Paul of Venice’s Disjunction Theses

Jonathan Lowe has argued that a particular variation on C.I. Lewis’ notion of strict implication avoids the paradoxes of strict implication. In this paper it is argued that Lowe’s notion of implication does not achieve this aim. More over, a general argument is given to the effect that no other variation on Lewis’ notion of constantly strict implication describes the logical behavior of natur al language conditionals in a satisfactory way. 1 Indicative conditionals and strict implication In reaction to Russell’s interpretation of conditional sen tences as classical material implications, C.I. Lewis suggested that it would be more appropriate to interpret natural language conditionals as strict implications. He claimed t hat the truth-conditions of a sentence of the form ‘if p then q’ are given by the sentence ‘Necessarily, not p or q’ Lewis (1912). Russell and Lewis were speaking at cross purposes. Russell was mainly interested in the logical meaning of conditional expressions in the restricted context of mathematical proofs, whereas C.I. Lewis wanted to express the logical meaning of indicative conditionals in natural language in general. And it is true that in mathematical proofs, the meaning of conditional assertions can be taken to be expressed by the corresponding material implications. But the so-called paradoxes of material implicationto which C.I. Lewis drew Russell’s attention do show that material im plications do not capture the truth-conditions of conditional expressions as th ey are generally used in daily speech. Attempts have been made to relegate the paradoxes of material implication to pragmatics, by classifying the paradoxes of material implication as true assertions which violate some of the Gricean conversational implicatures. But the attempts that are made so far are generally regarded as unsatisfactory. 1

Lewis, in the presentation of his calculus of strict implication, contends that this calculus accords with the usual meaning of “implies” such that “p strictly implies q” is synonymous with “q is deducible from p” (pp. 126–127, 235–262). It is the purpose of this paper to present, in 1, certain considerations in the light of which this statement does not hold, and in 2, a new implicative relation upon which a calculus of propositions can be based such that it will accord with the relation of deducibility and from which it is possible to derive the calculi of strict and material implications.1. Strict implication regarded as synonymous with deducibility. In the presentation of his calculus of strict implication Lewis observes that despite inclusion of both intensional and extensional propositions in this calculus, the meaning of its elements, p, q, r, etc., “remain fixed, whether it is their extensional or their intensional relations which are in question” (p. 120). But if the meaning or content of the elements of this calculus is irrelevant to the logical truth which it contains, then, as Weiss has already remarked, the propositions of this calculus “are really extensionally treated, involving multiple values,” e.g., possibility, on the analogy of truth and falsity. The importance of this fact is that, in consequence, the relation of strict implication is subject to certain paradoxes, such as (19.74) “a proposition which is impossible strictly implies any proposition” or (19.75) “a proposition which is necessarily true is strictly implied by any proposition” (p. 174) or (19.84) “any two necessarily true propositions are strictly equivalent” (p. 176).

In contemporary philosophy of language, mind and action, propositions are not only Senses of sentences with truth conditions but also contents of conceptual thoughts like illocutionary acts and attitudes that human agents perform and express. It is quite clear that propositions with the same truth conditions are not the senses of the same sentences, just as they are not the contents of the same thoughts. To account for that fact, the logic of propositions according to predication advocates finer criteria of propositional identity than logical equivalence and requires of competent speakers less than perfect rationality. Unlike classical logic it analyzes the structure of constituents of propositions. The logic is predicative in the very general sense that it analyzes the type of propositions by mainly taking into consideration the acts of predication that we make in expressing and understanding them. Predicative logic distinguishes strictly equivalent propositions whose expression requires different acts of predication or whose truth conditions are understood in different ways. It also explicates a new relation of strong implication between propositions much finer than strict implication and important for the analysis of psychological and illocutionary commitments. The main purpose of this work is to present and enrich the logic of propositions according to predication by analyzing elementary propositions that predicate all kinds of attributes (extensional or not) as well as modal propositions according to which it is necessary, possible or contingent that things are so and so. I will first explain how predicative logic analyzes the structure of constituents and truth conditions of propositions expressible in the modal predicate calculus without quantifiers. The ideal object language of my logic is a natural extension of that of the minimal logic of propositions. Next I will define the structure of a model and I will formulate an axiomatic system. At the end I will enumerate important valid laws. The present work on propositional logic is part of my next book Propositions, Truth and Thought which formulates a more general logic of propositions according to predication analyzing also generalization, ramified time, historic modalities as well as action and attitudes.

Although modern modal logic came about largely after Peirce's death, he anticipated some of its key aspects, including strict implication and possible worlds semantics. He developed the Gamma part of Existential Graphs with broken cuts signifying possible falsity, but later identified the need for a Delta part without ever spelling out exactly what he had in mind. An entry in his personal Logic Notebook is a plausible candidate, with heavy lines representing possible states of things where propositions denoted by attached letters would be true, rather than individual subjects to which predicates denoted by attached names are attributed as in the Beta part. New transformation rules implement various commonly employed formal systems of modal logic, which are readily interpreted by defining a possible world as one in which all the relevant laws for the actual world are facts, each world being partially but accurately and adequately described by a closed and consistent model set of propositions. In accordance with pragmaticism, the relevant laws for the actual world are represented as strict implications with real possibilities as their antecedents and conditional necessities as their consequents, corresponding to material implications in every possible world.

On the standard interpretation of Lewis’s criticism of Russell, Lewis takes his own account of strict implication to be more intuitive than the paradoxes to which Russell’s material implication leads. This chapter argues against the standard interpretation by showing that Russell’s views are not counterintuitive, but that appeals to “intuition” are not the substance of Lewis’s criticism of Russell. Lewis’s debate with Russell undergoes two main phases. In the first phase, Lewis argues both that the logic of material implication fails to represent correctly the distinction between correct and incorrect inferences and that the logic of material implication is not useful. In the second phase, Lewis argues that Russell’s explication of implication neither coheres fully with Russell’s deductive practices nor can Russell avoid appealing to strict implications. A better understanding of the Russell-Lewis dispute points to a conception of the nature of logic: it must be possible for logical principles to play a practical role in reasoning, else logic becomes disconnected from rational thought and discourse.

This paper introduces the logics of super-strict implications, where a super-strict implication is a strengthening of C.I. Lewis' strict implication that avoids not only the paradoxes of material implication but also those of strict implication. The semantics of super-strict implications is obtained by strengthening the (normal) relational semantics for strict implication. We consider all logics of super-strict implications that are based on relational frames for modal logics in the modal cube. it is shown that all logics of super-strict implications are connexive logics in that they validate Aristotle's Theses and (weak) Boethius's Theses. A proof-theoretic characterisation of logics of super-strict implications is given by means of G3-style labelled calculi, and it is proved that the structural rules of inference are admissible in these calculi. It is also shown that validity in the S5-based logic of super-strict implications is equivalent to validity in G. Priest's negation-as-cancellation-based logic. Hence, we also give a cut-free calculus for Priest's logic.

Lewis has made an attempt to construct a logistic system containing an implication-relation ⊰ in such a way that p⊰q shall have the meaning: q is deducible from p. Lewis and Langford admit that “the one serious doubt which can arise concerning the equivalence of p⊰q to the relation of deducibility … arises from the fact that strict implication has its corresponding paradoxes:19.74 ˜⋄ p ⊰ · p ⊰ q,‘If p is impossible, then p strictly implies any proposition q’; and19.75 ˜⋄˜p ⊰ · q ⊰ p,‘If p is necessary, then any proposition q strictly implies p.’”Indeed, it is not obvious that 19.74 and 19.75 should hold. It is true that in many cases 19.75 (as well as 19.74) is valid; for in the system of Lewis and Langfordp · ˜p · ⊰ qis provable, and the conformity of this law with real deduction is shown by them. In all cases in which a proposition of the form p·∼p can be derived from an impossible proposition, 19.75 holds. But it is not obvious that there are no other kinds of impossible propositions.Fortunately it is easy to see what is the origin of the paradoxes. They are introduced into the system by the definition11.02 p ⊰ q · = ˜ ⋄(p · ˜q).Assume this definition. Then if q is necessary, p·∼q is impossible, and so, in accordance with 11.02, p⊰q is valid.

In this paper I undermine the Entailment Principle according to which if an entity is a truthmaker for a certain proposition and this proposition entails another, then the entity in question is a truthmaker for the latter proposition. I argue that the two most promising versions of the principle entail the popular but false Conjunction Thesis, namely that a truthmaker for a conjunction is a truthmaker for its conjuncts. One promising version of the principle understands entailment as strict implication but restricts the field of application of the principle to purely contingent truths (i.e. those that contain no necessary proposition at any level of analysis). But a conjunction of purely contingent truths strictly implies its conjuncts. So this version of the principle is committed to the Conjunction Thesis. The same is true of the version of the principle where entailment is understood in the sense of systems T, R, and E of relevant logic, since in these systems conjunctions entail their conjuncts. I argue that the Conjunction Thesis is false because a truthmaker is that in virtue of what a certain proposition is true and it is false that, for example, what the proposition that Peter is a man is true in virtue of is the conjunctive fact that Peter is man and Saturn is a planet (or the facts that Peter is a man and that Saturn is a planet taken together). I also argue against other versions of the principle.

In this paper, I argue against the view there are contingent a priori truths, and against the related view that there are contingent logical truths. I will suggest that in general, predicates ›a priori‹ and ›contingent‹ are implicitly relativized to circumstances, and argue that apriority entails necessity, whenever the two are relativized to the same circumstance. I will then criticize the idea, inspired by David Kaplan's framework, of contingent contents »knowable under a priori characters.« I will also argue, against Kaplan, that sentences of the form »The actual F is F« do not deserve the status of logical truths, since what they express is neither necessary nor a priori (pace Kaplan).

Previous articleNext article FreeBook ReviewLaurie Shannon The Accommodated Animal: Cosmopolity in Shakespearean Locales The Accommodated Animal: Cosmopolity in Shakespearean Locales. Laurie Shannon. Chicago: University of Chicago Press, 2013. Pp. vii+290.Rhodri LewisRhodri LewisUniversity of Oxford Search for more articles by this author PDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinked InRedditEmailQR Code SectionsMoreI have no doubt that The Accommodated Animal will win an appreciative audience. In it, Laurie Shannon strongly disputes the Cartesian notion of the bête machine; in so doing, she adduces a number of texts produced (for the most part) in the century or so before Descartes wrote and seeks to expose the crudity, injustice, and intellectual poverty that inhere in Descartes’s denigration of animal life. If Descartes is the villain of the piece, then its hero is Montaigne, whose willingness to accept the permeability of the dividing lines between human and animal existence invests nonhuman beings with dignity, agency, and ontological significance. This terrain has been familiar since the publication of George Boas’s classic study of The Happy Beast in French Thought of the Seventeenth Century (1933). But in Shannon’s estimation, Boas “circumscribe[s] the entire logic of the happy beast within an imperturbably human-exceptionalist domain” (136); he is unduly determined to fit zoophilic discourse within the confines of a conventionally defined seventeenth-century literary and intellectual worldview. By contrast, Shannon seeks to unpack the implicit logic of early modern zoophilia and to reconstruct the status of animals as active participants in pre-Cartesian political culture—or what she elaborates as the “constitutionalist sense of legitimated capacities, authorities, and rights that set animals within the scope of justice and the span of political imagination” (3). Shannon conceives of this state as embodying the “cosmopolity” of her subtitle and proposes that “The Accommodated Animal tracks a particular tradition that accommodates the presence of animals and conceives them as actors and stakeholders endowed by their creator with certain subjective interests” (18).Since the advent of the New Historicism nearly forty years ago, students of English literature have grown used to assertions of the special congruity between the critical present and the sixteenth and seventeenth centuries. The Accommodated Animal is another case in point. Quite aside from its numerous intersections with the burgeoning subfield of early modern ecocriticism (in particular the work of Bruce Boehrer and Erica Fudge, generously acknowledged throughout), its arguments are directly addressed to contemporary discussions of animal welfare: for instance, Thomas More’s Utopia (1516, cited as 1517 by Shannon) is a “prescient intimation” of the brutalities of industrial farming in present-day North America (23–24). For anyone who follows the leads provided by Shannon in her footnotes (or with the will and inclination to search the internet for themselves), it is hard not to endorse the tenor and assumptions of her argument. Further, her argument is in itself advanced with intelligence, verve, and discursive passion. It is thus unfortunate that as an exercise in literary, cultural, or intellectual history—and as a piece of literary criticism—The Accommodated Animal too often falls short of the mark.Some of the responsibility for this belongs to Descartes. In seeking to repudiate so thoroughly his ill-grounded but influential theorizing, Shannon has projected its mirror image onto the cultural world that preceded it. On the one hand, doing so flattens out and misrepresents the nature of human-animal relations in the century or so before Descartes; on the other, it fails fully to account for the emergence of Descartes’s ideas as a reaction to the received early seventeenth-century wisdom. Contra Descartes, biological and religious comparativism had been a staple of accounts of the natural world and of the place of humankind within it, since ancient Athens at the latest. (Think of the scala naturae, or chain of being.) The rub is that the comparability of human beings and elephants, dogs, cats, parrots, apes, ants, or bees does not connote their identity. Nor does such comparability mean that animals are praised for attributes, whether physical or mental, that are anything other than anthropoid: elephants, like humankind, are social and have excellent memories; parrots, like humankind, have well-developed vocal organs; bees, like humankind, live in complex hierarchical societies. On the standard model of animate life derived from Aristotle, elaborated by Galen, and comprehensively baptized over the course of the Middle Ages, all living beings had vegetative and sensitive souls. Most obviously, the sensitive soul comprised the external senses of vision, hearing, taste, smell, and touch. But it also comprised the “internal senses”: these varied in number and nature from theorist to theorist but were usually between three and seven and included the faculties of imagination, memory, judgment, and common sense. In all of these areas, animal life stood in contiguous relation to its human equivalent. In many cases, animal abilities could even exceed those found in humankind: the vision of a cat or a hawk, like the olfactory sensitivity of a bloodhound, far outstripped those of any human being. What made humankind different was the possession of an immortal, immaterial, and intellective soul; it was in virtue of this that human life could claim deiformity and dominion over the created world. It was also through the intellect that humankind could think and communicate through language. As animals lacked an intellective soul, so even those of them with well-developed vocal organs could only express their passions: fear, pain, hunger, pleasure, affection, desire, satiation, and so on.1 What is more, biological comparativism made possible a species of moral comparativism—of which, as Shannon discusses at length, Montaigne was the principal exponent. Animals accepted their cosmic station with (the anthropomorphic) virtues of humility and patience and stood as a lesson and rebuke to the pride, dissatisfaction, anger, doubt, ambition, and cruelty of so much human life. For Descartes, this comparativism failed to put enough distance between human and animal life, thereby threatening the distinctive status of humankind and philosophical truth alike. The cogito and its concomitant bête machine were his response.By maintaining that “Montaigne and Descartes, pro et contra, stage a historically pivotal debate” on the subject of “animal stakeholdership” (13), Shannon elides the reactive quality of Descartes’s thinking just as fundamentally as she distorts Montaigne. As Shannon notes immediately after situating Montaigne and Descartes alongside one another, Montaigne turns to animals as a way in which to excoriate human pride and in which to probe the limitations of human intellection. For him, a vivid marker of humankind’s fallen insufficiency is the inability to comprehend the makeup of the natural world and its animal inhabitants, while even without the benefits of intellective discourse, animals often possess an innate wisdom to which humankind might well aspire. Shannon’s account of Montaigne is vibrant but loses sight of the fact that he nowhere suggests that animals might be political stakeholders, whether in virtue of their physical, mental, or habitual attributes. An example might better encapsulate the point. Montaigne was unquestionably both squeamish and an animal lover, but he was also a keen huntsman—he took pleasure in pursuing and killing (or, rather, having his hounds kill for him) animals like the hare, boar, and deer. By contrast, and in common with the overwhelming majority of his peers in sixteenth-century Europe, Montaigne seems never to have hunted human beings, just as he seems never to have eaten them, worn their skins, or burned them for fuel. Howsoever praiseworthy Montaigne may have taken animals to be, he viewed them as ontologically lesser forms of life. Hunting also provides some clues as to how early modern animals were valorized more generally. I quote from Oliver St John’s attempt to have Thomas Wentworth (Earl of Strafford) tried by bill of attainder rather than with benefit of law: “we give law to Hares and Deeres, because they be beasts of Chase; It was never accounted either cruelty or foul play to knock Foxes and Wolves on the head, as they can be found, because these be beasts of prey.”2 Remarkable though they were often thought to be, foxes and wolves were acknowledged as dangerous vermin. Unlike peaceable and purportedly noble creatures such as the deer and hare (which were to be given the chance to escape while being hunted), they were to be eradicated with violence and the minimum of fuss. Here as elsewhere, early modern animals were defined within a moral economy—call it a polis, if you will—that they had no role in framing or regulating.Although Shannon’s subtitle draws attention to “Shakespearean locales,” her spaces of exploration are textual and overlook the quotidian reality of human-animal interactions in the early modern world. It is good to see Shannon attending so thoughtfully to humanist natural history and the traditions of hexameral commentary (the twin pillars of her contextualizations), but in advancing an argument about early modern animal cosmopolity, one must surely seek to get in behind the learned world and its various idealizations. I have already mentioned hunting; a more surprising omission is that Shannon offers no sustained account of early modern farming, the site of human interactions with domesticated rather than wild animals. I kept hearing Keith Thomas’s admonition that changing attitudes to animals were the province of “well-to-do townsmen, remote from the agricultural process and inclined to think of animals as pets rather than as working livestock.”3 How did animals fare on the pre-Cartesian farm? About the closest we get to finding out comes in Shannon’s observation that the dimensions of Noah’s Ark proposed in John Wilkins’s Essay Towards a Real Character and a Philosophical Language (1668) “uncannily reflect the exact demands made in contemporary calls for minimum standards concerning livestock confinement” (280). This is ingenious and charmingly arcane but shirks the question. Notwithstanding her attention to Montaigne’s bookish thought experiments, Shannon by no means establishes that her pre-Cartesian stakeholder zootopia existed.There is comparatively little Shakespeare in The Accommodated Animal, and when Shannon does discuss his texts, it is generally as a peg on which to hang her broader arguments or as a pendant with which to adorn them. Chapter 1, for instance, begins with a discussion of A Midsummer Night’s Dream (29–31) and concludes some fifty pages later with As You Like It (80–81). Initially, these readings are unsatisfactory. It does not inspire confidence in Shannon’s critical ear to find her discussing “Jacques” throughout her account of As You Like It (80–81, also 69), but more concerning is her determination to set the play’s action within a prefabricated interpretative mold. After granting that Duke Senior is allowing his fancies to get the better of him by imagining the Forest of Arden as another Eden, Shannon asks us to read his assertion that deer are the “native burghers of this desert city” as evidence of animal political stakeholdership (80). Such a reading leaves no space for the critic to acknowledge Shakespeare’s riff on pastoral convention: the Duke’s metaphor belongs to the overwrought speech of one used to dwelling in an urban court and who is not yet habituated to the realities of country life. He and his cohort will hunt, kill, and eat the deer anyway. Here as elsewhere, it is disappointing that Shannon does not scrutinize the politics and dynamics of the pastoral mode.Things pick up in the book’s final two chapters, particularly in discussions of King Lear (127–37, 165–76) and A Midsummer Night’s Dream (180–84, 212–17). Here, instead of constraining her critical faculties to the demands of her thesis, Shannon works with patience and care, deftly teasing out the ways in which early modern writing on animals can help us to comprehend Shakespeare’s dramatic writing. This is historicism at its best. She reads Lear as “part of a larger zoographic critique of man” (133), in which humankind is contrasted unfavorably to the animals who know and contentedly inhabit their place in the order of things; Lear and his extended family are by contrast unaccommodated, disconnected from themselves and their place in the cosmos. As for the Dream, Shannon traces in it anxiety about the nocturnal weakness of humankind; in the darkness of the night, nonhuman animals establish temporary dominion of the worlds they inhabit. Through its “comparatist sensibility, the play engages in the zoographic mode of critique…[and] actually goes further and enforces human identity as a constraint” (180). Aside from their impressively analytical edginess, what these readings have in common is that they are epiphenomenal to Shannon’s stated thesis about pre-Cartesian cosmopolity and animal stakeholding. Animal discourse emerges as something though which Shakespeare was able to develop the thick weave and texture of his dramatic vision and against which he was able to measure certain aspects of the human condition. Near the start of The Accommodated Animal, Shannon announces her intention to move on from the critical habit of thinking “with” animals and to think “about” them instead (5). The problem is that Shakespeare, like Montaigne, resists this categorization: for the non-Cartesian early moderns, animals and the attempt to understand animal life were valuable to the extent that they helped to illuminate the human condition. Descartes’s revolution was to claim that such illuminations were illusory, harmful, and therefore to be discarded. Regrettably, he prevailed. Shannon, like many other broadly ecocritical writers, is to be praised for seeking to push back against him.University of Chicago Press has produced an attractive and affordable volume; it is especially welcome that Shannon’s references are printed in footnote rather than endnote form. Two cavils: copyediting is undistinguished (proper names, e.g., are a lottery); given the breadth and interest of the sources Shannon discusses, I cannot understand why her readers have been deprived of a bibliography.Notes1 The bibliography on such questions is massive. Three good starting points are Katharine Park, “The Organic Soul,” in The Cambridge History of Renaissance Philosophy, ed. Charles B. Schmitt, Quentin Skinner, and Eckhard Kessler (Cambridge: Cambridge University Press, 1988), 464–84; Peter Harrison, “Descartes on Animals,” Philosophical Quarterly 42 (1992): 219–27; and Richard W. Serjeantson, “The Passions and Animal Language, 1540–1700,” Journal of the History of Ideas 62 (2001): 425–44.2 Oliver St John, An Argument of Law Concerning the Bill of Attainder of High-Treason of Thomas, Earle of Strafford (London, 1641), 72.3 Keith Thomas, Man and the Natural World: Changing Attitudes in England, 1500–1800 (London: Lane, 1983), 182. Previous articleNext article DetailsFiguresReferencesCited by Modern Philology Volume 112, Number 4May 2015 Article DOIhttps://doi.org/10.1086/679328 Views: 281Total views on this site For permission to reuse, please contact [email protected]PDF download Crossref reports no articles citing this article.

One of the main controversies of the Logic Schools of the 12th century centered on the question: What follows from the impossible? In this paper arguments for two diametrically opposed positions are examined. The author of the ‘Avranches Text’ who probably belonged to the school of the Parvipontani defended the view that from an impossible proposition everything follows (‘Ex impossibili quodlibet’). In particular he developed a proof to show that by means of so-called ‘disjunctive syllogism’ any arbitrary proposition B can be logically derived from a pair of contradictory propositions A and Not-A. The author of the Ars Meliduna instead argued that nothing follows from an impossible proposition (‘ex falso nihil sequitur’). This view is supported by various counterexamples which aimed to show that the admission of impossible premises would give rise to inconsistent conclusions. Upon closer analysis these inconsistencies do not, however, have the formal structure of a real contradiction like A and Not-A, but rather the structure of two rivalling conditionals like ‘If B then A’ and ‘If B then Not-A’. Hence these counterexamples rather have to be considered as refutations of the basic principles of ‘connexive logic’.

This paper explores trivalent truth conditions for indicative conditionals, examining the “defective” truth table proposed by de Finetti (1936) and Reichenbach (1935, 1944). On their approach, a conditional takes the value of its consequent whenever its antecedent is true, and the value Indeterminate otherwise. Here we deal with the problem of selecting an adequate notion of validity for this conditional. We show that all standard validity schemes based on de Finetti’s table come with some problems, and highlight two ways out of the predicament: one pairs de Finetti’s conditional (DF) with validity as the preservation of non-false values (TT-validity), but at the expense of Modus Ponens; the other modifies de Finetti’s table to restore Modus Ponens. In Part I of this paper, we present both alternatives, with specific attention to a variant of de Finetti’s table (CC) proposed by Cooper (Inquiry 11, 295–320, 1968) and Cantwell (Notre Dame Journal of Formal Logic 49, 245–260, 2008). In Part II, we give an in-depth treatment of the proof theory of the resulting logics, DF/TT and CC/TT: both are connexive logics, but with significantly different algebraic properties.

This article investigates the prospect of giving de dicto- and de re-necessity a uniform treatment. The historical starting point is a puzzle raised by Aristotle's claim, advanced in one of the modal chapters of his Prior Analytics, that universally privative apodeictic premises simply convert. As regards the Prior and the Posterior Analytics, the data suggest a representation of propositions of the type in question by doubly modally qualified formulae of modal predicate logic that display a necessity operator in two distinct positions. Can the N-operator occurring in these positions be given a unified semantical treatment (which would justify dispensing with a notational differentiation)? A positive answer, based on a suitably shaped truth condition for N-formulae, is given, and is supported in the final section with an alternative proof theoretically based conception of a property's essential belonging to an individual.

The paper suggests a revival of the 17th century distinction between truths of reason and truths of fact. Some points are made which seem to me show it obviously false that a fact is merely a true proposition. Truths of fact, contingent truths, are rightly seen as corresponding to facts. Other truths, including ethical truths of right and wrong are, if true, necessarily true. In general, necessarily true statements, including those of mathematics are wrongly construed as factual.Ethics and aesthetics, it is maintained, can be construed as noncognitive; but not because claims in these domains are other than claims to truth. They are, in large part, not claims to knowledge, which does not bar them from being claims to truth.

Science progresses if we succeed in rendering the objects of scientific inquiry more comprehensively or more precisely. Popper tries to formalize this venerable idea. According to him the most comprehensive and most precise description of the world is given by the set T of all true statements. A hypothesis comes the closer to T, or has the more verisimilitude, the more true consequences and the fewer false consequences it implies. Popper proposes to order hypotheses by the inclusion relations between the sets of their true and of their false consequences (“truth contents” and “falsity contents”). A partial ordering would permit one to decide whether the substitution of theory t1 by t2 represents scientific progress. But because of the logical relations between the elements of the sets of logical consequences, or contents, false hypotheses cannot be compared. As our theories usually turn out to be false sooner or later, they can seldom be compared as to their verisimilitude and when they can, the result depends only on which theory implies the other and on their truth-values. Popper even tries to define a measure of verisimilitude on the partial ordering. It has to fail for the same reason. In addition he tries to relativize the concept of a content and fails. What is more, any attempt at defining a measure of better or worse correspondence to the whole truth must fail, as there is no justification for saying that any true primitive sentence asserts as much about reality as some other primitive sentence, or more.