Understandably absorbed in technical details, discussion of the semantic paradoxes risks losing sight of broad methodological principles. This chapter sketches a general approach to the comparison of rival logics, and applies it to argue that revision of classical propositional logic has much higher costs than its proponents typically recognize.

I use the principle of truth-maker maximalism to provide a new solution to the semantic paradoxes. According to the solution, AUS, its undecidable whether paradoxical sentences are grounded or ungrounded. From this it follows that their alethic status is undecidable. We cannot assert, in principle, whether paradoxical sentences are true, false, either true or false, neither true nor false, both true and false, and so on. AUS involves no ad hoc modification of logic, denial of the T-schema's validity, (...) or obvious revenge. (shrink)

Both in dealing with the semantic paradoxes and in dealing with vagueness and indeterminacy, there is some temptation to weaken classical logic: in particular, to restrict the law of excluded middle. The reasons for doing this are somewhat different in the two cases. In the case of the semantic paradoxes, a weakening of classical logic (presumably involving a restriction of excluded middle) is required if we are to preserve the naive theory of truth without inconsistency. In (...) the case of vagueness and indeterminacy, there is no worry about inconsistency; but a central intuition is that we must reject the factual status of certain sentences, and it hard to see how we can do that while claiming that the law of excluded middle applies to those sentences. So despite the different routes, we have a similar conclusion in the two cases. (shrink)

The semantic paradoxes are often associated with self-reference or referential circularity. Yablo (Analysis 53(4):251–252, 1993), however, has shown that there are infinitary versions of the paradoxes that do not involve this form of circularity. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality. In this essay, we lay the groundwork for a general investigation into the nature of reference structures that support the semantic paradoxes and (...) the semantic hypodoxes. We develop a functionally complete infinitary propositional language endowed with a denotation assignment and extract the reference structural information in terms of graph-theoretic properties. We introduce the new concepts of dangerous and precarious reference graphs, which allows us to rigorously define the task: classify the dangerous and precarious directed graphs purely in terms of their graph-theoretic properties. Ungroundedness will be shown to fully characterize the precarious reference graphs and fully characterize the dangerous finite graphs. We prove that an undirected graph has a dangerous orientation if and only if it contains a cycle, providing some support for the traditional idea that cyclic structure is required for paradoxicality. This leaves the task of classifying danger for infinite acyclic reference graphs. We provide some compactness results, which give further necessary conditions on danger in infinite graphs, which in conjunction with a notion of self-containment allows us to prove that dangerous acyclic graphs must have infinitely many vertices with infinite out-degree. But a full characterization of danger remains an open question. In the appendices we relate our results to the results given in Cook (J Symb Log 69(3):767–774, 2004) and Yablo (2006) with respect to more restricted sentences systems, which we call $\mathcal{F}$ -systems. (shrink)

If semantic paradoxes such as the Liar arise because ‘true’ and other metalinguistic expressions can change their reference with changes of linguistic context, is that due to indexicality (they have the same linguistic meaning as reference changes) or ambiguity (their linguistic meaning itself changes)? An argument from communication that appears to favour the indexicality interpretation is not compelling. This paper defends the ambiguity interpretation. It is left open whether its considerations generalize to other kinds of paradox.

The object of the present discussion is to show that the analysis of indirect discourse obtained when the concept of assertion is construed as a relationship that obtains between the asserting person and the asserted proposition—along the familiar lines proposed by Church [3, 4]—is entirely adequate of itself to circumvent the semantical paradoxes in which indirect discourse is involved.

In this paper we show how Dummett-Prawitz-style proof-theoretic semantics has to be modified in order to cope with paradoxical phenomena. It will turn out that one of its basic tenets has to be given up, namely the definition of the correctness of an inference as validity preservation. As a result, the notions of an argument being valid and of an argument being constituted by correct inference rules will no more coincide. The gap between the two notions is accounted for by (...) introducing the distinction between sense and denotation in the proof-theoretic-semantic setting. (shrink)

It is argued that a certain form of the view that the semantic paradoxes show that natural languages are "inconsistent" provides the best response to the semantic paradoxes. After extended discussions of the views of Kirk Ludwig and Matti Eklund, it is argued that in its strongest formulation the view maintains that understanding a natural language is sharing cognition of an inconsistent semantic theory for that language with other speakers. A number of aspects of this (...) approach are discussed and a few objections are entertained. (shrink)

This article contains an overview of the main problems, themes and theories relating to the semantic paradoxes in the twentieth century. From this historical overview I tentatively draw some lessons about the way in which the field may evolve in the next decade.

In this essay (for undergraduates) I introduce three of the famous semantic paradoxes: the Liar, Grelling’s, and the No-No. Collectively, they seem to show that the notion of truth is highly paradoxical, perhaps even contradictory. They seem to show that the concept of truth is a bit akin to the concept of a married bachelor—it just makes no sense at all. But in order to really understand those paradoxes one needs to be very comfortable thinking about how (...) lots of interesting sentences talk about not dogs or cats or elections or baseball but sentences. That is, we need to get familiar analyzing sentences that talk about sentences. (shrink)

Douglas Patterson argues that the best way to respond to the semantic paradoxes that arise in natural language is to take natural language semantics to be (explosively) inconsistent. According to Patterson, to understand a natural language is to share with others cognition of a false semantic theory. Patterson’s main argument runs as follows. English is expressively rich. So, the first sentence occurring in this review could be.

John Mair was an influential post-medieval scholar. This paper focuses on his Tractatus insolubilium, in which he proposed semantic analysis of self-referential phenomena, in particular on his solution to alethic and correspondence paradoxes and his treatment of their general semantic aspects as well as particular applications. His solution to paradoxes is based on the so-called “network evaluation”, i.e. on a semantics which defines the concepts of truth and correspondence with reality in contextual terms. Consequently, the relation (...) between semantic valuation, synonymy and contradiction must be redefined. (shrink)

Conceptual analysis of logical consequence can be regarded as a crucial part of any logical theory. The present paper focuses on John Mair’s approach to this issue from both historical and systematic point of view. Mair’s task is to analyse the concepts of modality and validity in universal token-based languages with non-compositional semantics based on network evaluation. To fulfil it, Mair addresses modal paradoxes, validity paradoxes and inferences with paradoxical components. Both truth and modality and truth and validity, (...) when conceived as semantic properties, turn out to be mutually independent as a result of Mair’s semantics: there are true impossible propositions and possible propositions which cannot be true and truth-preservation turns out to be neither a sufficient nor a necessary condition for validity. (shrink)

John Mair (1467–1550) was an influential post-medieval scholar. This paper focuses on his Tractatus insolubilium, in which he proposed semantic analysis of self-referential phenomena, in particular on his solution to alethic and correspondence paradoxes and his treatment of their general semantic aspects as well as particular applications. His solution to paradoxes is based on the so-called “network evaluation”, i.e. on a semantics which defines the concepts of truth and correspondence with reality in contextual terms. Consequently, the (...) relation between semantic valuation, synonymy and contradiction must be redefined. (shrink)

John Mair (1467–1550) was an influential post-medieval scholar. This paper focuses on his Tractatus insolubilium, in which he proposed semantic analysis of self-referential phenomena, in particular on his solution to alethic and correspondence paradoxes and his treatment of their general semantic aspects as well as particular applications. His solution to paradoxes is based on the so-called “network evaluation”, i.e. on a semantics which defines the concepts of truth and correspondence with reality in contextual terms. Consequently, the (...) relation between semantic valuation, synonymy and contradiction must be redefined. (shrink)

We introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook and Rabern, Rabern, and Macauley. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth (...) values. It is shown that this notion of paradoxicality coincides with that of Kripke. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs. (shrink)

The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema “True(A)↔A”, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in “ordinary” contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(A) (...) with A within the language.The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are “defective”. We can in fact define a hierarchy of “defectiveness” predicates within the language. Contrary to claims that any solution to the paradoxes just breeds further paradoxes (“revenge problems”) involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various “levels of defectiveness” can all be made coherent together within a single object language. (shrink)

Bernard Williams's essay ‘Wittgenstein and Idealism’ argues that that the conventionality of language entails the dependence of the truth of sentences and ultimately of corresponding states of affairs as truth‐makers on the existence of thinking subjects. Peter Winch and Colin Lyas try to avoid William's paradox by distinguishing between the existence conditions of a sentence and its assertion. The Winch‐Lyas solution is criticized and a stronger Winch‐Lays resistant version of Williams's paradox is proposed. A more satisfactory countercriticism is given, involving (...) an ineliminable modal fallacy occurring in the paradox inference, that arises because of the argument's invalid combination of categorical and counterfactual assumptions and conclusions. (shrink)

In this essay we introduce a new tool for studying the patterns of sentential reference within the framework introduced in [2] and known as the language of paradox $\mathcal {L}_{\mathsf {P}}$ : strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms. In particular, we show that (i) strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms between $\mathcal {L}_{\mathsf {P}}$ constructions preserve paradoxicality, (ii) many (but not all) earlier results regarding the paradoxicality of $\mathcal {L}_{\mathsf {P}}$ constructions can be recast as special cases of our central result regarding (...) strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms, and (iii) that we can use strong $\mathcal {L}_{\mathsf { P}}$ -homomorphisms to provide a simple demonstration of the paradoxical nature of a well-known paradox that has not received much attention in this context: the McGee paradox. In addition, along the way we will highlight how strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms highlight novel connections between the graph-theoretic analyses of paradoxes mobilized in the $\mathcal {L}_{\mathsf {P}}$ framework and the methods and tools of category theory. (shrink)

A number of philosophers have argued that the key to understanding the semantic paradoxes is to recognize that truth is essentially relative to context. All of these philosophers have been motivated by the idea that once a liar sentence has been uttered we can 'step back' and, from the point of view of a different context, judge that the liar sentence is true. This paper argues that this 'stepping back' idea is a mistake that results from failing to (...) relativize truth to context in the first place. Moreover, context-relative liar sentences, such as 'This sentence is not true in any context' present a paradox even after truth has been relativized to context. Nonetheless, the relativization of truth to context may offer us the means to avoid paradox, if we can justifiably deny that a sentence about a context can be true in the very context it is about. (shrink)

Substructural solutions to the semantic paradoxes have been broadly discussed in recent years. In particular, according to the non-transitive solution, we have to give up the metarule of Cut, whose role is to guarantee that the consequence relation is transitive. This concession—giving up a meta rule—allows us to maintain the entire consequence relation of classical logic. The non-transitive solution has been generalized in recent works into a hierarchy of logics where classicality is maintained at more and more metainferential (...) levels. All the logics in this hierarchy can accommodate a truth predicate, including the logic at the top of the hierarchy—known as CMω—which presumably maintains classicality at all levels. CMω has so far been accounted for exclusively in model-theoretic terms. Therefore, there remains an open question: how do we account for this logic in proof-theoretic terms? Can there be found a proof system that admits each and every classical principle—at all inferential levels—but nevertheless blocks the derivation of the liar? In the present paper, I solve this problem by providing such a proof system and establishing soundness and completeness results. Yet, I also argue that the outcome is philosophically unsatisfactory. In fact, I’m afraid that in light of my results this metainferential solution to the paradoxes can hardly be called a “solution,” let alone a good one. (shrink)

According to some philosophers, the Liar paradox arises because of a mistaken theory of truth. Its lesson is that we must reject some instances of the naive propositional truth-schema \It is true that \ if and only if \\. In this paper, I construct a novel semantic paradox in which no principle even analogous to the truth-schema plays any role. I argue that this undermines the claim that we ought to respond to the Liar by revising our theory of (...) truth. (shrink)

In the 1330s Roger Swyneshed formulated a solution to semantic paradoxes based on the distinction between correspondence with reality and self-falsification as truth-making factors. Since Swyneshed states that some valid inferences are not truth-preserving, his view implies the question of the general definition of validity which he does not address explicitly. Logical works attributed to Paul of Venice contain developments of Swyneshed's contextualist semantics substantially modified by the assumption that sentential meanings are objective propositional entities. The main goals (...) of this paper are to show the correlations between the ontological and logical developments of Swyneshed's semantics in the works of Paul of Venice and to outline a context-sensitive formal semantics that could serve as a model of this family of semantic theories. (shrink)

Plausible principles on truth seem to yield contradictory conclusions about paradoxical sentences such as the Strengthened Liar. Those who take the contextualist approach, such as Parsons and Burge, attempt to justify the seemingly contradictory conclusions by arguing that the natural reasoning that leads to them involves some kind of contextual shift that makes them compatible. This paper argues that one cannot take this approach to give a proper treatment of infinite descending chains of semantic attributions. It also examines a (...) related approach taken by Gaifman and argues that it has the same problem. (shrink)

In one of his key texts, the Mūlamadhyamakakārikā, Nāgārjuna famously sets out to refute the ontology of essence.1 He presents numerous arguments to show that things don’t exist essentially—that is, that things are empty of essence or inherent existence. The doctrine of emptiness has been variously understood by traditional and contemporary commentators. Most radical is the recent interpretation presented by Garfield and Priest. They have rationally reconstructed Nāgārjuna’s doctrine of emptiness as an endorsement of the contradictory nature of reality. According (...) to them, Nāgārjuna can be seen to be arguing that the way in which things exist in reality and what we can truly say about them must be... (shrink)

Philosophical Investigations, EarlyView.

Two semantic paradoxes, the Liar and Curry’s paradox, are analysed using a newly developed conception of procedural semantics (semantics according to which the truth of propositions is determined algorithmically), whose main characteristic is its departure from methodological realism. Rather than determining pre-existing facts, procedures are constitutive of them. Of this semantics, two versions are considered: closed (where the halting of procedures is presumed) and open (without this presumption). To this end, a procedural approach to deductive reasoning is developed, (...) based on the idea of simulation. As is shown, closed semantics supports classical logic, but cannot in any straightforward way accommodate the concept of truth. In open semantics, where paradoxical propositions naturally ‘belong’, they cease to be paradoxical; yet, it is concluded that the natural choice—for logicians and common people alike—is to stick to closed semantics, pragmatically circumventing problematic utterances. (shrink)