The conjunction fallacy occurs whenever probability compounds are thought of as more likely than its component probabilities alone. In the experiment we present, subjects chose between simple and compound lotteries after some practice. Depending on the condition, they were given more or less information about the nature of probability compounds. The conjunction fallacy was surprisingly robust. There was, however, a puzzling dissociation between verbal and behavioral learning: verbal responses were sensitive, but actual choices entirely insensitive, to the amount (...) of verbal instructions being provided. This might reflect a dichotomy between implicit and explicit learning. Caution must be exercised in generalizing results from what people say to what people do. (shrink)

Subjects were requested to choose between gambles, where the outcome of one gamble depended on a single elementary event, and the other depended on an event compounded of a series of such elementary events. The data supported the hypothesis that the subjective probability of a compound event is systematically biased in the direction of the probability of its components resulting in overestimation of conjunctive events and underestimation of disjunctive events. Studies pertaining to this topic are discussed.

There is wide support in logic, philosophy, and psychology for the hypothesis that the probability of the indicative conditional of natural language, P(if A then B), is the conditional probability of B given A, P(B|A). We identify a conditional which is such that P(if A then B)=P(B|A) with de Finetti’s conditional event, B | A. An objection to making this identification in the past was that it appeared unclear how to form compounds and iterations of conditional events. In (...) this paper, we illustrate how to overcome this objection with a probabilistic analysis, based on coherence, of these compounds and iterations. We interpret the compounds and iterations as conditional random quantities, which sometimes reduce to conditional events, given logical dependencies. We also show, for the first time, how to extend the inference of centering for conditional events, inferring B|A from the conjunction A ^ B, to compounds and iterations of both conditional events and biconditional events, B || A, and generalize it to n-conditional events. (shrink)

The problem of probabilities of conditionals is one of the long-standing puzzles in philosophy of language. We defend and update Adams' solution to the puzzle: the probability of an epistemic conditional is not the probability of a proposition, but a probability under a supposition. -/- Close inspection of how a triviality result unfolds in a concrete scenario does not provide counterexamples to the view that probabilities of conditionals are conditional probabilities: instead, it supports the conclusion that probabilities (...) of conditionals violate standard probability theory. -/- This does not call into question probability theory per se; rather, it calls for a more careful understanding of its role: probability theory is a theory of probabilities of propositions; but as conditionals do not express propositions, their probabilities are not subject to the standard laws. -/- We argue that both conditional probabilities and probabilities of conditionals are best understood in terms of the dynamics of supposing, modeled as a restriction operation on a probability space. This version of the suppositionalist view allows us to connect Adams' Thesis to the widely held restrictor view of the semantics of conditionals. -/- We address two common objections to Adams' view: that the relevant probabilities are ‘probabilities only in name’, and that giving up conditional propositions puts us at a disadvantage when it comes to interpreting compounds. -/- Finally, we argue that some putative counterexamples to Adams' Thesis can be diagnosed as fallacies of probabilistic reasoning: they arise from applying to conditionals laws of standard probability theory which are invalid for them. (shrink)

Plato's fondness for words compounded with παν- is obvious at the most cursory reading of his works; this characteristic of his style becomes even more striking when his use of these words is compared with their frequency in earlier authors. An investigation of Platonic usage in this respect, relatively easy since the publication of Leonard Brandwood's Word Index to Plato , yields interesting results. Whether the effect of the παν-prefix is intensive or determinative , Plato has a tendency to associate (...) these words with some sort of disapproval; this disapproval is sometimes explicit enough and can sometimes be inferred from the use of the word, or of a word related to it, in other contexts. The words may be used ironically, as πγκαλος often is and πσσοος always. Another sort of disapproval springs from what may be called Plato's general dislike of (promiscuous plurality, excess and variety; for a philosopher who believes in single, unchanging Forms there is something intrinsically objectionable in such words as πμπολυς and παντοδαπς. It also transpires that Plato may have coined a number, of these words and that he was probably the first prose writer to import others from poetry; in the face of the fragmentary nature of surviving Greek literature it would be unwise to be more dogmatic. The following, somewhat dry, survey will, it is hoped, throw some light on the usage of these interesting words. (shrink)

Plato's fondness for words compounded with παν- is obvious at the most cursory reading of his works; this characteristic of his style becomes even more striking when his use of these words is compared with their frequency in earlier authors. An investigation of Platonic usage in this respect, relatively easy since the publication of Leonard Brandwood's Word Index to Plato, yields interesting results. Whether the effect of the παν-prefix is intensive or determinative, Plato has a tendency to associate these words (...) with some sort of disapproval; this disapproval is sometimes explicit enough and can sometimes be inferred from the use of the word, or of a word related to it, in other contexts. The words may be used ironically, as πγκαλος often is and πσσοος always. Another sort of disapproval springs from what may be called Plato's general dislike of (promiscuous plurality, excess and variety; for a philosopher who believes in single, unchanging Forms there is something intrinsically objectionable in such words as πμπολυς and παντοδαπς. It also transpires that Plato may have coined a number, of these words and that he was probably the first prose writer to import others from poetry; in the face of the fragmentary nature of surviving Greek literature it would be unwise to be more dogmatic. The following, somewhat dry, survey will, it is hoped, throw some light on the usage of these interesting words. (shrink)

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single events, (...) for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

We present two approaches to investigate the validity of connexive principles and related formulas and properties within coherence-based probability logic. Connexive logic emerged from the intuition that conditionals of the form if not-A, thenA, should not hold, since the conditional’s antecedent not-A contradicts its consequent A. Our approaches cover this intuition by observing that the only coherent probability assessment on the conditional event $${A| \overline{A}}$$ A | A ¯ is $${p(A| \overline{A})=0}$$ p ( A | A ¯ ) (...) = 0. In the first approach we investigate connexive principles within coherence-based probabilistic default reasoning, by interpreting defaults and negated defaults in terms of suitable probabilistic constraints on conditional events. In the second approach we study connexivity within the coherence framework of compound conditionals, by interpreting connexive principles in terms of suitable conditional random quantities. After developing notions of validity in each approach, we analyze the following connexive principles: Aristotle’s theses, Aristotle’s Second Thesis, Abelard’s First Principle, and Boethius’ theses. We also deepen and generalize some principles and investigate further properties related to connexive logic (like non-symmetry). Both approaches satisfy minimal requirements for a connexive logic. Finally, we compare both approaches conceptually. (shrink)

Professor Linsay has called my attention to the need for an examination of these compounds, with a view to ascertaining whether they are ever used in the Singular by Latin writers, or always in the Plural. From a scholium, probably by Donatus, on Aen. X. 173: insula inexhaustis Chalybum generosa metallis, has come a gloss which appears in the Ansileubus Glossary in two forms: FR 205 Frofra : insula Tirreno mari in quo ferrifodina exercentur, RU 77 Rufa: insula Tirreni maris (...) in qua ferrifodina exercetur. (shrink)

Professor Linsay has called my attention to the need for an examination of these compounds, with a view to ascertaining whether they are ever used in the Singular by Latin writers, or always in the Plural. From a scholium, probably by Donatus, on Aen. X. 173: insula inexhaustis Chalybum generosa metallis, has come a gloss which appears in the Ansileubus Glossary in two forms: FR 205 Frofra : insula Tirreno mari in quo ferrifodina exercentur, RU 77 Rufa: insula Tirreni maris (...) in qua ferrifodina exercetur. (shrink)

Adams’ thesis is generally agreed to be linguistically compelling for simple conditionals with factual antecedent and consequent. We propose a derivation of Adams’ thesis from the Lewis- Kratzer analysis of if-clauses as domain restrictors, applied to probability operators. We argue that Lewis’s triviality result may be seen as a result of inexpressibility of the kind familiar in generalized quantifier theory. Some implications of the Lewis- Kratzer analysis are presented concerning the assignment of probabilities to compounds of conditionals.

Consider chemical elements as a system, we create an undirected chemical network with 99 elements and 1916 edges from Chemspider, a website that provide search engines to collect compounds. Using this network and the network that we used in our previous work with 97 elements and 2198 edges, we found that RootedPageRank, a link prediction tool in complex network, can be used to predict potential binary compounds, because the changing trend of PageRank probability of each element in these networks (...) all follow the periodic law, despite of the difference of scale of these networks. The accuracy test indicates that at least 7 among top 10 predicted compoundss in one network can be verified using the compoundss in the other network or in other chemical database, proving that this method can be used to provide guidance in finding potential binary compounds, suggesting that we can study chemical properties from the view of complex network. (shrink)

This paper studies the extent to which probability functions are recursively definable. It proves, in particular, that the (absolute) probability of a statement A is recursively definable from a certain point on, to wit: from the (absolute) probabilities of certain atomic components and conjunctions of atomic components of A on, but to no further extent. And it proves that, generally, the probability of a statement A relative to a statement B is recursively definable from a certain point (...) on, to wit: from the probabilities relative to that very B of certain atomic components and conjunctions of atomic components of A, but again to no further extent. These and other results are extended to the less studied case where A and B are compounded from atomic statements by means of `` ∀ '' as well as `` ∼ '' and "&". The absolute probability functions considered are those of Kolmogorov and Carnap, and the relative ones are those of Kolmogorov, Carnap, Renyi, and Popper. (shrink)

Due to new innovations and changes, every language needs new words simply because there is a need for new words to name new things. It is a common occurrence for a speaker to use some words in a way that has never been used before in order to communicate directly about certain facts or ideas. When new inventions and changes come into people’s lives, there is a need to name them and talk about them. If a new word is used (...) by many speakers of the language, it will probably survive, and the same word will one day become an everyday word and enter the vocabulary of a language. This paper looks at compounding as one of the most productive word formation process in English. The term compounding refers to a process in which two or more lexemes are combined into one new word. When a word is formed by merging two or more words, each of which can be used separately, it is called a compound word. The term “word formation” has no universally accepted use. Word formation is sometimes defined as a process associated with changing the form of a word, for example, affixation, which is, in fact, the subject of morphology. In a broader sense, word formation covers the processes of creating new lexical items. In English, word formation is of great importance because this phenomenon affects the English dictionary, which in addition to borrowing from various other languages is enriched in this way. The aim of this paper was to investigate the context based vs. non-context interpretation of English compounds by EFL students in legal discourse. The findings from the test run-questionnaire showed that students of English as a foreign language found it more difficult to apply compound words in context rather than choosing an appropriate definition for them, with or without a given context. Furthermore, students scored lower when 50% of the compounds were given in context. (shrink)

Coherent-ambiguity aversion is defined within the smooth-ambiguity model as the combination of choice-ambiguity and value-ambiguity aversion. Five ambiguous decision tasks are analyzed theoretically, where an individual faces two-stage lotteries with binomial, uniform, or unknown second-order probabilities. Theoretical predictions are then tested through a 10-task experiment. In tasks 1–5, risk aversion is elicited through both a portfolio choice method and a BDM mechanism. In tasks 6–10, choice-ambiguity aversion is elicited through the portfolio choice method, while value-ambiguity aversion comes about through the (...) BDM mechanism. The behavior of over 75 % of classified subjects is in line with the KMM model in all tasks 6–10, independent of their degree of risk aversion. Furthermore, the percentage of coherent-ambiguity-averse subjects is lower in the binomial than in the uniform and in the unknown treatments, with only the latter difference being significant. The most part of coherent-ambiguity-loving subjects show a high risk aversion. (shrink)

We present probabilistic approaches to check the validity of selected connexive principles within the setting of coherence. Connexive logics emerged from the intuition that conditionals of the form If ∼A, then A, should not hold, since the conditional’s antecedent ∼A contradicts its consequent A. Our approach covers this intuition by observing that for an event A the only coherent probability assessment on the conditional event A|~A is p(A|~A)=0 . Moreover, connexive logics aim to capture the intuition that conditionals should (...) express some “connection” between the antecedent and the consequent or, in terms of inferences, validity should require some connection between the premise set and the conclusion. This intuition is covered by a number of principles, a selection of which we analyze in our contribution. We present two approaches to connexivity within coherence-based probability logic. Specifically, we analyze connections between antecedents and consequents firstly, in terms of probabilistic constraints on conditional events (in the sense of defaults, or negated defaults) and secondly, in terms of constraints on compounds of conditionals and iterated conditionals. After developing different notions of negations and notions of validity, we analyze the following connexive principles within both approaches: Aristotle’s Theses, Aristotle’s Second Thesis, Abelard’s First Principle and selected versions of Boethius’ Theses. We conclude by remarking that coherence-based probability logic offers a rich language to investigate the validity of various connexive principles. (shrink)

Compound Gauss-Markov random field model is one of Markov random field models for natural image restorations. An optimization algorithm was constructed by means of mean-field approximation, which is a familiar techniques for analyzing massive probabilistic models approximately in the statistical mechanics. Cluster variation method was proposed as an extended version of the mean-field approximation in the statistical mechanics. Though the mean-field approximation treat only the marginal probability distribution for every single pixel, the cluster variation method can take acount into (...) the correlation between pixels by treating the marginal probability distribution for every nearest neighbor pair of pixels. In this paper, we propose a newstatistical-mechanical iterative algorithm by means of the cluster variation method for natural image restorations in the compound Gauss-Markov random field model. In some numerical experiments, it is investigate howthe proposed algorithm improves the quality of restored images by comparing it with the algorithm constructed from the mean-field approximation. (shrink)

This paper calls for a re-appraisal of McGee's analysis of the semantics, logic and probabilities of indicative conditionals presented in his 1989 paper Conditional probabilities and compounds of conditionals. The probabilistic measures introduced by McGee are given a new axiomatisation built on the principle that the antecedent of a conditional is probabilistically independent of the conditional and a more transparent method of constructing such measures is provided. McGee's Dutch book argument is restructured to more clearly reveal that it introduces a (...) novel contribution to the epistemology of semantic indeterminacy, and shows that its more controversial implications are unavoidable if we want to maintain the Ramsey Test along with the standard laws of probability. Importantly, it is shown that the counterexamples that have been levelled at McGee's analysis|generating a rather wide consensus that it yields `unintuitive' or `wrong' probabilities for compounds |fail to strike at their intended target; for to honour the intuitions of the counterexamples one must either give up the Ramsey Test or the standard laws of probability. It will be argued that we need to give up neither if we take the counterexamples as further evidence that the indicative conditional sometimes allows for a non-epistemic `causal' interpretation alongside its usual epistemic interpretation. (shrink)

The article familiarizes us with Salomon Maimon, a self-taught philosopher of Jewish descent, and with his attitude to the scientific views of Immanuel Kant. Although Maimon depicted himself as a follower of Kant, the reality contradicts this picture. The conclusions drawn in the article are based on Maimon’s letters and his manuscript of A Dialogue between Kant and One of His Students discovered only recently. The abovementioned correspondence with Kant clearly shows Maimon’s position in relation to the views of the (...) philosopher from Königsberg. The letters, most of which remained unanswered, are written in a characteristic, somewhat audacious manner. Maimon begins them with glorifying Kant’s philosophy but then suggests numerous corrections concerning not only its insignificant details but also some principal issues which would radically change the whole system of Kant’s philosophy. Presumably that justifies the fact that Maimon received but one answer in the whole correspondence. Kant answered it at the instigation of his friend, Marcus Herz, with whom Maimon won favour. Later, however, Kant bore a grudge against Herzto such an extent that their own correspondence, previously frequent and very personal in character, almost ceased, becoming scarce and official. This, however, did not discourage Maimon, who in his Autobiography claimed that Kant had a high regard of him. The Jewish self-taught philosopher kept writing letters to Kant, and as he could not bear the lack of any answer, he decided to respond to the lettershimself. Kant’s would-be answers are contained in the abovementioned manuscript of A Dialogue between Kant and One of His Students, which for Maimon was a kind of compensation for Kant’s repudiation. Thetitle dialogue is between the Königsberg philosopher and a student N, whose views are identical with those of Maimon. It is hard to overlook the importunity of his attacks on Kant, whose answers are far more polite perhaps due to the fact he cannot equal him. In the dialogue, Kant contradicts himself, is lost in the arguments, and finally agrees with the statements of N despite the fact that they are quite distant from his own critical philosophy. N – that is Maimon – casts doubts concerning various aspects of Kant’s philosophy, e.g. the notion of space as a form of pure intuition, and Kant’s attempts to defend his position are quite helpless. Frequent statements such as “I do not know” or “highly probable” compound Maimon’s picture of Kant. It turns out, however, that in some of the theses contained in the dialogue Maimon was close the views of Fichte, which Kant refused as well. Perhaps, then, Maimon was a link between Kantian philosophy and the classical German idealism, a link much more important than it is generally acknowledged. (shrink)

There is wide support in logic, philosophy, and psychology for the hypothesis that the probability of the indicative conditional of natural language, P(if A then B), is the conditional probability of B given A, P(B|A). We identify a conditional which is such that P(if A then B)=P(B|A) with de Finetti's conditional event, B|A. An objection to making this identification in the past was that it appeared unclear how to form compounds and iterations of conditional events. In this paper, (...) we illustrate how to overcome this objection with a probabilistic analysis, based on coherence, of these compounds and iterations. We interpret the compounds and iterations as conditional random quantities which, given some logical dependencies, may reduce to conditional events. We show how the inference to B|A from A and B can be extended to compounds and iterations of both conditional events and biconditional events. Moreover, we determine the respective uncertainty propagation rules. Finally, we make some comments on extending our analysis to counterfactuals. (shrink)

In this note, we consider a Merton model for default risk, where the firm’s value is driven by a Brownian motion and a compound Poisson process.

This article presents a fundamental advance in the theory of mental models as an explanation of reasoning about facts, possibilities, and probabilities. It postulates that the meanings of compound assertions, such as conditionals (if) and disjunctions (or), unlike those in logic, refer to conjunctions of epistemic possibilities that hold in default of information to the contrary. Various factors such as general knowledge can modulate these interpretations. New information can always override sentential inferences; that is, reasoning in daily life is defeasible (...) (or nonmonotonic). The theory is a dual process one: It distinguishes between intuitive inferences (based on system 1) and deliberative inferences (based on system 2). The article describes a computer implementation of the theory, including its two systems of reasoning, and it shows how the program simulates crucial predictions that evidence corroborates. It concludes with a discussion of how the theory contrasts with those based on logic or on probabilities. (shrink)

Conceptual combination performs a fundamental role in creating the broad range of compound phrases utilised in everyday language. This article provides a novel probabilistic framework for assessing whether the semantics of conceptual combinations are compositional, and so can be considered as a function of the semantics of the constituent concepts, or not. While the systematicity and productivity of language provide a strong argument in favor of assuming compositionality, this very assumption is still regularly questioned in both cognitive science and philosophy. (...) Additionally, the principle of semantic compositionality is underspecifi ed, which means that notions of both "strong" and "weak" compositionality appear in the literature. Rather than adjudicating between different grades of compositionality, the framework presented here contributes formal methods for determining a clear dividing line between compositional and non-compositional semantics. In addition, we suggest that the distinction between these is contextually sensitive. Compositionality is equated with a joint probability distribution modeling how the constituent concepts in the combination are interpreted. Marginal selectivity is introduced as a pivotal probabilistic constraint for the application of the Bell/CH and CHSH systems of inequalities. Non-compositionality is equated with a failure of marginal selectivity, or violation of either system of inequalities in the presence of marginal selectivity. This means that the conceptual combination cannot be modeled in a joint probability distribution, the variables of which correspond to how the constituent concepts are being interpreted. The formal analysis methods are demonstrated by applying them to an empirical illustration of twenty-four non-lexicalised conceptual combinations. (shrink)

Showing that the arithmetic mean number of offspring for a trait type often fails to be a predictive measure of fitness was a welcome correction to the philosophical literature on fitness. While the higher mathematical moments of a probability-weighted offspring distribution can influence fitness measurement in distinct ways, the geometric mean number of offspring is commonly singled out as the most appropriate measure. For it is well-suited to a compounding process and is sensitive to variance in offspring number. (...) The geometric mean thus proves to be a predictively efficacious measure of fitness in examples featuring discrete generations and within- or between-generation variance in offspring output. Unfortunately, this advance has subsequently led some to conclude that the arithmetic mean is never a good measure of fitness and that the geometric mean should accordingly be the default measure of fitness. We show not only that the arithmetic mean is a perfectly reasonable measure of fitness so long as one is clear about what it refers to, but also that it functions as a more general measure when properly interpreted. It must suffice as a measure of fitness in any case where the geometric mean has been effectively deployed as a measure. We conclude with a discussion about why the mathematical equivalence we highlight cannot be dismissed as merely of mathematical interest. (shrink)

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on the Boolean (...) logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)--just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition. (shrink)

There is a new theory of information based on logic. The definition of Shannon entropy as well as the notions on joint, conditional, and mutual entropy as defined by Shannon can all be derived by a uniform transformation from the corresponding formulas of logical information theory. Information is first defined in terms of sets of distinctions without using any probability measure. When a probability measure is introduced, the logical entropies are simply the values of the probability measure (...) on the sets of distinctions. The compound notions of joint, conditional, and mutual entropies are obtained as the values of the measure, respectively, on the union, difference, and intersection of the sets of distinctions. These compound notions of logical entropy satisfy the usual Venn diagram relationships since they are values of a measure. The uniform transformation into the formulas for Shannon entropy is linear so it explains the long-noted fact that the Shannon formulas satisfy the Venn diagram relations--as an analogy or mnemonic--since Shannon entropy is not a measure on a given set. What is the logic that gives rise to logical information theory? Partitions are dual to subsets, and the logic of partitions was recently developed in a dual/parallel relationship to the Boolean logic of subsets. Boole developed logical probability theory as the normalized counting measure on subsets. Similarly the normalized counting measure on partitions is logical entropy--when the partitions are represented as the set of distinctions that is the complement to the equivalence relation for the partition. In this manner, logical information theory provides the set-theoretic and measure-theoretic foundations for information theory. The Shannon theory is then derived by the transformation that replaces the counting of distinctions with the counting of the number of binary partitions it takes, on average, to make the same distinctions by uniquely encoding the distinct elements--which is why the Shannon theory perfectly dovetails into coding and communications theory. (shrink)

Talk at University of Turin, 'Kant, oltre Kant, May 5th 2023. --- -/- It is useful, while keeping in mind a holistic approach, to concentrate on a common theme in Kant’s text, which it will turn out is the quintessential element of his novel ‘way of thinking’, as he himself put it in preface of the second edition of the Critique of Pure Reason. This common theme is the idea of synthesis, which is what holds together, and is the entryway (...) to, all the other familiar aspects of Kant’s thought: his concept of self-consciousness or how mind is involved in cognition, his ideas about the role of the imagination, his theory of knowledge and his metaphysics of experience, and not unimportantly, its relation to what is probably the most Kantian element here: the categories of the understanding as the first principles of knowledge. But the meaning of synthesis and its role in its various facets, though frequently the subject of discussion in papers and monographs alike, is hardly understood in contemporary Kant scholarship. Most readings of the text of the Deduction, the part in which Kant elucidates his concept of synthesis, exhibit a shallow, mechanical understanding of it. -/- A compounding factor is that synthesis can’t really be understood independently of the other aforementioned elements of Kant’s thought. Grasping the meaning of synthesis implies coming to grips with his concept of self-consciousness and the way both self-consciousness and the categories are intricately and intimately involved. Most interpretations consider these aspects too much as if they were indeed separable faculties or entities that serve separable functions, undercutting an important feature of Kant’s metaphysics: the possibility of a priori unified cognition, for which an indivisible self-legislating subject is responsible. Rather, they are expressions of various aspects of a singular function of the mind as the ground of possible unified knowledge. -/- One striking example of such a superficial reading of the text is the way the relation between figurative synthesis and intellectual synthesis—or in the case of the A-Deduction the threefold synthesis—is standardly interpreted, as if the two are independent or quasi-independent functions of different faculties of the mind, the imagination and the understanding. In this paper, I focus on the relation between figurative and intellectual synthesis, while arguing that failing to understand this properly, indeed the relation between the imagination and the understanding, leads to a vicious regress in the explanation of the ground of knowledge, for which a priori synthesis is specifically designed. In other words, failing to see the intimate relation between the imagination and the understanding, the very identity that lies at their root, risks losing sight of the primary aim of Kant’s thought: the possibility of a priori unified knowledge. (shrink)

This monograph offers a new foundation for information theory that is based on the notion of information-as-distinctions, being directly measured by logical entropy, and on the re-quantification as Shannon entropy, which is the fundamental concept for the theory of coding and communications. Information is based on distinctions, differences, distinguishability, and diversity. Information sets are defined that express the distinctions made by a partition, e.g., the inverse-image of a random variable so they represent the pre-probability notion of information. Then logical (...) entropy is a probability measure on the information sets, the probability that on two independent trials, a distinction or “dit” of the partition will be obtained. The formula for logical entropy is a new derivation of an old formula that goes back to the early twentieth century and has been re-derived many times in different contexts. As a probability measure, all the compound notions of joint, conditional, and mutual logical entropy are immediate. The Shannon entropy and its compound notions are then derived from a non-linear dit-to-bit transform that re-quantifies the distinctions of a random variable in terms of bits—so the Shannon entropy is the average number of binary distinctions or bits necessary to make all the distinctions of the random variable. And, using a linearization method, all the set concepts in this logical information theory naturally extend to vector spaces in general—and to Hilbert spaces in particular—for quantum logical information theory which provides the natural measure of the distinctions made in quantum measurement. Relatively short but dense in content, this work can be a reference to researchers and graduate students doing investigations in information theory, maximum entropy methods in physics, engineering, and statistics, and to all those with a special interest in a new approach to quantum information theory. (shrink)

Psychological research on people’s understanding of natural language connectives has traditionally used truth table tasks, in which participants evaluate the truth or falsity of a compound sentence given the truth or falsity of its components in the framework of propositional logic. One perplexing result concerned the indicative conditional if A then C which was often evaluated as true when A and C are true, false when A is true and C is false but irrelevant“ (devoid of value) when A is (...) false (whatever the value of C). This was called the “psychological defective table of the conditional.” Here we show that far from being anomalous the “defective” table pattern reveals a coherent semantics for the basic connectives of natural language in a trivalent framework. This was done by establishing participants’ truth tables for negation, conjunction, disjunction, conditional, and biconditional, when they were presented with statements that could be certainly true, certainly false, or neither. We review systems of three-valued tables from logic, linguistics, foundations of quantum mechanics, philosophical logic, and artificial intelligence, to see whether one of these systems adequately describes people’s interpretations of natural language connectives. We find that de Finetti’s (1936/1995) three-valued system is the best approximation to participants’ truth tables. (shrink)

Mammals have an XX:XY system of chromosomal sex determination in which a small heterochromatic Y controls male development. The Y contains the testis determining factor SRY, as well as several genes important in spermatogenesis. Comparative studies show that the Y was once homologous with the X, but has been progressively degraded, and now consists largely of repeated sequences as well as degraded copies of X linked genes. The small original X and Y have been enlarged by cycles of autosomal addition (...) to one partner, recombination onto the other and continuing attrition of the compound Y. This addition–attrition hypothesis predicts that the pseudoautosomal region of the human X is merely the last relic of the latest addition. Genes (including SRY) on the conserved or added region of the Y evolved functions in male sex determination and differentiation distinct from the general functions of their X‐linked partners. Although the gonadogenesis pathway is highly conserved in vertebrates, its control has probably changed radically and rapidly in vertebrate – even mammalian – evolution. (shrink)

Standard axioms of additively separable utility for choice over time and classic axioms of expected utility theory for choice under risk yield a generalized expected additively separable utility representation of risk-time preferences over probability distributions over sure streams of intertemporal outcomes. A dual approach is to use the analogues of the same axioms in a reversed order to obtain a generalized additively separable expected utility representation of time–risk preferences over intertemporal streams of probability distributions over sure outcomes. The (...) paper proposes an additional axiom, which is called risk-time reversal, for obtaining a special case of the two representations—expected discounted utility. The axiom of risk-time reversal postulates that if a risky lottery over streams of sure intertemporal outcomes and an intertemporal stream of risky lotteries yield the same probability distribution of possible outcomes in every point in time then a decision-maker is indifferent between the two. This axiom is similar to assumption 2 “reversal of order in compound lotteries” in Anscombe and Aumann :199–205, 1963, p. 201). (shrink)

The extended semantic realism (ESR) model proposes a new theoretical perspective which embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) into a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide in this review an overall view on the present status of our research on this topic. We attain in a new, shortened way a mathematical representation of the generalized observables introduced by the ESR model and a generalization of the projection postulate of elementary (...) QM. Basing on these results we prove that the Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, a modified BCHSH inequality and quantum predictions hold together in the ESR model because they refer to different parts of the picture of the physical world supplied by the model. Then we show that a new mathematical representation of mixtures must be introduced in the ESR model which does not coincide with the standard representation in QM and avoids some deep problems that arise from the representation of mixtures provided by QM. Finally we get a nontrivial generalization of the Lüders postulate, which is justified in a special case by introducing a reasonable physical assumption on the evolution of the compound system made up of the measured system and the measuring apparatus. (shrink)

We present a logical calculus for reasoning about information flow in quantum programs. In particular we introduce a dynamic logic that is capable of dealing with quantum measurements, unitary evolutions and entanglements in compound quantum systems. We give a syntax and a relational semantics in which we abstract away from phases and probabilities. We present a sound proof system for this logic, and we show how to characterize by logical means various forms of entanglement (e.g. the Bell states) and various (...) linear operators. As an example we sketch an analysis of the teleportation protocol. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:New Light from Arabic Sources on Galen and the Fourth Figure of the Syllogism NICHOLAS RESCHER The Problem of the Origin of the Fourth Figure FLYING IN THE FACE of the long-standing tradition--going back in Europe to Renaissance times--which credits Galen of Pergamon with the origination of the fourth syllogistic figure, recent authorities have almost to a man evinced doubt about Galen's claim to this innovation. Heinrieh Scholz speaks (...) of "the Galenian syllogistic figure which has been attributed to him, probably wrongly." i j. W. Stakelum concludes his careful discussion of the matter with the categorical assertion that "Galen did not teach the fourth figure." ~ I. M. Bochenski--following Lukasiewicz (see below)--says that the fourth figure "was only ascribed to Galen by a misunderstanding." : William Kneale (also following Lukasiewicz) writes that "it is easy to see how misunderstanding.., by some Arabian philosopher... could have given rise to the tradition that Galen added a fourth figure to Aristotle's syllogistic theory." 4 The currently received account of the matter is that of Jan Lukasiewicz who, in his book on Aristotle's Syllogistic (1951; 2d ed., 1957), presents the known reports about Galen and the fourth figure as follows (I list them in the order in which they came to light) : 1) Several passages in Averroes' (d. 1198) Middle Commentary on Prior Analytics credit Galen with introducing a fourth figure (ad Anal. Pr. I, 5; I, 8; I, 23). These passages, known in Europe through a Renaissance Latin translation of a Hebrew version, provided the basis upon which Zabarella in his work, De Quarta Syllogismorum Figura (in his Opera [Leiden: 1587], pp. 41-53), popularized the "Galenian figure" in European logic. 2) In 1844 Minoides Mynas published in the Preface to his edition of Galen's EisagSg~ dialektik~ an anonymous Greek fragment (late, perhaps 6th century) which states that certain "later scholars" transformed the indirect moods of the first figure added by Theophrastus and Eudeffms into a new fourth figure, citing Galen as the originator of this doctrine.~ 1Abriss der Geschichte der Logik (Miinster: 1931), trans, into English by K. F. Leidecker, ConciseHistory of Logic (NewYork: 1961);p. 38ofthe Englishversion. "Why 'Galenian' Figure?" The New Scholasticism, XVI (1942),289-296. s Formale Logik (Miinchen-Freiburg:1956; 2nd. ed., 1962), trans, into English by Ivo Thomas, A History of Formal Logic (Notre Dame: 1961);p. 142of the Englishversion. Williamand Martha Kneale, The Developmentof Logic (Oxford: 1962),p. 184. Alsoprinted in Karl Kalbfieisch, ~ber Galen s Einleitung in die Logik, 2~. Supplementband der Jahrbi~cherfi~r klassische Philologie (Leipzig:1897), p. 707. This datum casts some 28 HISTORY OF PHILOSOPHY 3) A Greek fragment was found around 1858 by Karl Prantl in a logical work of the Byzantine scholar Ioannes Italus (llth century) which says that Galen taught the existence of a fourth figure, adding sarcastically that he (Galen) thought thus to appear cleverer than the older logical commentators (presumably Theophrastus and Eudemus are intended), but fell far short.6 4) In 1899 Maximilian Wallies published7 an anonymous Greek scholium (of perhaps the 6th or the 7th century) on Ammonins' commentary on Prior Analytics which states that Galen "says in his Apodictic that there are four figures, because he looks at the compound syllogisms consisting of four terms (i.e., with three premisses )" rather than the simple (three-term, two-premiss) syllogisms of Aristotle. (The scholiast goes on to explain at some length how such compound syllogisms can be sorted into four groups.) It is readily seen that, given these data, two possibilities stand before us: a) Galen did actually invent the traditional fourth figure. Reports (1)-(3) are correct, and the anonymous scholiast of (4) was put on the wrong track by the fact that Galen also said that compound syllogisms of four terms can be classed into four groups. b) Galen did not invent the traditional fourth figure. Only after his time when once the indirect moods of the first figure had--somehow--become systematized into a separate "figure," was this credited to Galen by a mistake along the lines described by the anonymous scholiast. Thus (4) alone is right, and (1)-(3) are mistaken. Which of... (shrink)

Three-valued accounts of conditionals frequently promise (a) to conform to the probabilistic view that conditionals are evaluated by conditional probabilities, and (b) to yield a plausible account of compounds of conditionals. However, McGee (1981) shows that probabilistic validity, the conception of validity most naturally associated with the probabilistic view, cannot be characterized by a finite matrix. Adams (1995) indicates a further generalization of this result. Nevertheless, Adams (1986) provides a description of probabilistic validity in three-valued terms by going beyond the (...) standard framework. Yet the language Adams considers is severely restricted: it does not contain compounds of conditionals. Thus, a natural question arises: Is there a plausible three-valued account of compounds of conditionals which agrees with probabilistic validity on the restricted language? In this note, I develop a general framework in which to address this question. The answer will be negative. (shrink)