Buddhist Illogic. The 2nd Century CE Indian philosopher Nagarjuna founded the Madhyamika (Middle Way) school of Mahayana Buddhism, which strongly influenced Chinese, Korean and Japanese (Ch’an or Zen) Buddhism, as well as Tibetan Buddhism. Nagarjuna is regarded by many Buddhist writers to this day as a very important philosopher, who they claim definitively proved the futility of ordinary human cognitive means. His writings include a series of arguments purporting to show the illogic of logic, the absurdity of (...) reason. He considers this the way to verbalize and justify the Buddhist doctrine of “emptiness” (Shunyata). These arguments attack some of the basic tenets and techniques of reasoning, such as the laws of thought (identity, non-contradiction and the excluded middle), conceptualization and predication, our common assumptions of self, entities and essences, as well as our beliefs in motion and causation. The present essay demonstrates the many sophistries involved in Nagarjuna’s arguments. He uses double standards, applying or ignoring the laws of thought and other norms as convenient to his goals; he manipulates his readers, by giving seemingly logical form (like the dilemma) to his discourse, while in fact engaged in non-sequiturs or appealing to doubtful premises; he plays with words, relying on unclear terminology, misleading equivocations and unfair fixations of meaning; and he ‘steals concepts’, using them to deny the very percepts on which they are based. Although a critique of the Madhyamika philosophical interpretation and defense of “emptiness”, Buddhist Illogic is not intended to dissuade readers from Buddhism. On the contrary, its aim to enhance personal awareness of actual cognitive processes, and so improve meditation. It is also an excellent primer on phenomenological epistemology. (shrink)

A key facet of Valerie Plumwood’s feminist critique of logic is her analysis of classical negation. On Plumwood’s reading, the exclusionary features of classical negation generate hierarchical dualisms, i.e., dichotomies in which dominant groups’ primacy is reinforced while underprivileged groups are oppressed. For example, Plumwood identifies the system collapse following from ex contradictione quodlibet—that a theory including both φ and ∼φ trivializes—as a primary source of many of these features. Although Plumwood considers the principle of excluded middle to (...) be compatible with her goals, that she identifies relevant logics as systems lacking a hierarchical negation—whose first-degree fragments are both paraconsistent and paracomplete—suggests that excluded middle plays some role in hierarchical dualisms as well. In these notes, I examine the role of excluded middle in generating oppressive homogenization and try to clarify the relationship between Plumwood’s critique and this principle from several contemporary perspectives. Finally, I examine the matter of whether Plumwood’s critique requires relevance or whether a non-relevant logic could satisfy her criteria and serve as a liberatory logic of difference. (shrink)

Western philosophy has mainly developed in accordance with the three laws of identity, noncontradiction and excluded middle, also known as “laws of thought”. Since Zen Buddhism often violates these apparently indisputable logical principles, a superficial reading may induce the idea that Zen Buddhism is a completely irrational, illogical doctrine. In this essay, I argue that Zen Buddhism is not absurd or illogical. Conversely, it relies on a different logic, which is perfectly consonant with the Buddhist view of (...) the world. (shrink)

The principle of excluded middle is the logical interpretation of the law V ≤ A v ヿA in an orthocomplemented lattice and, hence, in the lattice of the subspaces of a Hilbert space which correspond to quantum mechanical propositions. We use the dialogic approach to logic in order to show that, in addition to the already established laws of effective quantum logic, the principle of excluded middle can also be founded. The dialogic approach is based on (...) the very conditions under which propositions can be confirmed by measurements. From the fact that the principle of. excluded middle can be confirmed for elementary propositions which are proved by quantum mechanical measurements, we conclude that this principle is inherited by all finite compound propositions. For this proof it is essential that, in the dialog-game about a connective, a finite confirmation strategy for the mutual commensurability of the subpropositions is used. (shrink)

As part of our small contribution in dialogue toward better peace development and reconciliation studies, and following Toffler & Toffler’s War and Antiwar (1993), the present article delves into a realm of logic beyond the traditional confines of negation and the excluded middle principle, exploring the nuances of "Otherness" that transcend classical and Nagatomo logics. Departing from the foundational premises of classical Aristotelian logic systems, this exploration ventures into alternative realms of reasoning, specifically examining Neutrosophic Logic (...) and Klein bottle logic (cf. Smarandache, 2005). The study challenges conventional boundaries and explores the implications of embracing paradoxes and self-reference in logic systems, aiming to redefine approaches to understanding truth and reasoning. The paper investigates how these alternative logics open avenues for philosophical inquiry, redefining entropy, and potentially influencing innovative perspectives in free energy systems. Through this exploration, it seeks to expand the discourse on logic, welcoming a broader spectrum of thought beyond established frameworks; and we also discuss shortly a number of possible implementations including in risk management and also Klein bottle entropy redefinition (Tang et al, 2018). (shrink)

This paper is a proposal of continuation of the work of C. Rauszer. The logic of falsehood created by her may constitute the starting point for construction of logic formalising reductive reasonings. The extension of Heyting-Brouwer logic to its deductive-reductive form sheds new light upon those classical tautologies which are rejected in intuitionism. It turns out that among HBtautologies there can be found all the classical ones. Some of them are characteristic for deductive reasoning and they are (...) accepted by intuitionism. Others formulate the laws of reductive reasoning. Many of them, including the law of excluded middle has been rejected in Heyting’s intuitionism. Intuitionism only permits for those reductive tautologies, which at the same time bear deductive character. Thus, the complete HB intuitionism does not reject any of the classical tautologies. Every classical tautology appears in HB logic in its deductive or reductive part. Some are even present in both parts. At the end it is shown that the law of excluded middle α∨¬α is a law of the reductive part of the traditional, Heyting’s intuitionistic propositional calculus. (shrink)

A unified and modular falsification-aware single-succedent Gentzen-style framework is introduced for classical, paradefinite, paraconsistent, and paracomplete logics. This framework is composed of two special inference rules, referred to as the rules of explosion and excluded middle, which correspond to the principle of explosion and the law of excluded middle, respectively. Similar to the cut rule in Gentzen’s LK for classical logic, these rules are admissible in cut-free LK. A falsification-aware single-succedent Gentzen-style sequent calculus fsCL for classical (...) class='Hi'>logic is formalized based on the proposed framework. The calculus fsCL is obtained from the existing falsification-aware single-succedent Gentzen-style sequent calculus GN4 for Nelson’s paradefinite (or paraconsistent) four-valued logic N4 by adding the rules of explosion and excluded middle. A falsification-aware single-succedent Gentzen-style sequent calculus GN3 for Nelson’s paracomplete three-valued logic N3 is also obtained from GN4 by adding the rule of explosion. The cut-elimination theorems for fsCL, GN3, and some of their neighbors as well as the Glivenko theorem for fsCL are proved. (shrink)

It is shown for an arbitrary topos that the Law of the Excluded Middle holds in its propositional logic iff it satisfies the limited choice principle that every epimorphism from 2 = 1 ⊕ 1 splits.

Conditional excluded middle (CEM) is the following principe of counterfactual logic: either, if it were the case that φ, it would be the case that ψ, or, if it were the case that φ, it would be the case that not-ψ. I will first show that CEM entails the identity of indiscernibles, the falsity of physicalism, and the failure of the modal to supervene on the categorical and of the vague to supervene on the precise. I will then (...) argue that we should accept these startling conclusions, since CEM is valid. (shrink)

The Jaina tradition is known for its distinctive approach to prima facie incompatible claims about the nature of reality. The Jaina approach to conflicting views is to seek an integration or synthesis, in which apparently contrary views are resolved into a vantage point from which each view can be seen as expressing part of a larger, more complex truth. Viewed by some contemporary Jaina thinkers as an extension of the principle of ahiṃsā into the realm of intellectual discourse, Jaina (...) class='Hi'>logic marks quite a distinctive stance toward the concept of logical consistency. While it does not directly violate the law of excluded middle, it does, one might say, navigate this principle in a highly and potentially useful way. The potential usefulness of Jaina logic includes the possibility of its use in arguing for the position known as religious pluralism or worldview pluralism. This is a view which many philosophers see as holding great promise in developing a way to think about differences across worldviews in ways that do not lead to the kind of conflict and polarization that all too often characterizes ideological differences in today’s world. (shrink)

Peirce’s principles of excluded middle and contradiction more resembled those of Aristotle than those of contemporary logicians. While the principles themselves are simple and straightforward, many of Peirce’s comments about them have been misunderstood by commentators. In particular, his belief that the principle of excluded middle does not apply to the general and that the principle of contradiction does not apply to the vague have been mistakenly connected to his eventual rejection of the principle of bivalence and development (...) of three-valued logical connectives. An understanding of Peirce’s view of those logical principles shows that those beliefs motivated neither his rejection of bivalence nor his work in triadic logic. (shrink)

There is a standard quantificational view of generic sentences according to which they have a tripartite logical form involving a phonologically null generic operator called 'Gen'. Recently, a number of theorists have questioned the standard view and revived a competing proposal according to which generics involve the predication of properties to kinds. This paper offers a novel argument against the kind-predication approach on the basis of the invalidity of Generic Excluded Middle, a principle according to which any sentence of (...) the form ⌜Either Fs are G or Fs are not G⌝ is true. I argue that the kind-predication approach erroneously predicts that GEM is valid, and that it can only avoid this conclusion by either collapsing into a form of the quantificational analysis or otherwise garnering unpalatable metaphysical commitments. I also show that, while the quantificational approach does not validate GEM as a matter of logical form, the principle may be validated on certain semantic analyses of the generic operator, and so, such theories should be rejected. (shrink)

It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent. Given a bivalent background logic, this is equivalent to asking about the conjunction of Conditional Excluded Middle (CEM, opposite conditionals are not both false) and Weak Boethius' Thesis (WBT, opposite conditionals are not both true). In the system CI.0 of consequential implication, which is intertranslatable with the modal logic KT, WBT is a theorem, so it is natural to ask which (...) instances of CEM are derivable. We also investigate the systems CIw and CI of consequential implication, corresponding to the modal logics K and KD respectively, with occasional remarks about stronger systems. While unrestricted CEM produces modal collapse in all these systems, CEM restricted to contingent formulas yields the Alt2 axiom (semantically, each world can see at most two worlds), which corresponds to the symmetry of consequential implication. It is proved that in all the main systems considered, a given instance of CEM is derivable if and only if the result of replacing consequential implication by the material biconditional in one or other of its disjuncts is provable. Several related results are also proved. The methods of the paper are those of propositional modal logic as applied to a special sort of conditional. (shrink)

In this essay I renew the case for Conditional Excluded Middle (CXM) in light of recent developments in the semantics of the subjunctive conditional. I argue that Michael Tooley’s recent backward causation counterexample to the Stalnaker-Lewis comparative world similarity semantics undermines the strongest argument against CXM, and I offer a new, principled argument for the validity of CXM that is in no way undermined by Tooley’s counterexample. Finally, I formulate a simple semantics for the subjunctive conditional that is consistent (...) with both CXM and Tooley’s counterexample. (shrink)

In this essay I renew the case for Conditional Excluded Middle in light of recent developments in the semantics of the subjunctive conditional. I argue that Michael Tooley's recent backward causation counterexample to the Stalnaker-Lewis comparative world similarity semantics undermines the strongest argument against CXM, and I offer a new, principled argument for the validity of CXM that is in no way undermined by Tooley's counterexample. Finally, I formulate a simple semantics for the subjunctive conditional that is consistent with (...) both CXM and Tooley's counterexample. (shrink)

I carry out in this paper a philosophical analysis of the principle of excluded middle (or, as it is often called in the version I favor here, principle of bivalence: any meaningful assertion is either true or false). This principle has been criticized, and sometimes rejected, on the charge that its validity depends on presuppositions that are not, some believe, universally obtainable; in particular, that any well-posed problem is solvable. My goal here is to show that, although excluded (...) middle does indeed rest on certain presuppositions, they do not have the character of hypotheses that may or may not be true, or matters of fact that may or may not be the case. These presuppositions have, I claim, a transcendental character. Hence, the acceptance of excluded middle does not necessarily require, as some have claimed, an allegiance to ontological realism or some sort of cognitive optimism, construed as factual theses concerning the ontological status of domains of objects and our capability of accessing them cognitively. DOI:10.5007/1808-1711.2011v15n2p333. (shrink)

Herman Cappelen and Ernie Lepore’s book is ultimately a defense of their self-styled Semantic Minimalism, but it’s mainly a protracted assault on semantic Contextualism, both moderate and radical. They argue at length that Moderate Contextualism leads inevitably to Radical Contextualism and at greater length that Radical Contextualism is misguided. Supposing that “[Radical Contextualism] is the logical consequence of denying Semantic Minimalism”, they think they have given an indirect argument for their version of Semantic Minimalism. But they overlook a third view, (...) one that splits the difference between the other two. Like Contextualism it rejects Propositionalism, the conservative dogma that every indexical-free declarative sentence expresses a proposition. Unlike Contextualism, it does not invoke context to fill semantic gaps and, indeed, denies that filling those gaps is a semantic matter. Like Cappelen and Lepore’s brand of Semantic Minimalism, it rejects the very idea of pragmatic intrusion into semantic content. However, in rejecting Propositionalism, it is more radical, indeed, more minimalist than their version of Semantic Minimalism. It does not imagine that sentences that intuitively seem not to express propositions at least express “minimal propositions.” Radical Semantic Minimalism, or simply Radicalism, says that the sentences in question are semantically incomplete—their semantic contents are not propositions but merely “propositional radicals.”. (shrink)

In this chapter we consider three philosophical perspectives (including those of Stalnaker and Lewis) on the question of whether and how the principle of conditional excluded middle should figure in the logic and semantics of counterfactuals. We articulate and defend a third view that is patterned after belief revision theories offered in other areas of logic and philosophy. Unlike Lewis’ view, the belief revision perspective does not reject conditional excluded middle, and unlike Stalnaker’s, it does not (...) embrace supervaluationism. We adduce both theoretical and empirical considerations to argue that the belief revision perspective should be preferred to its alternatives. The empirical considerations are drawn from the results of four empirical studies (which we report below) of non-experts’ judgments about counterfactuals and conditional excluded middle. (shrink)

In this paper, intuitionistic modal logics which do not admit the law of the excluded middle are studied. The main result is that there exista a continuum of such logics.

The law of excluded middle is usually considered as intrinsically connected with the realistic standpoint and incompatible with the idealistic position. This is just what Ajdukiewicz claims in his critique of transcendental idealism. The analysis of Ajdukiewicz's argumentation raises the problem of validity of the law of excluded middle for vague (or incomplete) languages. The problem is being solved by differentiating between the logical (or ontological) and the metalogical (or semantical) law of excluded middle: in contrast to (...) the former, the latter is claimed to be invalid for the languages in question, without thereby embracing the idealist position. (shrink)

Physical dimensions like “mass”, “length”, “charge”, represented by the symbols [M], [L], [Q], are not numbers, but used as numbers to perform dimensional analysis in particular, and to write the equations of physics in general, by the physicist. The law of excluded middle falls short of explaining the contradictory meanings of the same symbols. The statements like “m tends to 0”, “r tends to 0”, “q tends to 0”, used by the physicist, are inconsistent on dimensional grounds because “m”, (...) “r”, “q” represent quantities with physical dimensions of [M], [L], [Q] respectively and “0” represents just a number—devoid of physical dimension. Consequently, due to the involvement of the statement “q tends to 0'', where q is the test charge” in the definition of electric field leads to either circular reasoning or a contradiction regarding the experimental verification of the smallest charge in the Millikan–Fletcher oil drop experiment. Considering such issues as problematic, by choice, I make an inquiry regarding the basic language in terms of which physics is written, with an aim of exploring how truthfully the verbal statements can be converted to the corresponding physico-mathematical expressions, where “physico-mathematical” signifies the involvement of physical dimensions. Such investigation necessitates an explanation by demonstration of “self inquiry”, “middle way”, “dependent origination”, “emptiness/relational existence”, which are certain terms that signify the basic tenets of Buddhism. In light of such demonstration I explain my view of “definition”; the relations among quantity, physical dimension and number; meaninglessness of “zero quantity” and the associated logico-linguistic fallacy; difference between unit and unity. Considering the importance of the notion of electric field in physics, I present a critical analysis of the definitions of electric field due to Maxwell and Jackson, along with the physico-mathematical conversions of the verbal statements. The analysis of Jackson’s definition points towards an expression of the electric field as an infinite series due to the associated “limiting process” of the test charge. However, it brings out the necessity of a postulate regarding the existence of charges, which nevertheless follows from the definition of quantity. Consequently, I explain the notion of undecidable charges that act as the middle way to resolve the contradiction regarding the Millikan–Fletcher oil drop experiment. In passing, I provide a logico-linguistic analysis, in physico-mathematical terms, of two verbal statements of Maxwell in relation to his definition of electric field, which suggests Maxwell’s conception of dependent origination of distance and charge ) and that of emptiness in the context of relative vacuum. This work is an appeal for the dissociation of the categorical disciplines of logic and physics and on the large, a fruitful merger of Eastern philosophy and Western science. Nevertheless, it remains open to how the reader relates to this work, which is the essence of emptiness. (shrink)

The principle of excluded middle is more important than is commonly believed for understanding Kant's overall philosophical project. In the article, this principle is examined in the following contexts: kinds of judgments, concepts of opposition, negation, and determination, and apagogic proof. It is first explained how the principle of excluded middle is employed by Kant in distinguishing between the kinds of judgment. Also called the principle of division, it is the principle of disjunctive and apodictic judgments in Kant's (...) famous table of judgments. Next, the Author shows which kind of opposition is related to the principle of the excluded middle. As a merely logical criterion of truth, this principle grounds the logical necessity of a cognition and plays an important role in the use of apagogical proofs. Then Kant's account of logical negation will be investigated briefly. Negative judgment with a negative copula indicates that something is not contained within the sphere of a given concept. This process occurs in accordance with the principle of excluded middle. Finally, an analysis is made of how Kant presents the metaphysical principle of thoroughgoing determination and its differences from the logical principle of excluded middle. (shrink)

The logic of the weak law of excluded middleKC p is obtained by adding the formula A A as an axiom scheme to Heyting's intuitionistic logicH p . A cut-free sequent calculus for this logic is given. As the consequences of the cut-elimination theorem, we get the decidability of the propositional part of this calculus, its separability, equality of the negationless fragments ofKC p andH p , interpolation theorems and so on. From the proof-theoretical point of view, (...) the formulation presented in this paper makes clearer the relations betweenKC p ,H p , and the classical logic. In the end, an interpretation of classical propositional logic in the propositional part ofKC p is given. (shrink)

Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemic conception of truth and the principle of excluded middle. In PART II I give a historical overview of different attitudes regarding the problem. In PART III I sketch a possible holistic solution.

The one-page 1978 informal proof of Goodman and Myhill is regimented in a weak constructive set theory in free logic. The decidability of identities in general (⁠|$a\!=\!b\vee\neg a\!=\!b$|⁠) is derived; then, of sentences in general (⁠|$\psi\vee\neg\psi$|⁠). Martin-Löf’s and Bell’s receptions of the latter result are discussed. Regimentation reveals the form of Choice used in deriving Excluded Middle. It also reveals an abstraction principle that the proof employs. It will be argued that the Goodman–Myhill result does not provide the (...) constructive set theorist with a dispositive reason for not adopting (full) Choice. (shrink)

This is the culmination of a discussion on Berry's Paradox with Graham Priest, over an extended period from 1983 to 2019, the central point being whether the Paradox can be avoided or not by removal of the Law of Excluded Middle (LEM). Priest is of the view that a form of the Paradox can be derived without the LEM, whilst Brady disputes this. We start by conceptualizing negation in the logic MC of meaning containment and introduce the LEM (...) as part of the classical recapture. We then examine the usage of the LEM in some other paradoxes and see that it is applied to cases of self-reference. In relation to Priest's [2019] paper, we go on to find a similar use of the LEM in Priest's derivation of Berry's Paradox. However, it is found to be deeper and trickier than other paradoxes, requiring a special effort to untangle the relationships between the LEM, self-reference and meta-theoretic influence. We then examine Brady's previous formalization of Berry's Paradox, considering Brady's most recent view of restricted quantification and his recursive account of the least number satisfying a property. We show that neither of these methods can be used to formalize the paradox. We also examine the inductive shapes for the Substitution of Identity Rule for intensional and extensional identities, while conceding Priest's point, made in [2019], that his shape of Substitution of Identity is not relevant to his proof of Berry's Paradox. (shrink)

The law of excluded middle is usually considered as intrinsically connected with the realistic standpoint and incompatible with the idealistic position. This is just what Ajdukiewicz claims in his critique of transcendental idealism. The analysis of Ajdukiewicz's argumentation raises the problem of validity of the law of excluded middle for vague (or incomplete) languages. The problem is being solved by differentiating between the logical (or ontological) and the metalogical (or semantical) law of excluded middle: in contrast to (...) the former, the latter is claimed to be invalid for the languages in question, without thereby embracing the idealist position. (shrink)

It has long been recognized that negation in Aristotle’s term logic differs syntactically from negation in classical logic: modern external negation attaches to propositions fully formed, whereas Aristotelian internal negation forms propositions from sentential constituents. Still, modern external negation is used to render Aristotelian internal negation, as may be seen in formalizations of Aristotle’s semantic principles of non-contradiction and of excluded middle. These principles govern the distribution of truth values among pairs of contradictory propositions, and Aristotelian contradictories (...) always consist of an affirmation and a denial. So how should we formalize a false denial? In the literature, we find that a false denial is formalized by means of two negation signs attached to a one-place predicate. However, it can be shown that this rendering leads to an incorrect picture of Aristotle’s principles. In this paper, I propose a solution to this technical problem by devising a formal notation especially for Aristotelian propositions in which internal negation is differentiated from external negation. I will also analyze both principles, each of which has two logically equivalent forms, a positive and a negative one. The fact that Aristotle’s principles are distinct and complementary is reflected in my new formalizations. (shrink)

The usual rule used to obtain natural deduction formulations of classical logic from intuitionistic logic, namely.

We study a class of formulas generalizing the weak law of the excluded middle and provide a characterization of these formulas in terms of Kripke frames and Brouwer algebras. We use these formulas to separate logics corresponding to factors of the Medvedev lattice.

Since last year, major initiatives have been undertaken by the chair of theoretical philosophy at the University of Tübingen in order to enhance the reception of analytic metaphysics in the European landscape. Here we review the 2013 summer workshop, intended to be the first of an annual series, on “Existence, Truth and Fundamentality”, the invited speakers being Graham Priest (Melbourne), Stephan Leuenberger (Glasgow), Dan López de Sa (Barcelona), Francesco Berto (Aberdeen), Friederike Moltmann (Paris – Pantheon Sorbonne) and Jason Turner (Leeds). (...) The workshop’s scope is fairly large, focusing on three different aspects of contemporary research in metaphysics. However, the underlying idea of organizers Thomas Sattig and Alessandro Torza consisted in exploring their interconnections and even new areas which possibly share the same problem spectrum. Also, they thought of bringing together both young and senior protagonists of the contemporary debate. In this sense, the main thread followed by the speakers consisted in enquiring the ideas of fundamentality, dependence and grounding with respect to ontology (Berto, López de Sa, Turner) and the truthmaking debate (Priest, Moltmann, Leuenberger), thus pretty much in line with some latest tendencies in metaphysics, such as, prima inter paria, the works of Kit Fine. When we say that something is more fundamental than something else, we move into a non-conventional (for the standards of 20 th century philosophy) metodological framework, hardly focusing on the question about what there is, and rather tackling the issue of how fundamentally there is what there is, or how the existence of something is required as a ground for the existence of other things. Here is how Jonathan Schaffer explains the methodological transition in his influential “On What Grounds What” (2009): -/- Among the many assumptions Quine and Carnap share is that metaphysical questions are existence questions, such as whether numbers exist. They only disagree on the further issue of whether such questions are meaningful (at least as the metaphysician might pose them). But why think that metaphysical questions are existence question of this sort? Return to Aristotle’s Metaphysics. There are virtually no existence question posed. The whole discussion is about substances (fundamental units of being). At one point Aristotle pauses to ask if numbers exist, and his answer is a brief and dismissive yes. [. . . ] For Aristotle, the question about numbers is whether they are transcendent substances or grounded in concreta. The question is not whether numbers exist, but how. (Schaffer, 2009, p. 347) -/- The conference opened with Graham Priest’s intriguing and peculiar review of Buddhist cornerstone “The Fundamental Verses of the Middle Way” (Mūlamadhyakamakārikā) by Nāgārjuna (ca. 150-ca. 250 AD), seen as a unrecognized forerunner of multi-valued semantics and dialetheism. Priest shows that Nagarjuna’s thought indeed has valuable analytic grip (contrary to the traditional mystical dismiss of his works altogether) and is even in accord to some formal schemas rejecting both the laws of excluded middle and non-contradiction. Dan López de Sa was interested in the topic of grounding, raising concerns about how to explain the kind of “reality” exemplified by what is not fundamental, but rather derivative on something else. He proposes a conceptual and non-primitive analysis of grounding as linked to fundamentality and derivation, and argues that the view can be advocated along with the acceptance that at least something is not fundamental. Subsequently, he illustrates an impressive range of applications linked to contemporary debates in meta-ontology, truthmaking and philosophy of time. Stephan Leuenberger discussed a semantics for “total logic”, a variant of first order logic which is claimed to have several applications with respect to the issue of excluding negative existentials from the scope of theories accounting for a given ’totality’ of facts. For example, materialism is the claim that everything there is can be traced back to physical facts or to truth implied by those facts. What about the claim that “There are no angels”? According to Leuenberger, ordinary logic falls prey of modal arguments (if it is possible that there are angels, by modus tollens, that it is not metaphysically necessary that everything there is is physical). So, there is space for the introduction of a new operator, the totality operator, claiming of what enters its scope that “that’s the whole truth”. Consequently, total logic’s semantics tackles David Chalmers’ well-known conceivability-based argument in favour of zombie’s (subjects producing behaviour but experiencing no qualia) possible existence. Friederike Moltmann presented a massive array of natural language-based examples on the role played by case-constructions, such as “it is the case that Caesar crossed the Rubicon”, “the case of the stolen statue”, “the case in which it might rain”, and so forth. She suggests that we should take surface phenomena of our language very seriously with regard to the determination of ontological categories and truth-makers. Cases are provided by many natural languages with their own existence predicates (“to occur” and “to present itself ”), and these do not apply to other kind of entities, such as objects and events. Trying to derive a substantive thesis on their nature, she claims that they are filtered entities, a structure taking into account some features of the corresponding objects and events and preserving their relational pattern. Jason Turner discussed meta-ontological concerns against the claim that ontological debates are defective. Analysed the structure of a metaphysical theory and identified two different forms of defectiveness that can occur in ontological questions, the defection from without and the defection from within, the author provided an original solution in order to avoid such difficulties. He argued thus in favor of a logical constancy, a translation principle thanks to it would be possible reconceive an ontological defection. Lastly, Francesco Berto’s talk closed the workshop with a sagacious discussion of meinongian quantification. He substains that Meinongians and Quineans experience no conceptual equivocation. Even if the former claims that “There are things that do not exist”, something that the latter contrasts with the interpretation of existential quantifiers as all-including, they still rely upon the same concept of “existence” and they understand each other when they are outside the ontology room, so that it cannot be that they lack competence. Subsequently, Berto deals with the objection that Meinongians commit analytic falsehood in separating the meaning of “to be” from that of “exists”. The Quinean has no independent evidence for analytic equivalence aside from her emphatic stress on the “is” in “there is”. Meinongians, however, can retort to linguistic considerations and claim that existence predicates display fairly different functions. -/- As for what regards our internal division of the “philosophical labor”, Fabio Ceravolo reviewed Moltmann and Berto’s talks; whereas Mattia Cozzi took care of Priest’s and Leuenberger’s, and Mattia Sorgon of those given by López de Sa and Turner. Conclusively, we would like to thank Thomas Sattig and Alessandro Torza for all of their organizational efforts and the kind welcome they reserved to us. We hope we will have further opportunities to collaborate with Tübingen in supporting similarly innovative and valiant european initiatives. (shrink)

For (finitary) deductive systems, we formulate a signature‐independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of has a greatest proper ‐congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, (...) in a suitable form, to all protoalgebraic logics. A super‐intuitionistic logic possesses a WEML iff it extends. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of has a global consequence relation with a WEML iff it extends, while every axiomatic extension of with an IL has a WEML. (shrink)

A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.

We show that the fact that the first player wins every instance of Galvin’s “racing pawns” game is equivalent to arithmetic transfinite recursion. Along the way we analyze the satisfaction relation for infinitary formulas, of “internal” hyperarithmetic comprehension, and of the law of excluded middle for such formulas.

The central question of my paper is whether there is a coherent logical theory in which truth is construed in epistemic terms and in which also some version of the law of excluded middle is defended. Brentano in his later writings has such a theory.2 My first question is whether his theory is consistent. I also make a comparison between Brentano’s view and that of an intuitionist at the present day, namely Per Martin-Löf. Such a comparison might provide some (...) insight into what is essential to a theory that understands truth in epistemic terms. (shrink)

As regards the issue of futura contingentia, Aquinas's interpretation turns on a phrase that has sometimes been dismissed as irrelevant or even confused: "...non tamen haec vel illa determinatae...". It is shown that this clause is in complete conformity with Aquinas's interpretation of the principles of non-contradiction and of excluded middle. According to this interpretation the meaning of propositional negation derives from the first two principles considered collectively but is not presupposed by them. With respect to time-relative modalities this (...) implies that at the time of announcement, say, of a sea battle both parts of the contradiction could be true, provided one understands "could" in the sense of a counterfactual possibility with respect to the final outcome. It is also shown that Aquinas's interpretation allows one to avoid some perplexities in connection with the controversial beginning of Met. IX, 4. Some final considerations are meant to show the rationale of that interpretation. (shrink)