The Negation Problem states that expressivism has insufficient structure to account for the various ways in which a moral sentence can be negated. We argue that the Negation Problem does not arise for expressivist accounts of all normative language but arises only for the specific examples on which expressivists usually focus. In support of this claim, we argue for the following three theses: 1) a problem that is structurally identical to the Negation Problem (...) arises in non-normative cases, and this problem is solved once the hidden quantificational structure involved in such cases is uncovered; 2) the terms ‘required’, ‘permissible’, and ‘forbidden’ can also be analyzed in terms of hidden quantificational structure, and the Negation Problem disappears once this hidden structure is uncovered; 3) the Negation Problem does not arise for normative language that has no hidden quantificational structure. We conclude that the Negation Problem is not really a problem about expressivism at all but is rather a feature of the quantificational structure of the required, permitted, and forbidden. (shrink)

We develop a novel solution to the negation version of the Frege-Geach problem by taking up recent insights from the bilateral programme in logic. Bilateralists derive the meaning of negation from a primitive *B-type* inconsistency involving the attitudes of assent and dissent. Some may demand an explanation of this inconsistency in simpler terms, but we argue that bilateralism’s assumptions are no less explanatory than those of *A-type* semantics that only require a single primitive attitude, but must stipulate (...) inconsistency elsewhere. Based on these insights, we develop a version of B-type expressivism called *inferential expressivism*. This is a novel semantic framework that characterises meanings by inferential roles that define which *attitudes* one can *infer* from the use of terms. We apply this framework to normative vocabulary, thereby solving the Frege-Geach problem generally and comprehensively. Our account moreover includes a semantics for epistemic modals, thereby also explaining normative terms under epistemic modals. (shrink)

It has recently been alleged that expressivism cannot account for the obvious fact that normative sentences and their negations express inconsistent kinds of attitudes. I explain how the expressivist can respond to this objection. I offer an account of attitudinal inconsistency that takes it to be a combination of descriptive and normative relations. The account I offer to explain these relations relies on a combination of functionalism about normative judgments and expressivism about the norms governing them. It holds that the (...) inconsistency of normative judgments is primitive. One potential problem for this view is that the large number of normative primitives that the expressivist will allegedly need to accept will render the view grossly unparsimonious. In defending this thesis, I suggest that it is a mistake to hold the lack of normative parsimony of expressivism against its core psychological claims. (shrink)

This short paper has two loosely connected parts. In the first part, I discuss the difference between classical and intuitionist logic in relation to different the role of hypotheses play in each logic. Harmony is normally understood as a relation between two ways of manipulating formulas in systems of natural deduction: their introduction and elimination. I argue, however, that there is at least a third way of manipulating formulas, namely the discharge of assumption, and that the difference between classical and (...) intuitionist logic can be characterised as a difference of the conditions under which discharge is allowed. Harmony, as ordinarily understood, has nothing to say about discharge. This raises the question whether the notion of harmony can be suitably extended. This requires there to be a suitable fourth way of manipulating formulas that discharge can stand in harmony to. The question is whether there is such a notion: what might it be that stands to discharge of formulas as introduction stands to elimination? One that immediately comes to mind is the making of assumptions. I leave it as an open question for further research whether the notion of harmony can be fruitfully extended in the way suggested here. In the second part, I discuss bilateralism, which proposes a wholesale revision of what it is that is assumed and manipulated by rules of inference in deductions: rules apply to speech acts – assertions and denials – rather than propositions. I point out two problems for bilateralism. First, bilaterlists cannot, contrary to what they claim to be able to do, draw a distinction between the truth and assertibility of a proposition. Secondly, it is not clear what it means to assume an expression such as '+ A' that is supposed to stand for an assertion. Worse than that, it is plausible that making an assumption is a particular speech act, as argued by Dummett (Frege: Philosophy of Language, p.309ff). Bilaterlists accept that speech acts cannot be embedded in other speech acts. But then it is meaningless to assume + A or − A. (shrink)

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the (...) meanings of modal operators in terms of rules of inference. (shrink)

Expressivists have a problem with negation. The problem is that they have not, to date, been able to explain why ‘murdering is wrong’ and ‘murdering is not wrong’ are inconsistent sentences. In this paper, I explain the nature of the problem, and why the best efforts of Gibbard, Dreier, and Horgan and Timmons don’t solve it. Then I show how to diagnose where the problem comes from, and consequently how it is possible for expressivists to (...) solve it. Expressivists should accept this solution, I argue, because it is demonstrably the only way of avoiding the problem, and because it generalizes. Once we see how to solve the negation problem, I show, it becomes easy to state a constructive, compositional expressivist semantics for a purely normative language with the expressive power of propositional logic, in which we can for the first time give explanatory, formally adequate expressivist accounts of logical inconsistency, logical entailment, and logical validity. As a corollary, I give what I take to be the first real expressivist explanation of why Geach’s original moral modus ponens argument is genuinely logically valid. This proves that the problem with expressivism cannot be that it can’t account for the logical properties of complex normative sentences. But it does not show that the same solution can work for a language with both normative and descriptive predicates, let alone that expressivists are able to deal with more complex linguistic constructions like tense, modals, or even quantifiers. In the final section, I show what kind of constraints the solution offered here would place expressivists under, in answering these further questions. (shrink)

Frege-Geach worries about embedding and composition have plagued metaethical theories like emotivism, prescriptivism and expressivism. The sharpened point of such criticism has come to focus on whether negation and inconsistency have to be understood in descriptivist terms. Because they reject descriptivism, these theories must offer a non-standard account of the meanings of ethical and normative sentences as well as related semantic facts, such as why certain sentences are inconsistent with each other. This paper fills out such a solution to (...) the negation problems, following some of the original emotivist ideas about the interplay of interests in conversation. We communicate both to share information and coordinate our actions, and we use distinctively normative language like deontic ‘must’ and ‘may’ to negotiate what people are to do. The kinds of disagreement involved in such negotiation can illuminate the issues with negation and inconsistency. This paper outlines a dynamic semantic system in which these ideas can bear fruit, developing the scorekeeping model of conversation. The result is clarification about what Frege-Geach worries can mean for nondescriptive semantics. (shrink)

Many think that expressivists have a special problem with negation. I disagree. For if there is a problem with negation, I argue, it is a problem shared by those who accept some plausible claims about the nature of intentionality. Whether there is any special problem for expressivists turns, I will argue, on whether facts about what truth-conditions beliefs have can explain facts about basic inferential relations among those beliefs. And I will suggest that the (...) answer to this last question is, on most plausible attempts at solving the problem of intentionality, ‘no’. (shrink)

This paper advances three necessary conditions on a successful account of sentential negation. First, the ability to explain the constancy of sentential meaning across negated and unnegated contexts (the Fregean Condition). Second, the ability to explain why sentences and their negations are inconsistent, and inconsistent in virtue of the meaning of negation (the Semantic Condition). Third, the ability of the account to generalize regardless of the topic of the negated sentence (the Generality Condition). The paper discusses three accounts (...) of negation available to moral expressivists. The first—the dominant commitment account—fails to meet the Fregean Condition. The two remaining accounts—commitment semantics and the expression account—satisfy all three conditions. A recent argument that the dominant commitment account is the only option available to expressivists is considered and rejected. (shrink)

The recent literature on Nāgārjuna’s catuṣkoṭi centres around Jay Garfield’s (2009) and Graham Priest’s (2010) interpretation. It is an open discussion to what extent their interpretation is an adequate model of the logic for the catuskoti, and the Mūla-madhyamaka-kārikā. Priest and Garfield try to make sense of the contradictions within the catuskoti by appeal to a series of lattices – orderings of truth-values, supposed to model the path to enlightenment. They use Anderson & Belnaps's (1975) framework of First Degree Entailment. (...) Cotnoir (2015) has argued that the lattices of Priest and Garfield cannot ground the logic of the catuskoti. The concern is simple: on the one hand, FDE brings with it the failure of classical principles such as modus ponens. On the other hand, we frequently encounter Nāgārjuna using classical principles in other arguments in the MMK. There is a problem of validity. If FDE is Nāgārjuna’s logic of choice, he is facing what is commonly called the classical recapture problem: how to make sense of cases where classical principles like modus pones are valid? One cannot just add principles like modus ponens as assumptions, because in the background paraconsistent logic this does not rule out their negations. In this essay, I shall explore and critically evaluate Cotnoir’s proposal. In detail, I shall reveal that his framework suffers collapse of the kotis. Furthermore, I shall argue that the Collapse Argument has been misguided from the outset. The last chapter suggests a formulation of the catuskoti in classical Boolean Algebra, extended by the notion of an external negation as an illocutionary act. I will focus on purely formal considerations, leaving doctrinal matters to the scholarly discourse – as far as this is possible. (shrink)

I resolve the major challenge to an Expressivist theory of the meaning of normative discourse: the Frege–Geach Problem. Drawing on considerations from the semantics of directive language (e.g., imperatives), I argue that, although certain forms of Expressivism (like Gibbard’s) do run into at least one version of the Problem, it is reasonably clear that there is a version of Expressivism that does not.

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)

This article has one aim, to reject the claim that negation is semantically ambiguous. The first section presents the putative incompatibility between truth-value gaps and the truth-schema; the second section presents the motivation for the ambiguity thesis; the third section summarizes arguments against the claim that natural language negation is semantically ambiguous; and the fourth section indicates the problems of an introduction of two distinct negation operators in natural language.

This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means Hilbert-style calculi that are finite. On the negative side, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely axiomatizable; (...) we likewise establish that the order-preserving logic of the variety of all Ockham algebras is also not finitely axiomatizable. On the positive side, we show that an arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite number of equations if and only if the corresponding order-preserving logic is axiomatized by a finite Hilbert-style calculus. This equivalence also holds for every subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a new proof that the implication-free fragment of intuitionistic logic is finitely axiomatizable, as well as a new corresponding Hilbert-style calculus. Our proofs are constructive in that they allow us to effectively convert an equational presentation of a variety of algebras into a Hilbert-style calculus for the corresponding order-preserving logic, and vice versa. We also consider the assertional logics associated to the above-mentioned varieties, showing in particular that the assertional logics of finitely axiomatizable subvarieties of semi-De Morgan algebras are finitely axiomatizable as well. (shrink)

In bilateral logic formulas are signed by + and –, indicating the speech acts assertion and denial. I argue that making an assumption is also speech act. Speech acts cannot be embedded within other speech acts. Hence we cannot make sense of the notion of making an assumption in bilateral logic. Attempts to solve this problem are considered and rejected.

This paper considers negative triggers and the interpretation of simple sentences containing more than one occurrence of those items . In the most typical interpretations those sentences have more negative expressions than negations in their semantic representation. It is first shown that this compositionality problem remains in current approaches. A principled algorithm for deriving the representation of sentences with multiple negative quantifiers in a DRT framework is then introduced. The algorithm is under the control of an on-line check-in, keeping (...) the complexity of negation auto-embedding below a threshold of complexity. This mechanism is seen as a competence limitation imposing the ‘abrogation of compositionality’ observed in the so-called negative concord readings . A solution to the compositionality problem is thus proposed, which is based on a control on the processing input motivated by a limitation of the processing mechanism itself. (shrink)

I discuss the general form of arguments that profess to prove that the view that things endure in tensed time through causally produced change (the dynamic view) must be false because it involves contradictions. I argue that these arguments implicitly presuppose what has been called the temporal parity thesis, i.e. that all moments of time are equally existent and real, and that this thesis must be understood as the denial of the dynamic view. When this implicit premise is made explicit, (...) the arguments turn out to be either circular, they presuppose what they profess to prove, or mere demonstrations of the fact that the dynamic view is incompatible with its own negation. Furthermore, I discuss the metaphysical consequences of accepting the temporal parity thesis, arguing that it deprives us of the means to provide natural explanations to empirical phenomena. (shrink)

Peer disagreement presents religious believers, agnostics, and skeptics alike with an epistemological problem: how can confidence in any religious claims (including their negations) be epistemically justified? There seem to be rational, well-informed adherents among a variety of mutually incompatible religious and non-religious perspectives, and so the problem of disagreement arises acutely in the religious domain. In this paper, we show that the transformative nature of religious experience and identity poses more than just this traditional, epistemic problem of (...) conflicting religious beliefs. In encountering one another, believers, agnostics, and skeptics confront not just different beliefs, but different ways of being a person. -/- To transition between religious belief and skepticism is not just to adopt a different set of beliefs, but to transform into a different version of oneself. We argue that the transformative nature of religious identity intensifies the problem of pluralism by adding a new dimension to religious disagreement, for there are principled reasons to think we can lack epistemic and affective access to our potential religious, agnostic, or skeptical selves. Yet, access to these selves seems to be required for the purposes of decision-making that is to be both rational and authentic. Finally, we reflect on the relationship between the transformative problem of religious disagreement and what it shows about the epistemic status of religious conversion and de-conversion, in which one disagrees with one’s own (transformed) self. (shrink)

We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first two (...) kinds of import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle’s system appears then as a fragment of a broader system which can be developed by using FOL. (shrink)

The paradigm of referent tracking is based on a realist presupposition which rejects so-called negative entities (congenital absent nipple, and the like) as spurious. How, then, can a referent tracking-based Electronic Health Record deal with what are standardly called ‘negative findings’? To answer this question we carried out an analysis of some 748 sentences drawn from patient charts and containing some form of negation. Our analysis shows that to deal with these sentences we need to introduce a new ontological (...) relationship between a particular and a universal, which holds when no instance of the universal has a specific qualified ontological relation with the particular. This relation is found to be able to accommodate nearly all occurrences of negative findings in the examined sample, in ways which involve no reference to negative entities. (shrink)

A large number of essays address the Sleeping Beauty problem, which undermines the validity of Bayesian inference and Bas Van Fraassen's 'Reflection Principle'. In this study a straightforward analysis of the problem based on probability theory is presented. The key difference from previous works is that apart from the random experiment imposed by the problem's description, a different one is also considered, in order to negate the confusion on the involved conditional probabilities. The results of the analysis (...) indicate that no inconsistency takes place, whereas both Bayesian inference and 'Reflection Principle' are valid. (shrink)

Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Landau's conjecture implies the following unproven statement Φ: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆[2,f(7)]. Let B denote the system of equations: {x_j!=x_k: i,k∈{1,...,9}}∪{x_i⋅x_j=x_k: i,j,k∈{1,...,9}}. The system of equations {x_1!=x_1, x_1 \cdot x_1=x_2, x_2!=x_3, x_3!=x_4, x_4!=x_5, x_5!=x_6, x_6!=x_7, x_7!=x_8, x_8!=x_9} has exactly two solutions in positive integers x_1,...,x_9, namely (1,...,1) and (f(1),...,f(9)). No known system S⊆B with a finite (...) number of solutions in positive integers x_1,...,x_9 has a solution (x_1,...,x_9)∈(N\{0})^9 satisfying max(x_1,...,x_9)>f(9). For every known system S⊆B, if the finiteness/infiniteness of the set {(x_1,...,x_9)∈(N\{0})^9: (x_1,...,x_9) solves S} is unknown, then the statement ∃ x_1,...,x_9∈N\{0} ((x_1,...,x_9) solves S)∧(max(x_1,...,x_9)>f(9)) remains unproven. Let Λ denote the statement: if the system of equations {x_2!=x_3, x_3!=x_4, x_5!=x_6, x_8!=x_9, x_1 \cdot x_1=x_2, x_3 \cdot x_5=x_6, x_4 \cdot x_8=x_9, x_5 \cdot x_7=x_8} has at most finitely many solutions in positive integers x_1,...,x_9, then each such solution (x_1,...,x_9) satisfies x_1,...,x_9≤f(9). The statement Λ is equivalent to the statement Φ. It heuristically justifies the statement Φ . This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ. Algorithms always terminate. We explain the distinction between existing algorithms (i.e. algorithms whose existence is provable in ZFC) and known algorithms (i.e. algorithms whose definition is constructive and currently known). Assuming that the infiniteness of a set X⊆N is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. *** (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) No known algorithm with no input returns the logical value of the statement card(X)=ω. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. *** Conditions (2)-(5) hold for X=P(n^2+1). The statement Φ implies Condition (1) for X=P(n^2+1). The set X={n∈N: the interval [-1,n] contains more than 29.5+\frac{11!}{3n+1}⋅sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. 501893∈X. Condition (1) holds with n=501893. card(X∩[0,501893])=159827. X∩[501894,∞)= {n∈N: the interval [-1,n] contains at least 30 primes of the form k!+1}. We present a table that shows satisfiable conjunctions of the form #(Condition 1) ∧ (Condition 2) ∧ #(Condition 3) ∧ (Condition 4) ∧ #(Condition 5), where # denotes the negation ¬ or the absence of any symbol. No set X⊆N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove this assumption. (shrink)

The problem of negative truth is the problem of how, if everything in the world is positive, we can speak truly about the world using negative propositions. A prominent solution is to explain negation in terms of a primitive notion of metaphysical incompatibility. I argue that if this account is correct, then minimal logic is the correct logic. The negation of a proposition A is characterised as the minimal incompatible of A composed of it and the (...) logical constant ¬. A rule based account of the meanings of logical constants that appeals to the notion of incompatibility in the introduction rule for negation ensures the existence and uniqueness of the negation of every proposition. But it endows the negation operator with no more formal properties than those it has in minimal logic. (shrink)

Suppose Alice asserts p, and the Caterpillar wants to disagree. If the Caterpillar accepts classical logic, he has an easy way to indicate this disagreement: he can simply assert ¬p. Sometimes, though, things are not so easy. For example, suppose the Cheshire Cat is a paracompletist who thinks that p ∨ ¬p fails (in familiar (if possibly misleading) language, the Cheshire Cat thinks p is a gap). Then he surely disagrees with Alice's assertion of p, but should himself be unwilling (...) to assert ¬p. So he cannot simply use the classical solution. Dually, suppose the Mad Hatter is a dialetheist who thinks that p ∧ ¬p holds (that is, he thinks p is a glut). Then he may assert ¬p, but it should not be taken to indicate that he disagrees with Alice; he doesn't. So he too can't use the classical solution. The Cheshire Cat and the Mad Hatter, then, have a common problem, and philosophers with opinions like theirs have adopted a common solution to this problem: appeal to denial. Denial, these philosophers suppose, is a speech act like assertion, but it is not to be understood as in any way reducing to assertion. Importantly, denial is something different from the assertion of a negation; this is what allows it to work even in cases where assertion of negation does not. Just as importantly, denial must express disagreement, since this is the job it's being enlisted to do. (shrink)

Could there be a single logical system that would allow us to work simultaneously with classical, paraconsistent, and paracomplete negations? These three negations were separately studied in logics whose negations bear their names. Initially we will restrict our analysis to propositional logics by analyzing classical negation, ¬c, as treated by Classical Propositional Logic (LPC); the paraconsistent negation, ¬p, as treated through the hierarchy of Paraconsistent Propositional Calculi Cn (0 ≤ n ≤ ω); and the paracomplete negation, ¬q, (...) as treated by the hierarchy of Paracomplete Propositional Calculi Pn (0 ≤ n ≤ ω). In “Logics that are both paraconsistent and paracomplete” (1989), Newton da Costa proposed a system with approximate characteristics to what we are looking for. In the hierarchy of Non-Alethical Propositional Calculi Nn (0 ≤ n ≤ ω), only one negation is introduced (as primitive), called a “non-alethic” (¬n), whose operation preserves the properties of classical, or paraconsistent or paracomplete negation -- depending on the well or ill behavior of the formula connected to it. However, as we shall see, in the hierarchy Nn we can not reiterate negations with different behaviors in a same formula (e.g., ¬p¬cα or ¬q¬c¬p α), or even analyze a formula like ¬cα → ¬pα. In view of these problems, can we really say that the hierarchy Nn allows us to understand the relationships and interactions of the three types of negations? In order to deal with this, given the initial problem, we will present four axiomatic systems (KG) in which, unlike Nn, the three negations are directly introduced -- offering a semantics and a method of proofs by analytic tableaux. Through the KG Systems we will show how the negations interact, obtaining non-demonstrable theorems in LPC, Cn, Pn, and Nn (0 ≤ n ≤ ω). Finally, we will also offer a first-order extension for the KG Systems. (shrink)

A formal theory of oppositions and opposites is proposed on the basis of a non- Fregean semantics, where opposites are negation-forming operators that shed some new light on the connection between opposition and negation. The paper proceeds as follows. After recalling the historical background, oppositions and opposites are compared from a mathematical perspective: the first occurs as a relation, the second as a function. Then the main point of the paper appears with a calculus of oppositions, by means (...) of a non-Fregean semantics that redefines the logical values of various sorts of sentences. A num- ber of topics are then addressed in the light of this algebraic semantics, namely: how to construct value-functional operators for any logical opposition, beyond the classical case of contradiction; Blanché's "closure problem", i.e., how to find a complete structure connecting the sixteen binary sentences with one another. All of this is meant to devise an abstract theory of opposition: it encompasses the relation of consequence as subalternation, while relying upon the use of a primary "proto- negation" that turns any relatum into an opposite. This results in sentential negations that proceed as intensional operators, while negation is broadly viewed as a difference-forming operator without special constraints on it. (shrink)

Cheating is a major problem in society, especially in the educational system. From the viewpoint of subjective cost-benefit analysis, concerns about reputation damage as well as considerations of cultural values against unethical behavior can help increase the perceived costs of cheating. To explore deeper into the psychological processes in such assessments, we employ Bayesian Mindsponge Framework (BMF) analytics – an information-processingbased method. Conducting Bayesian analysis on 493 university students from Germany, Vietnam, China, Taiwan, and Japan, we found that reputation (...) concern is negatively associated with cheating behavior. If a student embraces Confucian values, the above negating effect is stronger. Our findings support the notion that mentally simulated negative consequences of cheating reduce the probability of carrying out such behavior. As the educational system is changing rapidly due to technological advancement, a better understanding of the influences of sociocultural factors can be helpful in cheating prevention efforts. (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)

The problem of merging several ontologies has important applications in the Semantic Web, medical ontology engineering and other domains where information from several distinct sources needs to be integrated in a coherent manner.We propose to view ontology merging as a problem of social choice, i.e. as a problem of aggregating the input of a set of individuals into an adequate collective decision. That is, we propose to view ontology merging as ontology aggregation. As a first step in (...) this direction, we formulate several desirable properties for ontology aggregators, we identify the incompatibility of some of these properties, and we define and analyse several simple aggregation procedures. Our approach is closely related to work in judgment aggregation, but with the crucial difference that we adopt an open world assumption, by distinguishing between facts not included in an agent’s ontology and facts explicitly negated in an agent’s ontology. (shrink)

At first sight the Theory of Computation i) relies on a kind of mathematics based on the notion of potential infinity; ii) its theoretical organization is irreducible to an axiomatic one; rather it is organized in order to solve a problem: “What is a computation?”; iii) it makes essential use of doubly negated propositions of non-classical logic, in particular in the word expressions of the Church-Turing’s thesis; iv) its arguments include ad absurdum proofs. Under such aspects, it is like (...) many other scientific theories, in particular the first theories of both mechanical machines and heat machines. A more accurate examination of Theory of Computation shows a difference from the above mentioned theories in its essentially including an odd notion, “thesis”, to which no theorem corresponds. On the other hand, arguments of each of the other theories conclude a doubly negative predicate which then, by applying the inverse translation of the ‘negative one’, is translated into the corresponding affirmative predicate. By also taking into account three criticisms to the current Theory of Computation a rational re-formulation of it is sketched out; to Turing-Church thesis of the usual theory corresponds a similar proposition, yet connecting physical total computation functions with constructive mathematical total computation functions. -/- . (shrink)

I here propose a hitherto unnoticed possibility of solving embedding problems for noncognitivist expressivists in metaethics by appeal to Conceptual Role Semantics. I show that claims from the latter as to what constitutes various concepts can be used to define functions from states expressed by atomic sentences to states expressed by complex sentences, thereby allowing an expressivist semantics that satisfies a rather strict compositionality constraint. The proposal can be coupled with several different types of concept individuation claim, and is shown (...) to pave the way to novel accounts for, e.g., negation. (shrink)

A general characterization of logical opposition is given in the present paper, where oppositions are defined by specific answers in an algebraic question-answer game. It is shown that opposition is essentially a semantic relation of truth values between syntactic opposites, before generalizing the theory of opposition from the initial Apuleian square to a variety of alter- native geometrical representations. In the light of this generalization, the famous problem of existential import is traced back to an ambiguous interpretation of assertoric (...) sentences in Aristotle's traditional logic. Following Abelard’s distinction between two alternative readings of the O-vertex: Non omnis and Quidam non, a logical difference is made between negation and denial by means of a more fine- grained modal analysis. A consistent treatment of assertoric oppositions is thus made possible by an underlying abstract theory of logical opposition, where the central concept is negation. A parallel is finally drawn between opposition and consequence, laying the ground for future works on an abstract operator of opposition that would characterize logical negation just as does Tarski’s operator of consequence for logical truth. (shrink)

The ambition of the paper is to provide a solution to the problem posed by Von Wright (1999): how is it possible that the two actions, one of producing P and the other of preventing P can have different deontic status, the former being obligatory and the latter being forbidden. The solution for the problem is sought for by an investigation into connections between imperative and deontic logic. First, it is asked whether a solution could be found in (...) Lemmon's (1965) system of "change logic", using his idea on connection between logic of orders being in force and deontic logic. The answer is the negative one. Next, the connection between Lemmon's imperative logic and deontic logic given in Aqvist's paper - "Next" and "Ought" (1965) - is analyzed. Than, the Lemmon's treatment of imperatives is restricted to the natural language imperatives and Aqvist's way of connecting imperative and deontic logic is modified accordingly. Some principles for the natural language imperatives are established (the negation rule ; the law of contraposition for imperative conditionals) and a simple "global" semantics is developed. The notion of "opposite action" is introduced and it is given an important role in semantics. Finally, a solution for von Wright's problem is given. In the closing sections some further topics for investigation are hinted: one of them being the connection between Aqvist's epistemic- imperative conception of interrogatives and "epistemic obligations", the other being formalization of the idea that imperatives create and re-create obligation patterns that can be described in deontic terms. (shrink)

This paper considers whether incompatibilism, the view that negation is to be explained in terms of a primitive notion of incompatibility, and Fregeanism, the view that arithmetical truths are analytic according to Frege’s definition of that term in §3 of Foundations of Arithmetic, can both be upheld simultaneously. Both views are attractive on their own right, in particular for a certain empiricist mind-set. They promise to account for two philosophical puzzling phenomena: the problem of negative truth and the (...) problem of epistemic access to numbers. For an incompatibilist, proofs of numerical non-identities must appeal to primitive incompatibilities. I argue that no analytic primitive incompatibilities are forthcoming. Hence incompatibilists cannot be Fregeans. (shrink)

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want (...) to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

Molnar argues that the problem of truthmakers for negative truths arises because we tend to accept four metaphysical principles that entail that all negative truths have positive truthmakers. This conclusion, however, already follows from only three of Molnar´s metaphysical principles. One purpose of this note is to set the record straight. I provide an alternative reading of two of Molnar´s principles on which they are all needed to derive the desired conclusion. Furthermore, according to Molnar, the four principles may (...) be inconsistent. By themselves, however, they are not. The other purpose of this note is to propose some plausible further principles that, when added to the four metaphysical theses, entail a contradiction. (shrink)

Sullivan surveys the philosophical problem-areas surrounding multiculturalism as an ideology of group-identity. While endorsing the claims of underrepresented minorities for recognition, the article sides with traditionalists in prioritizing the autonomy of the self-fashioning individual over ethnic or cultural affiliations. The multicultural challenge to Western logocentrism, its assertion of the implicit power structures embedded in truth claims, and the excesses of postmodern relativism are all subjected to measured criticism. Finally, the essay examines Habermas' role in postwar Germany's embrace of multiculturalism (...) as a national platform of reinvention, and as a means to negate the trauma of the monocultural chauvinism of the Nazi era. However, Sullivan argues, it has taken cosmopolitanism too far. While post-national strategies such as Habermas' Verfassungspatriotismus are well-intentioned, they are ultimately too sterile, universalist and abstract, to satisfy a deep human need for belonging, rooted in a common cultural heritage. -/- All rights reserved. For permission to reprint please contact Rowman & Littlefield. (shrink)

The Role of Negation in the Description of the World According to Aristote’s Metaphysics. The notions of ‘being’ and ‘non-being’ have entered philosophical language, forming the basis of ontology and meontology, as the counterparts of the Greek expressions to on and to me on (nominalised forms, affirmative and negative, of the participle of the verb einai). Originally, however, these expressions did not have any objectifying meaning, but played the role of meta-language names, representing the copula einai in all its (...) forms, most generally in its affirmative and negative forms. The copula itself, which in later philosophy took on the existential meaning, had functioned only as a semantically empty connective of predicates. Over time the participle on has been used as a universal name of all predicates. The above-mentioned expressions became central in Greek philosophical terminology thanks to the debates, initiated by Parmenides, on the role of negation in the description of the world. Parmenides himself proposed a complete excision of negative sentences as describing by elimination, and created a positive-monistic system which abandoned multiplicity, divisibility, and variability. Later philosophers defended negation, fighting back against the paradoxes formulated by the Eleatics and later by the Sophists. Plato observed that without negation it is impossible to describe the multiplicity of things. He also distinguished a relative negation which does not eliminate anything but makes it possible to confront some things with others. According to the atomists, the divisibility of physical things forces us to accept that they consist of a positive element in the form of an impenetrable body, and of another element lacking any characteristics, i.e. the void. Finally, Aristotle, when analysing the process of change, justified the consistency of the statement that something comes out of “not being” and “being”, under the assumption that the former is understood as being actual, and the latter as being potential. In all these conceptions there is nothing non-existent, there are only proposals of how to identify the aspects of reality whose explanation justifies the use of negation. The affirmative and negative forms of the above expressions have also provoked reflection on the problem of truth and falsity. It has been observed that they are used in everyday language not only to state an agreement or disagreement with the actual facts of the matter, but also to affirm or deny something. (shrink)

In this paper, we use antecedent-final conditionals to formulate two problems for parsing-based theories of presupposition projection and triviality of the kind given in Schlenker 2009. We show that, when it comes to antecedent-final conditionals, parsing-based theories predict filtering of presuppositions where there is in fact projection, and triviality judgments for sentences which are in fact felicitous. More concretely, these theories predict that presuppositions triggered in the antecedent of antecedent-final conditionals will be filtered (i.e. will not project) if the (...) class='Hi'>negation of the consequent entails the presupposition. But this is wrong: John isn’t in Paris, if he regrets being in France intuitively presupposes that John is in France, contrary to this prediction. Likewise, parsing-based approaches to triviality predict that material entailed by the negation of the consequent will be redundant in the antecedent of the conditional; but John isn’t in Paris, if he’s in France and Mary is with him is intuitively felicitous, contrary to these predictions. Importantly, given that the trigger appears in sentence-final position, both incremental (left-to-right) and symmetric versions of such theories make the same predictions. These data constitute a challenge to the idea that presupposition projection and triviality should be computed on the basis of parsing. This issue is important because it relates to the more general question as to whether presupposition and triviality calculation should be thought of as a pragmatic post-compositional phenomenon or as part of compositional semantics (as in the more traditional dynamic approaches). We discuss a solution which allows us to maintain the parsing-based pragmatic approach; it is based on an analysis of conditionals which incorporates a presupposition that their antecedent is compatible with the context, together with a modification to Schlenker’s (2009) algorithm for calculating local contexts so that it takes into account presupposed material. As we will discuss, this solution works within a framework broadly similar to that of Schlenker’s (2009), but it doesn’t extend in an obvious way to other parsing-based accounts (e.g. parsing-based trivalent approaches). We conclude that a parsing-based theory can be maintained, but only if we adopt a substantial change of perspective on the framework. (shrink)

This work studies some problems connected to the role of negation in logic, treating the positive fragments of propositional calculus in order to deal with two main questions: the proof of the completeness theorems in systems lacking negation, and the puzzle raised by positive paradoxes like the well-known argument of Haskel Curry. We study the constructive com- pleteness method proposed by Leon Henkin for classical fragments endowed with implication, and advance some reasons explaining what makes difficult to extend (...) this constructive method to non-classical fragments equipped with weaker implications (that avoid Curry's objection). This is the case, for example, of Jan Lukasiewicz's n-valued logics and Wilhelm Ackermann's logic of restricted implication. Besides such problems, both Henkin's method and the triviality phenomenon enable us to propose a new positive tableau proof system which uses only positive meta-linguistic resources, and to mo- tivate a new discussion concerning the role of negation in logic proposing the concept of paratriviality. In this way, some relations between positive reasoning and infinity, the possibilities to obtain a ¯first-order positive logic as well as the philosophical connection between truth and meaning are dis- cussed from a conceptual point of view. (shrink)

Recently, a problem is addressed while dealing the data with Non-Euclidean Geometry and its characterization. The mathematician found negation of fifth postulates of Euclidean geometry easily and called as Non-Euclidean geometry. However Riemannian provided negation of second postulates also which still considered as Non-Euclidean. In this case the problem arises what will happen in case negation of other Euclid Postulates exists. Same time total total or partial negation of Euclid postulates fails as hybrid Geometry. (...) It become more crucial in case the data is unknown, incomplete or exists beyond the three-way space as heteroclinic pattern. To understand this problem, the current paper tried to distinguish Euclidean, Non-Euclidean, Anti-Geometry, Neutrogeometry and Turiyam or Unknown geometry using the complement operator with an example. (shrink)

Cheating is widely considered a condemnable behavior in society and a big problem in the educational system. In this study, we employ the information-processing-based Bayesian Mindsponge Framework to explore deeper the subjective cost-benefit evaluation involving the perceived value of cheating. Conducting Bayesian analysis on 493 university students from Germany, Vietnam, China, Taiwan, and Japan, we found that students who have more positive attitudes toward cheating are more likely to cheat. However, a higher capability of cognitive reflection acts as a (...) moderator that negates the above effect. Re-evaluation of prior thoughts allows for more accurate assessments of the associated risks and possible costs, which help reduce the probability of carrying out cheating behavior. As rapid technological advancement may introduce new and efficient cheating methods, a better understanding of the psychology of cheating in students can help schools prepare better countermeasures. (shrink)

Data involving epistemic modals suggest that some classically valid argument forms, such as _reductio_, are invalid in natural language reasoning as they lead to modal collapses. We adduce further data showing that the classical argument forms governing the existential quantifier are similarly defective, as they lead to a _de re–de dicto_ collapse. We observe a similar problem for disjunction. But if the classical argument forms for negation, disjunction and existential quantification are invalid, what are the correct forms that (...) govern the use of these items? Our diagnosis is that epistemic modals interfere with hypothetical reasoning. We present a modal first-order logic and model theory that characterizes hypothetical reasoning with epistemic modals in a principled manner. One upshot is a sound and complete natural deduction system for reasoning with epistemic modals in first-order logic. (shrink)

Expressivist views of an area of discourse encourage us to ask not about the nature of the relevant kinds of values but rather about the nature of the relevant kind of evaluations. Their answer to the latter question typically claims some interesting disanalogy between those kinds of evaluations and descriptions of the world. It does so in hope of providing traction against naturalism-inspired ontological and epistemological worries threatening more ‘realist’ positions. This is a familiar position regarding ethical discourse; however, some (...) authors have recently defended a similar view regarding epistemic discourse. Others have argued that epistemic expressivism faces special problems, not necessarily attaching to expressivism about other areas. Their arguments differ in interesting ways, but the common strategy is an attempt to show that the very sort of meta-epistemological theorizing needed to articulate and establish epistemic expressivism involves the epistemic expressivist in some sort of internal incoherence or self-defeat. That is, they think that articulating or defending the position requires implicit commitment to the negation of one of the positions core tenets. This paper responds to those arguments on behalf of epistemic expressivism, suggesting that they each misunderstand what is crucial to epistemic expressivism. By responding to these arguments, we hope to achieve more clarity about what epistemic expressivism is and why one might want to endorse it in a meta-epistemology. (shrink)

The meaning of definite descriptions (like ‘the King of France’, ‘the girl’, etc.) has been a central topic in philosophy and linguistics for the past century. Indefinites (‘Something is on the floor’, ‘A child sat down’, etc.) have been relatively neglected in philosophy, under the Russellian assumption that they can be unproblematically treated as existential quantifiers. However, an important tradition, drawing from Stoic logic, has pointed to patterns which suggest that indefinites cannot be treated simply as existential quantifiers. The standard (...) dynamic semantic treatment of those phenomena, however, has well-known problems with negation and disjunction. -/- In this paper I develop a new approach to (in)definites. On my theory, truth-conditions are classical. But in addition to truth-conditions, meanings comprise a second dimension of what I call bounds. It is at the level of bounds, not truth-conditions, that I locate the characteristically dynamic coordination between indefinites and definites. The resulting system thus has a classical logic. This approach avoids dynamic semantics’ logical problems, and, more generally, yields a new perspective on the relation between truth-conditional and dynamic effects in natural language. (shrink)

Deflationists about truth hold that the function of the truth predicate is to enable us to make certain assertions we could not otherwise make. Pragmatists claim that the utility of negation lies in its role in registering incompatibility. The pragmatist insight about negation has been successfully incorporated into bilateral theories of content, which take the meaning of negation to be inferentially explained in terms of the speech act of rejection. We implement the deflationist insight in a bilateral (...) theory by taking the meaning of the truth predicate to be explained by its inferential relation to assertion. We combine this account of the meaning of the truth predicate with a new diagnosis of the Liar Paradox: its derivation requires the truth rules to preserve evidence, but these rules only preserve commitment. The result is a novel inferential deflationist theory of truth, which solves the Liar Paradox in a principled manner. We end by showing that our theory and simple extensions thereof have the resources to axiomatize the internal logic of several supervaluational hierarchies, including Cantini’s. This solves open problems of Halbach (2011) and Horsten (2011). (shrink)

All humans have an equal basic moral status. They possess the same fundamental rights, and the comparable interests of each person should count the same in calculations that determine social policy. Neither supposed racial differences, nor skin color, sex, sexual orientation, ethnicity, intelligence, nor any other differences among humans negate their fundamental equal worth and dignity. These platitudes are virtually universally affirmed. A white supremacist racist or an admirer of Adolf Hitler who denies them is rightly regarded as beyond the (...) pale of civilized dialogue. However, a very simple line of argument developed by Peter Singer challenges our understanding of these platitudes and forces us to rethink the basis and nature of the moral equality of all humans. What I call the “Singer Problem” arises if one accepts that the morally significant cognitive capacities that are relevant to the determination of the fundamental moral status of a being vary from individual to individual by degree. The problem is to specify a moral principle determining fundamental moral status that assigns a superior status to humans compared to other animals on the basis of the superior cognitive capacities of humans but also assigns all human persons an equal fundamental moral status regardless of their differing cognitive capacities. More broadly posed, the problem is to specify moral principles that yield intuitively satisfactory implications for the treatment of human individuals and other individual animals given that cognitive capacities differ across species and individuals. A search for a resolution of the Singer problem leads to disappointing results. (shrink)

Although arguments for and against competing theories of vagueness often appeal to claims about the use of vague predicates by ordinary speakers, such claims are rarely tested. An exception is Bonini et al. (1999), who report empirical results on the use of vague predicates by Italian speakers, and take the results to count in favor of epistemicism. Yet several methodological difficulties mar their experiments; we outline these problems and devise revised experiments that do not show the same results. We then (...) describe three additional empirical studies that investigate further claims in the literature on vagueness: the hypothesis that speakers confuse ‘P’ with ‘definitely P’, the relative persuasiveness of different formulations of the inductive premise of the Sorites, and the interaction of vague predicates with three different forms of negation. (shrink)

Dâwûd al-Qarisî (Dâvûd al-Karsî) was a versatile and prolific 18th century Ottoman scholar who studied in İstanbul and Egypt and then taught for long years in various centers of learning like Egypt, Cyprus, Karaman, and İstanbul. He held high esteem for Mehmed Efendi of Birgi (Imâm Birgivî/Birgili, d.1573), out of respect for whom, towards the end of his life, Karsî, like Birgivî, occupied himself with teaching in the town of Birgi, where he died in 1756 and was buried next to (...) Birgivî. Better known for his following works on Arabic language and rhetoric and on the prophetic traditions (hadith): Sharḥu uṣûli’l-ḥadîth li’l-Birgivî; Sharḥu’l-Ḳaṣîdati’n-nûniyya (two commentaries, in Arabic and Turkish); Şarḥu’l-Emsileti’l-mukhtalifa fi’ṣ-ṣarf (two commentaries, in Arabic and Turkish); Sharḥu’l-Binâʾ; Sharḥu’l-ʿAvâmil; and Sharḥu İzhâri’l-asrâr, Karsî has actually composed textbooks in quite different fields. Hence the hundreds of manuscript copies of his works in world libraries. Many of his works were also recurrently printed in the Ottoman period. One of the neglected aspects of Karsî is his identity as a logician. Although he authored ambitious and potent works in the field of logic, this aspect of him has not been subject to modern studies. Even his bibliography has not been established so far (with scattered manuscript copies of his works and incomplete catalogue entries). This article primarily and in a long research based on manuscript copies and bibliographic sources, identifies twelve works on logic that Karsî has authored. We have clarified the works that are frequently mistaken for each other, and, especially, have definitively established his authorship of a voluminous commentary on al-Kâtibî’s al-Shamsiyya, of which commentary a second manuscript copy has been identified and described together with the other copy. Next is handled his most famous work of logic, the Sharhu Îsâghûcî, which constitutes an important and assertive ring in the tradition of commentaries on Îsâghûcî. We describe in detail the nine manuscript copies of this work that have been identified in various libraries. The critical text of Karsî’s Sharhu Îsâghûcî, whose composition was finished on 5 March 1745, has been prepared based on the following four manuscripts: (1) MS Kayseri Raşid Efendi Kütüphanesi, No. 857, ff.1v-3v, dated 1746, that is, only one year after the composition of the work; (2) MS Bursa İnebey Yazma Eser Kütüphanesi, Genel, No.794B, ff.96v-114v, dated 1755; (3) MS Millet Kütüphanesi, Ali Emiri Efendi Arapça, No. 1752, ff.48v-58r, dated 1760; (4) MS Beyazıt Yazma Eser Kütüphanesi, Beyazıt, No. 3129, ff.41v-55v, dated 8 March 1772. While preparing the critical text, we have applied the Center for Islamic Studies (İslam Araştırmaları Merkezi, İSAM)’s method of optional text choice. The critical text is preceded by a content analysis. Karsî is well aware of the preceding tradition of commentary on Îsâghûcî, and has composed his own commentary as a ‘simile’ or alternative to the commentary by Mollâ Fanârî which was famous and current in his own day. Karsî’s statement “the commentary in one day and one night” is a reference to Mollâ Fanârî who had stated that he started writing his commentary in the morning and finished it by the evening. Karsî, who spent long years in the Egyptian scholarly and cultural basin, adopted the religious-sciences-centered ‘instrumentalist’ understanding of logic that was dominant in the Egypt-Maghrib region. Therefore, no matter how famous they were, he criticized those theoretical, long, and detailed works of logic which mingled with philosophy; and defended and favored authoring functional and cogent logic texts that were beneficial, in terms of religious sciences, to the seekers of knowledge and the scholars. Therefore, in a manner not frequently encountered in other texts of its kind, he refers to the writings and views of Muhammad b. Yûsuf al-Sanûsî (d.1490), the great representative of this logical school in the Egyptian-Maghrib region. Where there is divergence between the views of the ‘earlier scholars’ (mutaqaddimûn) like Ibn Sînâ and his followers and the ‘later scholars’ (muta’akhkhirûn), i.e., post-Fakhr al-dîn al-Râzî logicians, Karsî is careful to distance himself from partisanship, preferring sometimes the views of the earliers, other times those of the laters. For instance, on the eight conditions proposed for the realization of contradiction, he finds truth to be with al-Fârâbî, who proposed “unity in the predicative attribution” as the single condition for the realization of contradiction. Similarly, on the subject matter of Logic, he tried to reconcile the mutaqaddimûn’s notion of ‘second intelligibles’ with the muta’akhkhirûn’s notion of ‘apprehensional and declarational knowledge,’ suggesting that not much difference exists between the two, on the grounds that both notions are limited to the aspect of ‘known things that lead to the knowledge of unknown things.’ Karsî asserts that established and commonly used metaphors have, according to the verifying scholars, signification by correspondence (dalâlat al-mutâbaqah), adding also that it should not be ignored that such metaphors may change from society to society and from time to time. Karsî also endorses the earlier scholars’ position concerning the impossibility of quiddity (mâhiyya) being composed of two co-extensive parts, and emphasizes that credit should not be given to later scholars’ position who see it possible. According to the verifying scholars (muhaqqîqûn), it is possible to make definition (hadd) by mentioning only difference (fasl), in which case it becomes an imperfect definition (hadd nâqis). He is of the opinion that the definition of the proposition (qadiyya) in al-Taftâzânî’s Tahdhîb is clearer and more complete: “a proposition is an expression that bears the possibility of being true or false”. He states that in the division of proposition according to quantity what is taken into consideration is the subject (mawdû‘) in categorical propositions, and the temporal aspect of the antecedent (muqaddam) in hypothetical propositions. As for the unquantified, indefinite proposition (qadiyya muhmalah), Karsî assumes that if it is not about the problems of the sciences, then it is virtually/potentially a particular proposition (qadiyya juz’iyyah); but if it is about the problems of the sciences, then it is virtually/potentially a universal proposition (qadiyya kulliyyah). This being the general rule about the ambiguous (muhmal) propositions, he nevertheless contends that, because its subject (mawdû‘) is negated, it is preferable to consider a negative ambiguous (sâliba muhmalah) proposition like “human (insân) is not standing” to be a virtually/potentially universal negative (sâliba kulliyyâh) proposition. He states that a disjunctive hypothetical proposition (shartiyya al-munfasila) that is composed of more than two parts/units is only seemingly so, and that in reality it cannot be composed of more than two units. Syllogism (qiyâs), according to Karsî, is the ultimate purpose (al-maqsad al-aqsâ) and the most valuable subject-matter of the science of Logic. For him, the entire range of topics that are handled before this one are only prolegomena to it. This approach of Karsî clearly reveals how much the ‘demonstration (burhân)-centered’ approach of the founding figures of the Muslim tradition of logic like al-Fârâbî and Ibn Sînâ has changed. al-Abharî, in his Îsâghûjî makes no mention of ‘conversion by contradiction’ (‘aks al-naqîd). Therefore, Karsî, too, in his commentary, does not touch upon the issue. However, in his Îsâghûjî al-jadîd Karsî does handle the conversion by contradiction and its rules. Following the method of Îsâghûjî, in his commentary Karsî shortly touches on the four figures (shakl) of conjuctive syllogism (qiyâs iqtirânî) and their conditions, after which he passes to the first figure (shakl), which is considered ‘the balance of the sciences’ (mi‘yâr al-‘ulûm), explaining the four moods (darb) of it. In his Îsâghûjî al-jadîd, however, Karsî handles all the four figures (shakl) with all their related moods (darb), where he speaks of fife moods (darb) of the fourth figure (shakl). The topic of ‘modal propositions’ (al-muwajjahât) and of ‘modal syllogism’ (al-mukhtalitât), both of which do not take place in the Îsâghûjî, are not mentioned by Karsî as well, either in his commentary on Îsâghûjî or in his Îsâghûjî al-jadîd. Karsî proposes that the certainties (yaqîniyyât), of which demonstration (burhân) is made, have seven, not six, divisions. After mentioning (1) axioms/first principles (awwaliyyât), (2) observata/sensuals (mushâhadât), (3) experta/empiricals (mujarrabât), (4) acumenalia (hadthiyyât), (5) testata (mutawâtirât), and (6) instictives (fitriyyât), that is, all the ‘propositions accompanied by their demonstrations,’ Karsî states that these six divisions, which do not need research and reflection (nazar), are called badîhiyyât (self-evidents), and constitute the foundations (usûl) of certainties (yaqîniyyât). As the seventh division he mentions (7) the nazariyyât (theoreticals), which are known via the badîhiyyât, end up in them, and therefore convey certainty (yaqîn). For Karsî, the nazariyyât/theoreticals, which constitute the seventh division of yaqîniyyât/certainties, are too numerous, and constitute the branches (far‘) of yaqîniyyât. Every time the concept of ‘Mughâlata’ (sophistry) comes forth in the traditional sections on the five arts usually appended to logic works, Karsî often gives examples from what he sees as extreme sûfî sayings, lamenting that these expressions are so widespread and held in esteem. He sometimes criticizes these expressions. However, it is observed that he does not reject tasawwuf in toto, but excludes from his criticism the mystical views and approaches of the truth-abiding (ahl al-haqq), shârî‘â-observant (mutasharri‘) leading sufis who have reached to the highest level of karâmah. (shrink)

I show that the Lottery Paradox is just a version of the Sorites, and argue that this should modify our way of looking at the Paradox itself. In particular, I focus on what I call “the Cut-off Point Problem” and contend that this problem, well known by Sorites scholars, ought to play a key role in the debate on Kyburg’s puzzle. Very briefly, I show that, in the Lottery Paradox, the premises “ticket n°1 will lose”, “ticket n°2 will (...) lose”… “ticket n°1000 will lose” are equivalent to soritical premises of the form “~(the winning ticket is in {…, (tn)}) ⊃ ~(the winning ticket is in {…, tn, (tn + 1)})” (where “⊃” is the material conditional, “~” is the negation symbol, “tn” and “tn + 1” are “ticket n°n” and “ticket n°n + 1” respectively, and “{}” identify the elements of the lottery tickets’ set. The brackets in “(tn)” and “(tn + 1)” are meant to point out that in the antecedent of the conditional we do not always have a “tn” (and, as a result, a “tn + 1” in the consequent): consider the conditional “~(the winning ticket is in {}) ⊃ ~(the winning ticket is in {t1})”). As a result, failing to believe, for some ticket, that it will lose comes down to introducing a cut-off point in a chain of soritical premises. In this paper I explore the consequences of the different ways of blocking the Lottery Paradox with respect to the Cut-off Point Problem. A heap variant of the Lottery Paradox is especially relevant for evaluating the different solutions. One important result is that the most popular way out of the puzzle, i.e., denying the Lockean Thesis, becomes less attractive. Moreover, I show that, along with the debate on whether rational belief is closed under classical logic, the debate on the validity of modus ponens should play an important role in discussions on the Lottery Paradox. -/- . (shrink)