Our regimentation of Goodman and Myhill’s proof of Excluded Middle revealed among its premises a form of Choice and an instance of Separation.Here we revisit Zermelo’s requirement that the separating property be definite. The instance that Goodman and Myhill used is not constructively warranted. It is that principle, and not Choice alone, that precipitates Excluded Middle.Separation in various axiomatizations of constructive set theory is examined. We conclude that insufficient critical attention has been paid to how those (...) forms of Separation fail, in light of the Goodman–Myhill result, to capture a genuinely constructive notion of set. (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)

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)

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)

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)

This is a paper on borderline cases and the law of Excluded Middle. In it I try to make use of some long forgotten, but perhaps valuable, work on the topic – a bit of Hegel for instance.

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)

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)

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.

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)

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)

I consider two related objections to the claim that the law of excluded middle does not imply bivalence. One objection claims that the truth predicate captured by supervaluation semantics is not properly motivated. The second objection says that even if it is, LEM still implies bivalence. I show that LEM does not imply bivalence in a supervaluational language. I also argue that considering supertruth as truth can be reasonably motivated.

Lewis (1973) gave a short argument against conditional excluded middle, based on his treatment of ‘might’ counterfactuals. Bennett (2003), with much of the recent literature, gives an alternative take on ‘might’ counterfactuals. But Bennett claims the might-argument against CEM still goes through. This turns on a specific claim I call Bennett’s Hypothesis. I argue that independently of issues to do with the proper analysis of might-counterfactuals, Bennett’s Hypothesis is inconsistent with CEM. But Bennett’s Hypothesis is independently objectionable, so (...) we should resolve this tension by dropping the Hypothesis, not by dropping CEM. (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)

Higginbotham (1986) observed that quantified conditionals have a stronger meaning than might be expected, as attested by the apparent equivalence of examples like No student will pass if he goofs off and Every student will fail if he goofs off. Higginbotham's observation follows straightforwardly given the validity of conditional excluded middle (CEM; as observed by von Fintel & Iatridou 2002), and as such could be taken as evidence thereof (e.g. Williams forthcoming). However, the empirical status of CEM has (...) been disputed, and it is invalid under many prominent theories of conditionals—notably Lewis (1973) for counterfactuals, also Kratzer (1979, 1991). More acutely, Higginbotham's observation holds even for quantified counterparts of conditionals that appear not to obey CEM (Higginbotham 2003), and the standard way of explaining (away) such apparent counterexamples to the principle, à la Stalnaker (1981), does not directly yield an account of our apparent truth-conditional intuitions about the quantified counterparts (Leslie 2009). This article provides an explanation for the latter intuitions within Stalnaker's framework, the upshot being that CEM does remain a viable explanation, in principle, for Higginbotham's observation. (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)

Insensitive Semantics is mainly a protracted assault on semantic Contextualism, both moderate and radical. Cappelen and Lepore argue that Moderate Contextualism leads inevitably, like marijuana to heroin or masturbation to blindness, to Radical Contextualism and in turn that Radical Contextualism is misguided. Assuming that the only alternative to Contextualism is their Semantic Minimalism, they think they’ve given an indirect argument for it. 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 indexicalfree 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. In rejecting Propositionalism, it is more radical, indeed, more minimalist than Cappelen and Lepore’s brand 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)

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 (...) class='Hi'>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)

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)

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)

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)

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.

Determining whether the law of excluded middle requires bivalence depends upon whether we are talking about sentences or propositions. If we are talking about sentences, neither side has a decisive case. If we are talking of propositions, there is a strong argument on the side of those who say the excluded middle does require bivalence. I argue that all challenges to this argument can be met.

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

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 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)

I carry out in this paper a philosophical analysis of the principle of excluded middle. 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. (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)

Why is there so much distortion in ordinary, political, social, and ethical argument? Since we have a pervasive interest in reasoning well and corresponding abilities, the extent of distortion invites explanation. The leading candidates are the need to economize, widespread, fallacious heuristics or assumptions, and self-defensive biases. I argue that these are not sufficient. An additional force is the intellectual pressure generated by acceptance of norms of conversation and argument, which exclude ‘middles’ of, prominently, neither accept nor reject. I conjecture (...) that the distortion we find is due to intellectual and normative pressures generated by our commitment to these excluded-middle norms and if, or when, their force is lessened, there is likely to be less distortion. (shrink)

The historiography on American science and technology in the 1970s is still small, yet there are already three distinct strands of work: studies of countercultural scientists, portrayed as enacting or advocating ‘groovy’ research; studies of the politically polarized debate pitting conservative and libertarian ‘cornucopianists’ against environmentalists and modelers forecasting resource scarcity; and studies of the early commercialization of technoscience (e.g., biotechnology) that took off in the 1980s. Left out, I argue, are a class of ‘square scientists’ with little sympathy for (...) the counterculture, and yet open to (even eager for) a new kind of science oriented to the same problems activists said they wanted science to solve: pollution, mass transit, housing, biomedicine, disability technologies, pedagogical machines, etc. Square scientists at places like NASA and Texas Instruments adapted military-industrial-academic templates to a wide variety of socially ‘relevant’ topics in the 1970s. Yet square scientists still looked to the military-industrial complex for allies, rather than to countercultural colleagues. This potential middle ground remained excluded – contributing, in large part, to the failure of schemes to reorient US R&D to civilian social problems, and to the invisibility of the squares in today's historical accounts. (shrink)

The intersection of a linearly ordered set of total subrelations of a total relation with range 2 need not be total, constructively.