Quasi Indexicals

I argue that not all context dependent expressions are alike. Pure (or ordinary) indexicals behave more or less as Kaplan thought. But quasi indexicals behave in some ways like indexicals and in other ways not like indexicals. A quasi indexical sentence ϕ allows for cases in which one party utters ϕ and the other its negation, and neither party's claim has to be false. In this sense, quasi indexicals are like pure indexicals (think: “I am a doctor”/“I am not a doctor” as uttered by different individuals). In such cases involving a pure indexical sentence, it is not appropriate for the two parties to reject each other's claims by saying, “No.” However, in such cases involving a quasi indexical sentence, it is appropriate for the parties to reject each other's claims. In this sense, quasi indexicals are not like pure indexicals. Drawing on experimental evidence, I argue that gradable adjectives like “rich” are quasi indexicals in this sense. The existence of quasi indexicals raises trouble for many existing theories of context dependence, including standard contextualist and relativist theories. I propose an alternative semantic and pragmatic theory of quasi indexicals, negotiated contextualism, that combines insights from Kaplan 1989 and Lewis 1979. On my theory, rejection is licensed with quasi indexicals (even when neither of the claims involved has to be false) because the two utterances involve conflicting proposals about how to update the conversational score. I also adduce evidence that conflicting truth value assessments of a single quasi indexical utterance exhibit the same behavior. I argue that negotiated contextualism can account for this puzzling property of quasi indexicals as well.     

The Analytic Truth and Falsity of Disjunctions

Disjunctive inferences are difficult. According to the theory of mental models, it is because of the alternative possibilities to which disjunctions refer. Three experiments corroborated further predictions of the mental model theory. Participants judged that disjunctions, such as Either this year is a leap year or it is a common year are true. Given a disjunction such as Either A or B, they tended to evaluate the four cases in its ‘partition’: A and B, A and not‐B, not‐A and B, not‐A and not‐B, as ‘possible’ or ‘impossible’ in ways that bore out the difference between inclusive disjunctions (‘or both’) and exclusive disjunctions (‘but not both’). Knowledge usually concerns what is true, and so when participants judge that a disjunction is false, or contingent, and evaluate the cases in its partition, they depend on inferences that yield predictable errors. They tended to judge that disjunctions, such as follows: Either the food is cold or else it is tepid, but not both, are true, though in fact they could be false. They tended to infer ‘mirror‐image’ evaluations that yield the same possibilities for false disjunctions as those for true disjunctions. The article considers the implications of these results for alternative theories based on classical logic or on the probability calculus.     

Internal Negation and the Principles of Non-Contradiction and of Excluded Middle in Aristotle

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

The Compositionality of Logical Connectives in Child Italian

This paper investigates the interpretation that Italian-speaking children and adults assign to negative sentences with disjunction and negative sentences with conjunction. The aim of the study was to determine whether children and adults assign the same interpretation to these types of sentences. The Semantic Subset Principle (SSP) (Crain et al., in: Clifton, Frazer, Rayner (eds) Perspective on sentence processing, Lawrence Erlbaum, Hillside, 1994) predicts that children’s initial scope assignment should correspond to the interpretation that makes sentences true in the narrowest range of circumstances, even when this is not the interpretation assigned by adults. This prediction was borne out in previous studies in Japanese, Mandarin and Turkish. As predicted by the SSP, the findings of the present study indicate that Italian-speaking children and adults assign the same interpretation to negative sentences with conjunction (conjunction takes scope over negation). By contrast, the study revealed that some children differed from adults in the interpretation they assigned to negative sentences with disjunction. Adults interpreted disjunction as taking scope over negation, whereas children were divided into two groups: one group interpreted disjunction as taking scope over negation as adults did; another group interpreted negation as taking scope over disjunction, as predicted by the SSP. To explain the findings, we propose that Italian-speaking children initially differ from adults as dictated by the SSP, but children converge on the adult grammar earlier than children acquiring other languages due to the negative concord status of Italian, including the application of negative concord to sentences with disjunction.     

Some definitions of negation leading to paraconsistent logics

In positive logic the negation of a propositionA is defined byA X whereX is some fixed proposition. A number of standard properties of negation, includingreductio ad absurdum, can then be proved, but not the law of noncontradiction so that this forms a paraconsistent logic. Various stronger paraconsistent logics are then generated by putting in particular propositions forX. These propositions range from true through contingent to false.     

The Secret of My Success

In an information state where various agents have both factual knowledge and knowledge about each other, announcements can be made that change the state of information. Such informative announcements can have the curious property that they become false because they are announced. The most typical example of that is ‘fact p is true and you don’t know that’, after which you know that p, which entails the negation of the announcement formula. The announcement of such a formula in a given information state is called an unsuccessful update. A successful formula is a formula that always becomes common knowledge after being announced. Analysis of information systems and ‘philosophical puzzles’ reveals a growing number of dynamic phenomena that can be described or explained by unsuccessful updates. This increases our understanding of such philosophical problems. We also investigate the syntactic characterization of the successful formulas. An erratum to this article is available at .     

Fuzzy intuitionistic quantum logics

Fuzzy intuitionistic quantum logics (called also Brouwer-Zadeh logics) represent to non standard version of quantum logic where the connective not is split into two different negation: a fuzzy-like negation that gives rise to a paraconsistent behavior and an intuitionistic-like negation. A completeness theorem for a particular form of Brouwer-Zadeh logic (BZL ³) is proved. A phisical interpretation of these logics can be constructed in the framework of the unsharp approach to quantum theory.     

Answer Sets and Qualitative Decision Making

Logic programs under answer set semantics have become popular as a knowledge representation formalism in Artificial Intelligence. In this paper we investigate the possibility of using answer sets for qualitative decision making. Our approach is based on an extension of the formalism, called logic programs with ordered disjunction (LPODs). These programs contain a new connective called ordered disjunction. The new connective allows us to represent alternative, ranked options for problem solutions in the heads of rules: A × B intuitively means: if possible A, but if A is not possible then at least B. The semantics of logic programs with ordered disjunction is based on a preference relation on answer sets. We show that LPODs can serve as a basis for qualitative decision making.     

The Craig interpolation theorem for prepositional logics with strong negation

This paper deals with, prepositional calculi with strong negation (N-logics) in which the Craig interpolation theorem holds. N-logics are defined to be axiomatic strengthenings of the intuitionistic calculus enriched with a unary connective called strong negation. There exists continuum of N-logics, but the Craig interpolation theorem holds only in 14 of them.     

Field's Paradox and Its Medieval Solution

Hartry Field's revised logic for the theory of truth in his new book, Saving Truth from Paradox, seeking to preserve Tarski's T-scheme, does not admit a full theory of negation. In response, Crispin Wright proposed that the negation of a proposition is the proposition saying that some proposition inconsistent with the first is true. For this to work, we have to show that this proposition is entailed by any proposition incompatible with the first, that is, that it is the weakest proposition incompatible with the proposition whose negation it should be. To show that his proposal gave a full intuitionist theory of negation, Wright appealed to two principles, about incompatibility and entailment, and using them Field formulated a paradox of validity (or more precisely, of inconsistency).

The medieval mathematician, theologian and logician, Thomas Bradwardine, writing in the fourteenth century, proposed a solution to the paradoxes of truth which does not require any revision of logic. The key principle behind Bradwardine's solution is a pluralist doctrine of meaning, or signification, that propositions can mean more than they explicitly say. In particular, he proposed that signification is closed under entailment. In light of this, Bradwardine revised the truth-rules, in particular, refining the T-scheme, so that a proposition is true only if everything that it signifies obtains. Thereby, he was able to show that any proposition which signifies that it itself is false, also signifies that it is true, and consequently is false and not true. I show that Bradwardine's solution is also able to deal with Field's paradox and others of a similar nature. Hence Field's logical revisions are unnecessary to save truth from paradox.     

Mistaking the instance for the rule: A critical analysis of the truth-table evaluation paradigm

Many studies probe for interpretations of < if A then C> by having people evaluate truth-table cases (<A and C>, < A and not-C>, < not-A and C>, < not-A and not-C>) as making the rule true or false, or being irrelevant. We argue that a single case can never prove a general rule to be true, as philosophy of science has taught any researcher. Giving participants the impossible “true” option would therefore bias results away from this response. In Experiment 1 people judged instead whether cases make a rule false, do not make the rule false, or are irrelevant to the rule. The experimental group (N = 44) showed a significant increase in not-false responses compared with true responses of the control group (N = 39). In Experiments 2 and 3 the experimental groups judged whether cases make a rule true, corroborate it (i.e., make the rule more plausible, but neither true nor false), make it false, or are irrelevant. There was a significant reduction of irrelevant responses as compared to the default true/false/irrelevant task for the control groups. Even < A and C> cases were often no longer considered to make an < if A then C> rule true and were correctly judged to corroborate (vs. verify) rules. Results corroborate our conceptual analyses of the unsuitable “true” response option and put into question arguments that hinge on the presumed likelihood by which people consider truth contingencies to make a rule “true”.     

Drawing what lies ahead: False intentions are more abstractly depicted than true intentions

The aim of this study was to examine how people mentally represent and depict true and false statements about claimed future actions—so‐called true and false intentions. On the basis of construal level theory, which proposes that subjectively unlikely events are more abstractly represented than likely ones, we hypothesized that false intentions should be represented at a more abstract level than true intentions. Fifty‐six hand drawings, produced by participants to describe mental images accompanying either true or false intentions, were rated on level of abstractness by a second set of participants (N = 117) blind to the veracity of the intentions. As predicted, drawings of false intentions were rated as more abstract than drawings of true intentions. This result advances the use of drawing‐based deception detection techniques to the field of true and false intentions and highlights the potential for abstractness as a novel cue to deceit.     

An Extended Lewis/Stalnaker Semantics and the New Problem of Counterpossibles

Abstract

Closest-possible-world analyses of counterfactuals suffer from what has been called the ‘problem of counterpossibles’: some counterfactuals with metaphysically impossible antecedents seem plainly false, but the proposed analyses imply that they are all (vacuously) true. One alleged solution to this problem is the addition of impossible worlds. In this paper, I argue that the closest possible or impossible world analyses that have recently been suggested suffer from the ‘new problem of counterpossibles’: the proposed analyses imply that some plainly true counterpossibles (viz., ‘counterlogicals’) are false. After motivating and presenting the ‘new problem’, I give reasons to think that the most plausible objection to my argument is not compelling.     

Fuzzy logic and approximate reasoning

The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, ?, of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in ? as a fuzzy subset of [0, 1]. Since ? is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in ? requires, in general, a linguistic approximation by a truth-value in ?. As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc. Approximate reasoning is viewed as a process of approximate solution of a system of relational assignment equations. This process is formulated as a compositional rule of inference which subsumes modus ponens as a special case. A characteristic feature of approximate reasoning is the fuzziness and nonuniqueness of consequents of fuzzy premisses. Simple examples of approximate reasoning are: (a) Most men are vain; Socrates is a man; therefore, it is very likely that Socrates is vain. (b) x is small; x and y are approximately equal; therefore y is more or less small, where italicized words are labels of fuzzy sets.     

Modus Tollens,Modus Shmollens: Contrapositive reasoning and the pragmatics of negation

The utterance of a negative statement invites the pragmatic inference that some reason exists for the proposition it negates to be true; this pragmatic inference paves the way for the logically unexpected Modus Shmollens inference: “If p then q; not-q; therefore, p.” Experiment 1 shows that a majority of reasoners endorse Modus Shmollens from an explicit major conditional premise and a negative utterance as a minor premise: e.g., reasoners conclude that “the soup tastes like garlic” from the premises “If a soup tastes like garlic, then there is garlic in the soup; Carole tells Didier that there is no garlic in the soup they are eating.” Experiment 2 shows that this effect is mediated by the derivation of a pragmatic inference from negation. We discuss how theories of conditional reasoning can integrate such a pragmatic effect.     

Complexity Results for Modal Dependence Logic

Modal dependence logic was introduced recently by Väänänen. It enhances the basic modal language by an operator = (). For propositional variables p ₁, . . . , p _n , = (p ₁, . . . , p _n-1, p _n ) intuitively states that the value of p _n is determined by those of p ₁, . . . , p _n-1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfiability for poor man’s dependence logic, the language consisting of formulas built from literals and dependence atoms using ${\wedge, \square, \lozenge}$ (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend Väänänen’s language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satisfiability is complete for the second level of the polynomial hierarchy. Additionally we consider the restriction of modal dependence logic where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the satisfiability problem for this bounded arity dependence logic is PSPACE-complete and that the complexity drops to the third level of the polynomial hierarchy if we then disallow disjunction. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Väänänen and Sevenster.     

"This is Simply What I Do"

Wittgenstein's discussion of rule‐following is widely regarded to have identified what Kripke called “the most radical and original sceptical problem that philosophy has seen to date”. But does it? This paper examines the problem in the light of Charles Peirce's distinctive scientific hierarchy. Peirce identifies a phenomenological inquiry which is prior to both logic and metaphysics, whose role is to identify the most fundamental philosophical categories. His third category, particularly salient in this context. pertains to general predication. Rule‐following scepticism, the paper suggests, results from running together two questions: “How is it that I can project rules?”, and, “What is it for a given usage of a rule to be right?”. In Peircean terms the former question, concerning the irreducibility of general predication (to singular reference), must be answered in phenomenology, while the latter, concerning the difference between true and false predication, is answered in logic. A failure to appreciate this distinction, it is argued, has led philosophers to focus exclusively on Wittgenstein's famous public account of rule‐following rightness, thus overlooking a private, phenomenological dimension to Wittgenstein's remarks on following a rule which gives the lie to Kripke's reading of him as a sceptic.     

Double-Negation Elimination in Some Propositional Logics

This article answers two questions (posed in the literature), each concerning the guaranteed existence of proofs free of double negation. A proof is free of double negation if none of its deduced steps contains a term of the formn(n(t)) for some term t, where n denotes negation. The first question asks for conditions on the hypotheses that, if satisfied, guarantee the existence of a double-negation-free proof when the conclusion is free of double negation. The second question asks about the existence of an axiom system for classical propositional calculus whose use, for theorems with a conclusion free of double negation, guarantees the existence of a double-negation-free proof. After giving conditions that answer the first question, we answer the second question by focusing on the Lukasiewicz three-axiom system. We then extend our studies to infinite-valued sentential calculus and to intuitionistic logic and generalize the notion of being double-negation free. The double-negation proofs of interest rely exclusively on the inference rule condensed detachment, a rule that combines modus ponens with an appropriately general rule of substitution. The automated reasoning program Otter played an indispensable role in this study.     

Combining linear-time temporal logic with constructiveness and paraconsistency

It is known that linear-time temporal logic (LTL), which is an extension of classical logic, is useful for expressing temporal reasoning as investigated in computer science. In this paper, two constructive and bounded versions of LTL, which are extensions of intuitionistic logic or Nelson's paraconsistent logic, are introduced as Gentzen-type sequent calculi. These logics, $\mathrm{IB}[l]$ and $\mathrm{PB}[l]$, are intended to provide a useful theoretical basis for representing not only temporal (linear-time), but also constructive, and paraconsistent (inconsistency-tolerant) reasoning. The time domain of the proposed logics is bounded by a fixed positive integer. Despite the restriction on the time domain, the logics can derive almost all the typical temporal axioms of LTL. As a merit of bounding time, faithful embeddings into intuitionistic logic and Nelson's paraconsistent logic are shown for $\mathrm{IB}[l]$ and $\mathrm{PB}[l]$, respectively. Completeness (with respect to Kripke semantics), cut–elimination, normalization (with respect to natural deduction), and decidability theorems for the newly defined logics are proved as the main results of this paper. Moreover, we present sound and complete display calculi for $\mathrm{IB}[l]$ and $\mathrm{PB}[l]$.In [P. Maier, Intuitionistic LTL and a new characterization of safety and liveness, in: Proceedings of Computer Science Logic 2004, in: Lecture Notes in Computer Science, vol. 3210, Springer-Verlag, Berlin, 2004, pp. 295–309] it has been emphasized that intuitionistic linear-time logic (ILTL) admits an elegant characterization of safety and liveness properties. The system ILTL, however, has been presented only in an algebraic setting. The present paper is the first semantical and proof-theoretical study of bounded constructive linear-time temporal logics containing either intuitionistic or strong negation.     

Measurement theory in linguistics

This paper presents a novel semantic analysis of unit names (like pound and meter) and gradable adjectives (like tall, short and happy), inspired by measurement theory (Krantz et al. In Foundations of measurement: Additive and Polynomial Representations, 1971). Based on measurement theory’s four-way typology of measures, I claim that different adjectives are associated with different types of measures whose special characteristics, together with features of the relations denoted by unit names, explain the puzzling limited distribution of measure phrases, as well as unit-based comparisons between predicates (as in the table is longer than it is wide). All considered, my analyses support the view that the grammar of natural languages is sensitive to features of measurement theory.