled to tell the truth if and when they answer questions, but also compelled to answer Wonder Woman's questions in the first place. The second thing to notice is that answering truthfully does not, in this context, (...) necessarily mean uttering a true sentence. Rather, if a person is tied up with the golden lasso, then they are compelled to answer questions with what they believe is the truth. The chapter shows the examples that contemplating Wonder Woman's golden lasso, and Bizarra's golden lasso, can provide us with a wealth of interesting and entertaining paradoxes and puzzles. (shrink)

Alexius Meinong developed a notion of defective objects in order to account for various logical and psychological paradoxes. The notion is of historical interest, since it presages recent work on the logical paradoxes by Herzberger and Kripke. But it fails to do the job it was designed for. However, a technique implicit in Meinong's investigation is more successful and can be adapted to resolve a similar paradox discovered by Romane Clark in a revised version of Meinong's Theory (...) of Objects due to Rapaport. One family of paradoxes remains, but it is argued that they are unavoidable and relatively harmless. (shrink)

This work is, in large part, a series of refutations; it is also the author's Ph.D. thesis. First to be refuted is Russell's vicious circle principle as a general remedy for the solution of the paradoxes. The author rejects the classification of paradoxes into syntactic and semantic, since in his view there are no purely syntactic paradoxes. The distinction in logic between the uninterpreted syntactical aspect of a system and the system when given a determinate interpretation is held to be (...) untenable. Tarski's distinction between object-language and meta-language and his concept of semantically closed language are considered irrelevant for the solution of the Liar paradox. The author claims that the usual versions of the Liar paradox have the same structure as the Barber paradox, viz., [S ↔ ~S]. The author solves the Liar paradox by pointing out that it does not have a proper reference. Cantor's diagonal argument for the indenumerability [[sic]] of the real numbers is labeled as unsatisfactory. Since the diagonal number is dependent upon the real numbers in the constructed list, the author claims that this makes the diagonal number to be of a different nature and status than the real numbers in the list; thus we have what the author calls the dependence fallacy. The author also refuses to accept Cantor's nested interval proof of the indenumerability [[sic]] of the real numbers. Within the proof, two infinite sequences are constructed, each of which converges to a limit. Because the proof does not give a "definite rule of convergence," the author is not satisfied that the infinite sequences converge. Also rejected is Cantor's theorem. Other paradoxes analyzed are the Berry, Richard, heterological, "Richardian", Russell, Cantor, and Burali-Forti paradoxes.--T. G. N. (shrink)

We argued [Since this argument appeared in other journals, I am reprising it here, almost verbatim.] (Fulda in J Law Info Sci 2:230–232, 1991/AI & Soc 8(4):357–359, 1994) that the paradox of the preface suggests a reason why machines cannot, will not, and should not be allowed to judge criminal cases. The argument merely shows that they cannot now and will not soon or easily be so allowed. The author, in fact, now believes that when—and only when—they are ready (...) they actually should be so allowed, in the interests of justice. Both the original argument applied and this detailed reconsideration applies exclusively to trial courts, and both specifically exclude(d) sentencing. The argument highlights some key relevant differences between minds and machines and attempts, also, to explain why automation is of far greater import for the first-level justice system (trial courts) than for higher courts. A final section discusses why sentencing was, is, and should be excluded. (shrink)

Graham Priest 2002 argues that all logical paradoxes that include set-theoretic paradoxes and semantic paradoxes share a common structure, the Inclosure Schema, so they should be treated as one family. Through a discussion of Berry's Paradox and the semantic notion ?definable?, I argue that (i) the Inclosure Schema is not fine-grained enough to capture the essential features of semantic paradoxes, and (ii) the traditional separation of the two groups of logical paradoxes should be retained.

The property common to three kinds of paradoxes (logical, semantic, and cultural) is the underlying presence of an exclusive disjunction: even when it is put to a check by the paradox, it is still invoked at the level of implicit discourse. Hence the argumentative strength of paradoxical propositions is derived. Logical paradoxes (insolubilia) always involve two contradictory, mutually exclusive, truths. One truth is always perceived to the detriment of the other, in accordance with a succession which is (...) endlessly repetitive. A check is put on the principle of the excluded middle by the logical paradoxes, because self-reference leads to an endlessly repeating circle, out of which no resolution is conceivable. Logical paradoxes are to be compared with the `objective ambiguity' prevalent in oracles (Gallet, 1990). Semantic paradoxes are contextually-determined occurrences, whose resolution at the metalinguistic level is made possible by the discovery of a middle term. They express a wilful ambiguity, in which the interlocutor is invited to take an active part in the construction of sense, since what must be found is the unexpected sense thanks to which A and not-A can be asserted simultaneously. Cultural paradoxes play about doxa (`common sense') and openly challenge common opinion because of their character as inopinata (`unexpected'). My aim is to show that even cultural paradoxes hide sometimes a flaw of argumentation similar to logical or semantic paradox; they too imply an exclusive disjunction leading to the disappearance of the middle terms. Finally, basing myself on the theory of topoi (Anscombre and Ducrot, 1983), a tentative resolution of the cultural paradoxes will be suggested. (shrink)

Alexius Meinong developed a notion of defective objects in order to account for various logical and psychological paradoxes. The notion is of historical interest, since it presages recent work on the logical paradoxes by Herzberger and Kripke. But it fails to do the job it was designed for. However, a technique implicit in Meinong's investigation is more successful and can be adapted to resolve a similar paradox discovered by Romane Clark in a revised version of Meinong's Theory (...) of Objects due to Rapaport. One family of paradoxes remains, but it is argued that they are unavoidable and relatively harmless. (shrink)

Quantum physics exhibits various non-classical and paradoxical features. Among them are non-locality and contextuality (e.g. Bell’s theorem or the Einstein-Podolsky-Rosen paradox). Since they are expected to constitute a key resource in quantum computation, several approaches have been proposed to provide high-level expressions for them. In one of these approaches, Abramsky and others use the mathematics of algebraic topology and characterize non-locality and contextuality as the same type of phenomena as M. C. Escher’s impossible figures. This article expands this topological (...) insight and demonstrates that logical paradoxes arising from circular references (of the sort formalized by Gödel’s fixed-point or diagonalization lemma) share the same topological structure as the quantum paradoxes, by reformatting the topological model of contextuality into a semantics of logical paradoxes. This topological semantics indeed provides a unifying perspective from which previous approaches of philosophers and logicians to logical paradoxes can be understood as diverse ways of fine-tuning topologies to model paradoxes. (shrink)

We group the existing variants of the familiar set-theoretical and truth-theoretical paradoxes into two classes: connective paradoxes, which can in principle be ascribed to the presence of a contracting connective of some sort, and structural paradoxes, where at most the faulty use of a structural inference rule can possibly be blamed. We impute the former to an equivocation over the meaning of logical constants, and the latter to an equivocation over the notion of consequence. Both equivocation sources are tightly (...) related, and can be cleared up by adopting a particular substructural logic in place of classical logic. We then argue that our perspective can be justified via an informational semantics of contraction-free substructural logics. (shrink)

Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered (...) sets. Burali-Forti’s paradox, like Russell’s paradox, is portable. I offer the following explanation for this fact: Burali-Forti’s paradox, like Russell’s, is purely logical. Concretely, I show that if we enrich the language \ of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal. Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti’s. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti’s paradox is explained in the same way as the portability of Russell’s: both paradoxes involve an inconsistent logical form—Russell’s involves an inconsistent form expressible in \ and Burali-Forti’s involves an inconsistent form expressible in \. (shrink)

Epistemic logic in the tradition of Hintikka provides, as one of its many applications, a toolkit for the precise analysis of certain epistemological problems. In recent years, dynamic epistemic logic has expanded this toolkit. Dynamic epistemic logic has been used in analyses of well-known epistemic “paradoxes”, such as the Paradox of the Surprise Examination and Fitch’s Paradox of Knowability, and related epistemic phenomena, such as what Hintikka called the “anti-performatory effect” of Moorean announcements. In this paper, we explore (...) a variation on basic dynamic epistemic logic—what we call sequential epistemic logic—and argue that it allows more faithful and fine-grained analyses of those epistemological topics. (shrink)

Paradox Lost covers ten of philosophy’s most fascinating paradoxes, in which seemingly compelling reasoning leads to absurd conclusions. The following paradoxes are included: The Liar Paradox, in which a sentence says of itself that it is false. Is the sentence true or false? The Sorites Paradox, in which we imagine removing grains of sand one at a time from a heap of sand. Is there a particular grain whose removal converts the heap to a non-heap? The Puzzle (...) of the Self-Torturer, in which a series of seemingly rational choices has us accepting a life of excruciating pain, in exchange for millions of dollars. Newcomb’s Problem, in which we seemingly maximize our expected profit by taking an unknown sum of money, rather than taking the same sum plus $1000. The Surprise Quiz Paradox, in which a professor finds that it is impossible to give a surprise quiz on any particular day of the week... but also that if this is so, then a surprise quiz can be given on any day. The Two Envelope Paradox, in which we are asked to choose between two indistinguishable envelopes, and it is seemingly shown that each envelope is preferable to the other. The Ravens Paradox, in which observing a purple shoe provides evidence that all ravens are black. The Shooting Room Paradox, in which a deadly game kills 90% of all who play, yet each individual’s survival turns on the flip of a fair coin. Each paradox is clearly described, common mistakes are explored, and a clear, logical solution offered. Paradox Lost will appeal to professional philosophers, students of philosophy, and all who love intellectual puzzles. (shrink)

This paper aims to present how logic may undermine a parliamentary assault on democratic institutions based on the analysis conducted with reference to the so-called Polish constitutional crisis. I analyse whether a law can be reviewed on the basis of this law itself. The Polish Constitutional Tribunal faced such a problem while passing the verdict of 9th March, 2016, regarding the constitutionality of the amendment to the Statute on the Constitutional Tribunal from 22nd December, 2015. This problem, called a ‘verdict (...) paradox’, was claimed to be a type of the Liar paradox. I argue that, contrary to the common view, the problem of the verdict paradox is not based on the Liar paradox; for this purpose, a logical analysis is applied to four variants of a reasoning with regard to the constitutionality of the said amendment. The distinction between two levels of analysis concerning emerging reasonings, namely an abstract level and a concrete level placed in the context of the legal system, is also introduced. This paper demonstrates that although only two variants of the reasoning concerning a law’s judicial review based on the law itself involve logical contradiction, the possibility of employing reasonings from other variants must be excluded, albeit due to alternative reasons. Therefore, the Constitutional Tribunal’s decision to avoid the verdict paradox by passing over the reviewed provisions in the review process was correct and might be perceived as an example of how logic and reason can defend the institutional guarantees of political freedom. (shrink)

The main aim of this paper is to elucidate, from a logical point of view, the phenomenon of Simpson reversal — the paradox of a statistical reasoning. We define a binary relation of supporting in the following way: a sentence A supports a sentence B if and only if the probability of B is higher when A is true, than when A is false. It appears that a statistical argument occurring in Simpson paradox cannot be formalized by (...) means of a binary relation. We generalize the relation of support introducing the third parameter. Then we argue that it properly mirrors main features of the statistical argument occurring in Simpson paradox. (shrink)

The Schrödinger’s cat paradox has troubled physicists for many years and many ways to solve it has been provided so far. In this paper I will try to provide a different method of solving the Paradox with the help of logical machine and it will look like a hybrid of Wigner’s no-go theorem and the machine of Gödel in his Incompleteness Theorems. And due to the machine the paradox will not require any human intervention to measure (...) the state of the cat, whether it is alive or dead, while the measurement will be made by humans with regard to the statement given out by the machine. (shrink)

In this paper, I analyze the "surprise exam paradox". I think that the paradox can be avoided and I am going to focus on three points: 1) A conflict arises between reasoning and the confidence in the person that makes the original statement. If we examine the situation by reasoning we conclude that the statement is not going to come true, because we trust the person that states it. However, if it is not possible to happen, it happens, (...) and the person told the truth; 2) There is a disjunction among the days of the week: “or it is the first day or it is the second day or … it is the last day” (or Monday or Tuesday or Wednesday or Thursday or Friday). If I still have not been given the exam by Friday, the only possible conclusion is that the exam will be given on Friday. On Thursday, however, thesituation is completely different; 3) It seems that this paradox is a case of contingent futures, which branching time logics usually solve. The truth-value of the sentences is only relative to the branch that takes place in the end. (shrink)