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 (...) class='Hi'>will then argue that we should accept these startling conclusions, since CEM is valid. (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)

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 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)

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)

This article explores the connection between two theses: the principle of conditional excluded middle for the counterfactual conditional, and the claim that it is a contingent matter which (coarse grained) propositions there are. Both theses enjoy wide support, and have been defended at length by Robert Stalnaker. We will argue that, given plausible background assumptions, these two principles are incompatible, provided that conditional excluded middle is understood in a certain modalized way. We then show that (...) some (although not all) arguments for conditional excluded middle can in fact be extended to motivate this modalized version of the principle. (shrink)

There are essentially two ways to develop the Peircean idea that future contingents are all false. One is to provide a quantificational semantics for "will," as is usually done. The other is to define a quantificational postsemantics based on a linear semantics for "will." As we will suggest, the second option, although less conventional, is more plausible than the first in some crucial respects. The postsemantic approach overcomes three major troubles that have been raised in connection with (...) Peirceanism: the apparent scopelessness of "will" with respect to negation, the failure of Future Excluded Middle, and the so-called zero credence problem. (shrink)

Various philosophers have long since been attracted to the doctrine that future contingent propositions systematically fail to be true—what is sometimes called the doctrine of the open future. However, open futurists have always struggled to articulate how their view interacts with standard principles of classical logic—most notably, with the Law of Excluded Middle. For consider the following two claims: Trump will be impeached tomorrow; Trump will not be impeached tomorrow. According to the kind of open futurist (...) at issue, both of these claims may well fail to be true. According to many, however, the disjunction of these claims can be represented as p ∨ ~p—that is, as an instance of LEM. In this essay, however, I wish to defend the view that the disjunction these claims cannot be represented as an instance of p ∨ ~p. And this is for the following reason: the latter claim is not, in fact, the strict negation of the former. More particularly, there is an important semantic distinction between the strict negation of the first claim [~] and the latter claim. However, the viability of this approach has been denied by Thomason, and more recently by MacFarlane and Cariani and Santorio, the latter of whom call the denial of the given semantic distinction “scopelessness”. According to these authors, that is, will is “scopeless” with respect to negation; whereas there is perhaps a syntactic distinction between ‘~Will p’ and ‘Will ~p’, there is no corresponding semantic distinction. And if this is so, the approach in question fails. In this paper, then, I criticize the claim that will is “scopeless” with respect to negation. I argue that will is a so-called “neg-raising” predicate—and that, in this light, we can see that the requisite scope distinctions aren’t missing, but are simply being masked. The result: a under-appreciated solution to the problem of future contingents that sees and as contraries, not contradictories. (shrink)

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no (...) matter whether the proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

Short abstract: this is a reply to Schoubye and Rabern's 2017 paper, in Mind, to my own 2016 paper, also in Mind, "Future Contingents are All False! On Behalf of a Russellian Open Future." -/- Long abstract: There is a familiar philosophical position – sometimes called the doctrine of the open future ­– according to which future contingents (claims about underdetermined aspects of the future) systematically fail to be true. For well over 2000 years, however, open futurists have been accused (...) of denying certain logical laws – bivalence, excluded middle, or both – for entirely ad hoc reasons, most notably, that their denials are required for the preservation of something we hold dear. In a recent paper, however, I sought to argue that this deeply entrenched narrative ought to be overturned. My thought was this: given a popular, plausible approach to the semantics of future contingents, we can reduce the question of their status to the Russell/Strawson debate concerning presupposition failure, definite descriptions, and bivalence. In that case, we will see that open futurists in fact needn’t deny bivalence (Russell), or, if they do, they will do so for perfectly general (Strawsonian) reasons – reasons for which we all must deny bivalence. Of course, the metaphysical objections to the open futurist’s model of the future will remain just as they were. However, the millennia-old “semantic” or “logical” objections to the doctrine would be answered. In developing this approach, however, I came in fact to see the deep attractions of what I saw as the Russellian approach to these questions, and in my paper, I sought to articulate and display those attractions. The principal advantage of the view I wished to defend is that it is simple (no determinacy operators, no supervaluations, no third truth values) and it is classical. The view I developed, however, has recently come in for criticism in a paper by Anders Schoubye and Brian Rabern. The aim of this paper is to defend this view against their criticisms. The result, I believe, is an improved version of the theory, and a renewed defense of a radical claim: future contingents are systematically false. (shrink)

The classical response to Zeno’s paradoxes goes like this: ‘Motion cannot properly be defined within an instant. Only over a period’ (Vlastos.) I show that this ob-jection is exactly what it takes for Zeno to be right. If motion cannot be defined at an instant, even though the object is always moving at that instant, motion cannot be defined at all, for any longer period of time identical in content to that instant. The nonclassical response introduces discontinuity, to evade the (...) paradox of infinite proximity of any point of a distance with any ‘next’. But it introduces the wrong sort of discontinuity because, rather than assuming the discontinuity of motion, as Quantum Theory does, it assumes the discontinuity of space. Due then to the resulting spacetime disorder, though all else is certainly lost, the Tortoise now turns up at least as fast as Achilles and hence not even this much is rescued. Zeno rejects motion because he shows that a moving object must be where it is not. Hence motion, if to occur, must violate the Law of Contradiction (LNC). Applying the concept of quantum discontinuity, I produce an alternative. If an object is to move discontinuously between two boundary points, A and B, what actually obtains is, rather, that it is nowhere at all in-between A and B. And cannot therefore be at two places in-between A and B. And cannot therefore be where it is not. Thus, LNC is conserved. However, in these conditions, the Law of the Excluded Middle (LEM) fails. To mitigate the undesirability of this effect, I show that LEM fails because LNC holds. Thus, the resulting nonbivalent logic, which is also appropriate for quantized transitions of all kinds, will always turn up nonbivalent, because consistent. And this is not too bad, considering. (shrink)

Increasingly, connections are being made between neuroscience and education. At their interface is the attempt to ‘bridge the gap between conscious minds and living brains’. All too often, the two sides pursue a reductionist strategy of excluding the other. A middle way, promoted by Sankey in the context of values education, is emergentism: our conscious mental states are the product of brain processes but are not reducible to them. This paper outlines Sankey’s emergentist position and raises two objections: What (...) are emergent properties and What is the process of emergence? Sankey accepts neuroscience but rejects an eliminative materialist version on three grounds: materialism, reduction and determinism. All three are rebutted. His account of the neuronal, synaptic self is considered and found wanting. Finally, some positive remarks are made about the contribution eliminativism might make to values education. The conclusion reached is this: attempts to bridge the gap between mind and brain have not been successful and Sankey’s emergentism is no exception. This is one divide in education which will not be crossed. (shrink)

This book launches a sustained defense of a radical interpretation of the doctrine of the open future. Patrick Todd argues that all claims about undetermined aspects of the future are simply false.

In debates concerning the consequence argument, it has long been claimed that [McKay, T. J., and D. Johnson. 1996. “A Reconsideration of an Argument Against Compatibilism.” Philosophical Topics 24 (2): 113–122] demonstrated the invalidity of rule (β). Here, I argue that their result is not as robust as we might like to think. First, I argue that McKay and Johnson's counterexample is successful if one adopts a certain interpretation of ‘no choice about’ and if one is willing to deny the (...) conditional excluded middle principle. In order to make this point I demonstrate that (β) is valid on Stalnaker's theory of counterfactuals. This result is important and should not be neglected, I argue, because there is a particular line of objection to the revised formulations of the consequence argument that does not succeed against the original version. (shrink)

In De interpretatione 9, Aristotle argues against the fatalist view that if statements about future contingent singular events (e.g. ‘There will be a sea battle tomorrow,’ ‘There will not be a sea battle tomorrow’) are already true or false, then the events to which those statements refer will necessarily occur or necessarily not occur. Scholars have generally held that, to refute this argument, Aristotle allows that future contingent statements are exempt from either the principle of bivalence, or (...) the law of excluded middle. This paper offers a new interpretation of Aristotle’s refutation of fatalism. According to this interpretation, each member of a pair of contradictory future contingent statements, in virtue of expressing modal necessity, is simply false. (shrink)

In different papers, Carnielli, W. & Rodrigues, A., Carnielli, W. Coniglio, M. & Rodrigues, A. and Rodrigues & Carnielli, present two logics motivated by the idea of capturing contradictions as conflicting evidence. The first logic is called BLE and the second—that is a conservative extension of BLE—is named LETJ. Roughly, BLE and LETJ are two non-classical logics in which the Laws of Explosion and Excluded Middle are not admissible. LETJ is built on top of BLE. Moreover, LETJ is (...) a Logic of Formal Inconsistency. This means that there is an operator that, roughly speaking, identifies a formula as having classical behavior. Both systems are motivated by the idea that there are different conditions for accepting or rejecting a sentence of our natural language. So, there are some special introduction and elimination rules in the theory that are capturing different conditions of use. Rodrigues & Carnielli’s paper has an interesting and challenging idea. According to them, BLE and LETJ are incompatible with dialetheia. It seems to show that these paraconsistent logics cannot be interpreted using truth-conditions that allow true contradictions. In short, BLE and LETJ talk about conflicting evidence avoiding to talk about gluts. I am going to argue against this point of view. Basically, I will firstly offer a new interpretation of BLE and LETJ that is compatible with dialetheia. The background of my position is to reject the one canonical interpretation thesis: the idea according to which a logical system has one standard interpretation. Then, I will secondly show that there is no logical basis to fix that Rodrigues & Carnielli’s interpretation is the canonical way to establish the content of logical notions of BLE and LETJ. Furthermore, the system LETJ captures inside classical logic. Then, I am also going to use this technical result to offer some further doubts about the one canonical interpretation thesis. (shrink)

Todd (2016) proposes an analysis of future-directed sentences, in particular sentences of the form 'will(φ)', that is based on the classic Russellian analysis of definite descriptions. Todd's analysis is supposed to vindicate the claim that the future is metaphysically open while retaining a simple Ockhamist semantics of future contingents and the principles of classical logic, i.e. bivalence and the law of excluded middle. Consequently, an open futurist can straightforwardly retain classical logic without appeal to supervaluations, determinacy operators, (...) or any further controversial semantical or metaphysical complication. In this paper, we will show that this quasi-Russellian analysis of 'will' both lacks linguistic motivation and faces a variety of significant problems. In particular, we show that the standard arguments for Russell's treatment of definite descriptions fail to apply to statements of the form 'will(φ)'. (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)

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)

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.

My aim in this essay is largely defensive. I aim to discuss some problems for moral error theory and to offer plausible solutions. A full positive defense of moral error theory would require substantial investigations of rival metaethical views, but that is beyond the scope of this essay. I will, however, try to motivate moral error theory and to clarify its commitments. Moral error theorists typically accept two claims – one conceptual and one ontological – about moral facts. The (...) conceptual claim is that moral facts are or entail facts about categorical reasons (and correspondingly that moral claims are or entail claims about categorical reasons); the ontological claim is that there are no categorical reasons – and consequently no moral facts – in reality. I accept this version of moral error theory and I try to unpack what it amounts to in Section 2.1 In the course of doing so I consider two preliminary objections: that moral error theory is (probably) false because its implications are intuitively unacceptable (what I call the Moorean objection) and that the general motivation for moral error theory is self-undermining in that it rests on a hidden appeal to norms. The above characterization seems to entail the standard formulation of moral error theory, according to which first-order moral claims are uniformly false. Critics have argued that the standard formulation is incoherent since – by the law of excluded middle – the negation of a false claim is true. Hence if ‘Torture is wrong’ is false, ‘Torture is not wrong’ is true. Contrary to what moral error theorists contend, then, moral error theory seems to carry first-order moral implications that by the theory’s own lights are uniformly false. In Section 3 I suggest a formulation that is consistent with the standard formulation of moral error theory, free of first-order moral implications, and subject to no logical difficulties. In Section 4 I consider and rebut Stephen Finlay’s recent attack on moral error theory. According to Finlay the conceptual claim is false because all moral claims – and indeed all normative claims – are, or should be understood as, relativized to some moral standard or system of ends. Moral error theorists thus attribute to ordinary speakers an error that simply isn’t there. I argue that Finlay’s view has some very implausible implications and that it does not avoid commitment to various forms of error theory. This becomes especially clear when we focus on fundamental moral claims. In Section 5 I consider the worry that error theorists’ rejection of categorical reasons proves too much; in particular, the worry that error theorists’ qualms about categorical reasons apply equally to claims about hypothetical reasons, that is, claims to the effect that there is reason to take the means to one’s ends. In my view error theorists such as Mackie and Joyce have failed to pay due consideration to this problem. What the challenge establishes, I submit, is that error theorists cannot just take for granted that hypothetical reasons are metaphysically unproblematic; they must offer an account of hypothetical reasons that shows that they are. I argue that the only plausible account available to error theorists is one according to which claims about hypothetical reasons reduce to non-normative claims about relations between means and ends. (shrink)

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)

We are concerned with two modes of describing the dynamics of natural systems. Global descriptions require simultaneous global coordination of all dynamical operations. Global dynamics, including mechanics, remain invariant in the absence of external perturbation. But, failing impossible global coordination, dynamical operations could actually become coordinated only locally. In local records, as in global ones, the law of the excluded middle would be strictly observed, but without global coordination it could only be fullfilled sequentially by passing causative factors (...) forward onto subsequent contiguous operations.The local dynamics of sequential operations would be indefinite with regard to how commitments will be made which will avoid violating the law of the excluded middle, but any resulting record will be as definite as if there had been global coordination. While maintaining an agential capacity for making contingent choices internally, local dynamics could be cumulated into a global record of seemingly simultaneous operations. Natural selection within a framework of local dynamics would have a capacity for making opportunistic commitments, but its effects in a posterior record can be reduced to the mechanistic neodarwinian version as if there had been a global dynamics. However, the resulting global description falsifies the actual material nature of the dynamics. (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 idea of classical recapture has played a prominent role for non-classical logicians. In the specific case of non-classical theories of truth, although we know that it is not possible to retain classical logic for every statement involving the truth predicate, it is clear that for many such statements this is in principle feasible, and even desirable. What is not entirely obvious or well-known is how far this idea can be pushed. Can the non-classical theorist retain classical logic for every (...) non-paradoxical statement? If not, is she forced to settle for a very weak form of Classical Recapture, or are there robust versions of classical recapture available to her? These are the main questions that I will address in this paper. As a test case I will consider a paracomplete account of the truth-theoretic paradoxes and I will argue for two claims. First, that it is not possible to retain the law of excluded middle for every non-paradoxical statement. Secondly, that there are no robust versions of classical recapture available to the paracomplete logician. (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)

The principles that AN INSTANCE OF A CLASS IS THE CLASS and A CLASS IS AN INSTANCE OF ITSELF allow for the so called LAWS OF THOUGHTIDENTITY - WHAT IS, IS.CONTRADICTION - NOTHING BOTH IS and IS NOT.EXCLUDED MIDDLE - EVERYTHING IS or IS NOT.and allow us to adopt a bivalent system. Everything essential for primary logic is provided.Though this is not the place to discuss it, it should be noted that the development of general logic with its (...) current theories of class, identity and natural numbers could take a very different course from the one it currently pursues.Human thought would have had to evolve to a high level of development before any systematic attempt to analyse it could have been made. It seems it was in the floruit of the Greek civilisation that this attempt was first made. The issues in this paper were of prime concern to those ancient thinkers and their achievements were considerable. Since those times further developments have occurred, many of which they would not have dreamed, but to this day there are still great issues that have not been resolved. The state of the art leaves much to be desired and it is my hope that this paper will contribute to its progress.If the ideas in this paper are true, then it seems we have come across the very origins of human rationality. (shrink)

The paper consists of two parts. The first part begins with the problem of whether the original three-valued calculus, invented by J. Łukasiewicz, really conforms to his philosophical and semantic intuitions. I claim that one of the basic semantic assumptions underlying Łukasiewicz's three-valued logic should be that if under any possible circumstances a sentence of the form "X will be the case at time t" is true (resp. false) at time t, then this sentence must be already true (resp. (...) false) at present. However, it is easy to see that this principle is violated in Lukasiewicz's original calculus (as the cases of the law of excluded middle and the law of contradiction show). Nevertheless it is possible to construct (either with the help of the notion of "supervaluation", or purely algebraically) a different three-valued, semi-classical sentential calculus, which would properly incorporate Łukasiewicz's initial intuitions. Algebraically, this calculus has the ordinary Boolean structure, and therefore it retains all classically valid formulas. Yet because possible valuations are no longer represented by ultrafilters, but by filters (not necessarily maximal), the new calculus displays certain non-classical metalogical features (like, for example, nonextensionality and the lack of the metalogical rule enabling one to derive "p is true or q is true" from" 'p ∨ q' is true"). The second part analyses whether the proposed calculus could be useful in formalizing inferences in situations, when for some reason (epistemological or ontological) our knowledge of certain facts is subject to limitation. Special attention should be paid to the possibility of employing this calculus to the case of quantum mechanics. I am going to compare it with standard non-Boolean quantum logic (in the Jauch-Piron approach), and to show that certain shortcomings of the latter can be avoided in the former. For example, I will argue that in order to properly account for quantum features of microphysics, we do not need to drop the law of distributivity. Also the idea of "reading off" the logical structure of propositions from the structure of Hilbert space leads to some conceptual troubles, which I am going to point out. The thesis of the paper is that all we need to speak about quantum reality can be acquired by dropping the principle of bivalence and extensionality, while accepting all classically valid formulas. (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.

We start by noting that the set-theoretic and semantic paradoxes are framed in terms of a definition or series of definitions. In the process of deriving paradoxes, these definitions are logically represented by a logical equivalence. We will firstly examine the role and usage of definitions in the derivation of paradoxes, both set-theoretic and semantic. We will see that this examination is important in determining how the paradoxes were created in the first place and indeed how they are (...) to be solved in a uniform way. There are three features that are special about these definitions used in the derivation of most of the above paradoxes. The first is the use of self-reference between the definiens and the definiendum, the second is the generality of the definiendum, and the third is the under-determination and over-determination of concepts that usually occur as a result of these definitions. We will examine the impact of these three features on the logical representation of definitions and show how this representation then leads to a uniform paradox solution using an appropriate logic that is both paraconsistent and paracomplete. However, it is the paracompleteness, exhibited through the rejection of the Law of Excluded Middle, together with the rejection of contraction principles, that enables the solution of the paradoxes to go through. We characterize definitions as involving syntactic identity and/or meaning identity. We point out that some paradoxes do not have explicit self-reference or circularity and some may not utilize the generality of the definiendum, but the general characterizations of definitions that we give will still apply. We also look beyond all this to paradoxes that rely on illicit definitions between objects that are essentially different to start with. (shrink)

Here we shall employ the term "classical" to denote situations in science in which all propositions satisfy all the laws of classical logic, including that of excluded middle. This means that if x is a proposition of a given field of science, then the following statements will be true in regard to it: "either x or non-x," "either x is true or non-x is true," "x is either true or false." The emphasis here upon the law of (...) excluded middle is associated with the fact that criticism of this law is the basic content of nonclassical criticism of classical logic. (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 lieu of an abstract, here is a brief excerpt of the content:RUSSELL TO FREGE, 24 MAY 1903: "I BELIEVE I HAVE DISCOVERED THAT CLASSES ARE ENTIRELY SUPERFLUOUS" GREGORY LANDINI Philosophy / University of Iowa Iowa City, IA 52242, USA It was his consideration of Cantor's proof that there is no greatest cardinal, Russell recalls in My Philosophical Development, that led in the spring of 1901 to the discovery of the paradox of the class of all classes not members of (...) themselves. "Never glad confident morning again", were Whitehead's reported words (MPD, p. 58). Whitehead was, of course, wrong. Russell had many new confident mornings. One in particular apparently occurred on 19 May 1903. On 23 May Russell wrote in his journal: "Four days ago 1 solved the Contradiction -the relief of this is unspeakable" (Papm 12: 24). The "solution" was communicated to Whitehead, who responded by telegram: "Heartiest congratulations Aristoteles secundus. 1 am delighted" (Clark, p. III). But it was not long before Russell knew his proposal was inadequate. On the form he would scrawl: "A propos of solVing the Contradiction. [But the solution was wrong]" (ibid; Russell's brackets). The episode is intriguing. What was the proposal? Russell's correspondence with Frege sheds some light. On 20 October 1902 Frege had sent a letter suggesting a way to avoid the contradiction. Frege's suggestion, formulated in a great hurry so that it might appear in the Appendix of Vol. II of the Grundgesetze der Arithmetik, was to abandon his Basic Law V; zcj>z =zez. ==. (x)(x. ==. ex), and to replace it by Russell to Frege, 24 May I903 161 V' zz =zez. ==. (x)(x *zz & x* zez: ~ : x. ==. ex). The underlying idea was to accept that one concept may have the same extension as another even though the two concepts are not coextensive. Though he thought the underlying idea "probably correct",! Russell expressed reservation in a reply of 12 December:... I find it difficult to accept your solution even though it is probably correct. Do you deny, e.g., that all classes form a class? And if this is admitted, then it is possible that - ana. Moreover, the class of non-humans is a. non-human. Otherwise it must be admitted that not ail objects fall either under a or not-a; namely, if a is a range of values, then a fails neither under a nor under not-a. This contradicts the law of excluded middle, which will be inconvenient to say the least.2 (Frege I980, p. 151) Frege's response of 28 December does not mention the contradiction, and Volume II of his Grundgesetze appeared early in 1903. Russell's reservations concerning Frege's solution· continue in a letter to Frege of 20 February 1903: What you say about my contradiction is of the greatest interest to me. Do you believe that the range of values remains unchanged if some subclass of the class is assigned to it as a new member? Extension seems to fit this view better than intension. Bur I fe;el far from clear about this question. (Frege I980, p. 155) Frege replied on 21 May, explaining that in general a class does not remain unchanged when a particular subclass is added to it, but that two concepts may have the same extension (the same class) when the only difference between them is that this class falls under the first concept but not under the second (p. 157). Then on 24 May 1903 Russell excitedly wrote Frege of a solution of the paradox of classes: "I received your letter this morning, and 1 am replying to it at once, for russell: the Journal of the Berrrand Russdl Archives McMasrer University Library Press n.S.•2 (wincer 1992-93): 160-85 ISSN 0036-01631 • The same evaluation appears in Russell's Appendix to The Principles ofMathematics (p. 522). 2 Frege's '\- a II b" is Peano and Russell's "a E b". Russell should have put "~ana". 162 GREGORY LANDINI I believe I have discovered that classes are entirely superfluous. Your designation i(z) can be used for itself, and x E i(z) for (x).... this seems to me to avoid the contradiction" (p. 159). The proposal is also mentioned in a... (shrink)

This paper and its sequel “look under the hood” of the usual sorts of proof-theoretic systems for certain well-known intuitionistic and classical propositional modal logics. Section 1 is preliminary. Of most importance: a marked formula will be the result of prefixing a formula in a propositional modal language with a step-marker, for this paper either 0 or 1. Think of 1 as indicating the taking of “one step away from 0.” Deductions will be constructed using marked formulas. Section (...) 2 presents the model-theoretic concepts, based on those in [7], that guide the rest of this paper. Section 3 presents Natural Deduction systems IK and CK, formalizations of intuitionistic and classical one-step versions of K. In these systems, occurrences of step-markers allow deductions to display deductive structure that is covered over in familiar “no step” proof-theoretic systems for such logics. Box and Diamond are governed by Introduction and Elimination rules; the familiar K rule and Necessitation are derived (i.e. admissible) rules. CK will be the result of adding the 0-version of the Rule of Excluded Middle to the rules which generate IK. Note: IK is the result of merely dropping that rule from those generating CK, without addition of further rules or axioms (as was needed in [7]). These proof-theoretic systems yield intuitionistic and classical consequence relations by the obvious definition. Section 4 provides some examples of what can be deduced in IK. Section 5 defines some proof-theoretic concepts that are used in Section 6 to prove the soundness of the consequence relation for IK (relative to the class of models defined in Section 2.) Section 7 proves its completeness (relative to that class). Section 8 extends these results to the consequence relation for CK. (Looking ahead: Part 2 will investigate one-step proof-theoretic systems formalizing intuitionistic and classical one-step versions of some familiar logics stronger than K.). (shrink)

The principle of identity is one of the three basic principles of reason on which classical logic rests. The other two basic principles of reason are principle of noncontradiction and of the excluded middle. These principles were accepted almost unquestionably until the beginning of the twentieth century. Therefore, classical logic, which adhered to these principles, continued to prevail. Today, however, one of the most interesting debates in the field of science is whether these fundamental principles on which classical (...) logic rests can be seen as necessary and invariable truths. With this discussion, a logic emerged in the twentieth century, rejecting the basic principles of reason. In this study, we will examine the logics -non-reflexive logics- that rejects or limits the principle of identity, from which the other two principles of classical logic are derived. First of all, we will talk about the origins of non-reflexive logic. Next, the basic assumptions of the identity principle of classical logic will be discussed. Finally, we will discuss the reasons for which non-reflexive logic rejects or limits the principle of identity, in other words, which assumptions of the identity principle are considered problematic. (shrink)

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)

Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in his strict finitistic metatheory, of equations between numerals on which concrete arithmetical sentences are evaluated. As a first step towards classical formalisation of strict finitism, we propose their counterparts in the classical metatheory with one additional assumption, and then extract the propositional part of ‘strict finitistic (...) logic’ from it and investigate. We will provide a sound and complete pair of a Kripke-style semantics and a sequent calculus, and compare with other logics. The logic lacks the law of excluded middle and Modus Ponens and is weaker than classical logic, but stronger than any proper intermediate logics in terms of theoremhood. In fact, all the other well-known classical theorems are found to be theorems. Finally, we will make an observation that models of this semantics can be seen as nodes of an intuitionistic model. (shrink)

It will be argued that Wittgenstein did not outright reject the law of excluded middle for mathematics or the proof-techniques that constructivists reject in connection with the law of excluded middle. Wittgenstein can be seen to be critical of the dogmatic claims of Brouwer and Weyl concerning how proofs should be constructed. Rather than himself laying down a requirement concerning what is and is not a proof, Wittgenstein can be read as exploring the differences between (...) constructive and non-constructive proofs. I will read Wittgenstein as arguing that Brouwer’s rejection of excluded middle is based upon an over-extension from examples that Wittgenstein reads as special cases. Wittgenstein is certainly interested in the difference between constructive and non-constructive proof but only for the purposes of exploring the easily-missed differences between the two, not in order to reject non-constructive approaches. (shrink)

Educational philosophy in English-speaking countries tends to be informed mainly by analytic philosophy common to Western thinking. A welcome alternative is provided by pragmatism in the tradition of Peirce, James and Dewey. Still, the habit of the so-called linguistic turn has a firm grip in terms of analytic philosophy based on the logic of non-contradiction as the excluded middle. A body∼mind approach pertains to the edusemiotic turn that this article elucidates. Importantly, semiotics is not illogical but is informed (...) by the paradoxical logic of the included middle. The process of reasoning is however indirect or mediated; it involves active interpretation (in a variety of forms) versus direct representation; it is analogical and connects what are otherwise doomed to remain isolated substances of body versus mind with a separation of knowledge and action. Analysing and synthesising the philosophies of Charles Sanders Peirce and Gilles Deleuze, together with a brief excursion into the cutting-edge science of coordination dynamics, this article will demonstrate how the body∼mind assemblage is created in practice, and what may be the implications of such a stance for educational philosophy and pedagogical practice. (shrink)

The aim of this paper is to present and analyze the truthmaker monism , a result according to which any truthmaker makes true any true proposition. Truthmaker monism depends on the characterization of the truthmaking relation in terms of strict implication, the principle of excluded middle and the so called disjunction thesis. I will restrict the discussion to a theory of truthmakers of empirical truths and I will argue that, in the scope of such a theory, (...) the problem does not lie in the use of strict implication neither in the excluded middle but, rather, in the disjunction thesis, that must be rejected. (shrink)

Emily Elizabeth Constance Jones was an English logician and contemporary of Bertrand Russell, as well as Mistress of Girton College, Cambridge. In this book, originally published in 1911, she argues for the existence of another fundamental law of thought to join the Law of Contradiction and the Law of Excluded Middle: the Law of Significant Assertion. This book will be of value to anyone with an interest in logic or in Jones' work.

Resumen La pretensión de que la violencia es un fenómeno apto para el abordaje objetivo es altamente cuestionable. En este artículo se indicarán algunos aspectos que subyacen en los enfoques más clásicos sobre tal tópico y se destacará el potencial violentogénico que encapsulan. El núcleo de las ideas expuestas apunta a plantear que la epistemología objetivista induce a una violencia simbólica enquistada en el principio del tercero excluido. En consecuencia, los esfuerzos por convertir a la violencia en un tema de (...) investigación objetiva terminan transformándose en una flagrante retroalimentación de la misma violencia que se proponen observar. Ahora bien, en contraste con esa anquilosada postura objetivista, se explorará un enfoque epistemológico de corte socio-construccionista que complejiza el abordaje de la violencia al enmarcarla dentro de una "investigación de segundo orden".The claim that violence is a phenomenon suitable for the objective approach is highly questionable. In this article, some aspects underlying the more classical approaches about such topic will be indicated, and their intrinsic violent potential will be emphasized. The core of the ideas that aims to raise the objectivist epistemology leads to a symbolic violence entrenched in the principle of the excluded middle. Consequently, efforts to make violence a topic of objective research end up becoming a flagrant feedback of the same violence they intend to observe. However, in contrast to the ossified objectivist position, a socio-constructionist epistemological approach will be explored. It focuses on violence as a complex reality, by framing it within a "Second-Order Research". (shrink)

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, (...) contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. One rather conspicuous example is that the process of refining logical terminology has been productive. Future logic students will no longer be burdened by obscure terminology and they will be able to read, think, talk, and write about logic in a more careful and more rewarding manner. Closely related is increased use and study of variable-enhanced natural language as in “Every proposition x that implies some proposition y that is false also implies some proposition z that is true”. Another welcome development is the culmination of the slow demise of logicism. No longer is the teacher blocked from using examples from arithmetic and algebra fearing that the students had been indoctrinated into thinking that every mathematical truth was a tautology and that every mathematical falsehood was a contradiction. A fifth welcome development is the separation of laws of logic from so-called logical truths, i.e., tautologies. Now we can teach the logical independence of the laws of excluded middle and non-contradiction without fear that students had been indoctrinated into thinking that every logical law was a tautology and that every falsehood of logic was a contradiction. This separation permits the logic teacher to apply logic in the clarification of laws of logic. This lecture expands the above points, which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

Those inquiring into the nature of mind have long been interested in the foundations of mathematics, and conversely this branch of knowledge is distinctive in that our access to it is purely through thought. A better understanding of mathematical thought should clarify the conceptual foundations of mathematics, and a deeper grasp of the latter should in turn illuminate the powers of mind through which mathematics is made available to us. The link between conceptions of mind and of mathematics has been (...) a central theme running through the great competing philosophies of mathematics of the twentieth century, though each has refashioned the connection and its import in distinctive ways. The present collection will be of interest to students of both mathematics and of mind. Contents include: "Introduction" by Alexander George; "What is Mathematics About?" by Michael Dummett; "The Advantages of Honest Toil over Theft" by George Boolos; "The Law of Excluded Middle and the Axiom of Choice" by W.W. Tait; "Mechanical Procedures and Mathematical Experience" by Wilfried Sieg; "Mathematical Intuition and Objectivity" by Daniel Isaacson; "Intuition and Number" by Charles Parsons; and "Hilbert's Axiomatic Method and the Laws of Thought" by Michael Hallett. (shrink)

1. Logic, determinism and free will. The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological and logical character; my concern here is to limit attention to two arguments from logic. To begin with, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded (...) Middle, according to which every proposition is either true or false, no matter whether the proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. Surely no such argument could really establish determinism, but one is hard pressed to explain where it goes wrong. One now classic dismantling of it has been given by Gilbert Ryle, in the chapter ‘What was to be’ of his fine book, Dilemmas (Ryle 1954). We leave it to the interested reader to pursue that and the subsequent literature. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:FATALISM AND TRUTH ABOUT THE FUTURE }AMES w. FELT, S.J. Santa Clara University Santa Clara, California WHEN WE SPEAK of future events, does today's ruth mean tomorrow's necessity? The question is as old as Aristotle's sea battle tomorrow. The last ships should have been sunk long ago, but after two thousand years the textual analysis of this passage is still controverted. Yet I think something new can be said (...) about it if we consider afresh the philosophic issues themselves. What philosophic consequences must we accept if we suppose that predictions, that is, propositions referring to future events, are either true or false antecedently to the events themselves? In particular, if a prediction be true now, does its present truth imply a fixity inherent in the future such that fatalism is unavoidable? 1 It has been argued that it does. Aristotle sketched such an argument in the sea battle passage already alluded to:... [If anything] is white now it was true to say earlier that it would be white; so that it was always true to say of anything that has happened that it would be so. But if it was always true to say that it was so, or would be so, it could not not be so, or not be going to be so. But if something cannot not happen it is impossible for it not to happen; and if it is impossible for something not to happen it is necessary for it to happen. Everything that will be, therefore, happens necessarily. So nothing will come about as chance has it or by chance; for if by chance, not of necessity.2 1 Whether it makes any sense to speak of a proposition as true " now " or at any other time will be considered below. 2 De Interpretatione (trans. J. L. Ackrill; Oxford University Press: 1963), ch. 9, 18b9 ff. 209 210 JAMES W. FELT, S.J. If, then, a prediction is true in the present or false in the present, its very truth value today seems to create an ineluctable fixity upon tomorrow's event such that fatalism would be unavoidable. And as is implied in Aristotle's example, it seems natural to suppose that a prediction must, after all, be either true or, if not true, then false. Does not the law of excluded middle demand this? Thus the logic of truth relations seems to impose a fatalistic view of events. Indeed, fatalism is sometimes defined precisely in terms of logic, so that a contemporary author writes : " Fatalism is the thesis that the laws of logic alone suffice to prove that no man has free will, suffice to prove that the only actions which a man can perform are the actions which he does, in fact, perform, and suffice to prove that a man can bring about only those events which do, in fact, occur and can prevent only those events which do not, in fact, occur." 3 I shall, however, argue that ( 1) neither the law of excluded middle nor any other logical consideration requires that predictions be true or else false when they are asserted; (2) the antecedent truth (or falsity) of predictions would not necessitate fatalism by reason of any logical considerations, (3) though it could necessitate fatalism for causal reasons; and (4) predictions are never, absolutely speaking, true or false before the occurrence of the events to which they refer, though they may be true or false in an attenuated, relative sense. Thesis 1: Neither the law of excluded middle nor any other logical consideration requires that a prediction be, prior to the event, true or, if not true, then false. I understand the law of excluded middle (LEM) to mean that, for any meaningful proposition p, it is (logically) necessary that p be true or, if not true, then false. By ordinary usage a proposition referring to a state of affairs in the world, as distinguished from one referring to logical re3 Steven M. Cahn. Fate, Logic, and Time (New Haven: Yale University Press, 1967), p. 8. FATALISM AND TRUTH ABOUT THE FUTURE 211 lationships, is called ' true' if and only if that state of affairs obtains. The... (shrink)

then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be (...) constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory3, the following.. (shrink)

The classical response to Zeno’s paradoxes goes like this: ‘Motion cannot properly be defined within an instant. Only over a period’ (Vlastos.) I show that this ob-jection is exactly what it takes for Zeno to be right. If motion cannot be defined at an instant, even though the object is always moving at that instant, motion cannot be defined at all, for any longer period of time identical in content to that instant. The nonclassical response introduces discontinuity, to evade the (...) paradox of infinite proximity of any point of a distance with any ‘next’. But it introduces the wrong sort of discontinuity because, rather than assuming the discontinuity of motion, as Quantum Theory does, it assumes the discontinuity of space. Due then to the resulting spacetime disorder, though all else is certainly lost, the Tortoise now turns up at least as fast as Achilles and hence not even this much is rescued. Zeno rejects motion because he shows that a moving object must be where it is not. Hence motion, if to occur, must violate the Law of Contradiction (LNC). Applying the concept of quantum discontinuity, I produce an alternative. If an object is to move discontinuously between two boundary points, A and B, what actually obtains is, rather, that it is nowhere at all in-between A and B. And cannot therefore be at two places in-between A and B. And cannot therefore be where it is not. Thus, LNC is conserved. However, in these conditions, the Law of the Excluded Middle (LEM) fails. To mitigate the undesirability of this effect, I show that LEM fails because LNC holds. Thus, the resulting nonbivalent logic, which is also appropriate for quantized transitions of all kinds, will always turn up nonbivalent, because consistent. And this is not too bad, considering. (shrink)