Dynamic Probabilistic Entailment. Improving on Adams' Dynamic Entailment Relation

The inferences of contraposition (A ⇒ C ∴ ¬C ⇒ ¬A), the hypothetical syllogism (A ⇒ B, B ⇒ C ∴ A ⇒ C), and others are widely seen as unacceptable for counterfactual conditionals. Adams convincingly argued, however, that these inferences are unacceptable for indicative conditionals as well. He argued that an indicative conditional of form A ⇒ C has assertability conditions instead of truth conditions, and that their assertability ‘goes with’ the conditional probability p(C|A). To account for inferences, Adams developed the notion of probabilistic entailment as an extension of classical entailment. This combined approach (correctly) predicts that contraposition and the hypothetical syllogism are invalid inferences. Perhaps less well-known, however, is that the approach also predicts that the unconditional counterparts of these inferences, e.g., modus tollens (A ⇒ C, ¬C ∴ ¬A), and iterated modus ponens (A ⇒ B, B ⇒ C, A ∴ C) are predicted to be valid. We will argue both by example and by calling to the results from a behavioral experiment (N = 159) that these latter predictions are incorrect if the unconditional premises in these inferences are seen as new information. Then we will discuss Adams’ (1998) dynamic probabilistic entailment relation, and argue that it is problematic. Finally, it will be shown how his dynamic entailment relation can be improved such that the incongruence predicted by Adams’ original system concerning conditionals and their unconditional counterparts are overcome. Finally, it will be argued that the idea behind this new notion of entailment is of more general relevance.

Parmenides’ First Attack on the Forms

This paper provides a case study for the use of syllogistic reconstructions in the commentaries on Plato by the fifth-century commentator Proclus. The paper discusses Proclus’ reconstruction of the argument about the range of the Forms in Plato’s Parmenides (130b–e). In his commentary on this dialogue, Proclus reports a syllogistic reconstruction of the argument proposed by some of his predecessors. In this reconstruction, the argument as a whole is interpreted as a straightforward attack on the existence of Forms, while the different premises of the hypothetical syllogism represent the respective positions of Parmenides and Socrates in the discussion. For Proclus, however, the argument about the range of Forms is not meant to be critical of the Forms, but rather provides a positive instruction about their range of application. I argue that while Proclus finds the syllogism a useful tool to reconstruct the different positions in the exegetical history of the argument, he does not accept it as an adequate reconstruction on his own account. The argument can be traced back most likely to the so-called ‘logical’ interpretations of the Parmenides that Proclus discusses – and dismisses – in the prologue to his commentary.

Argumentasi dalam Dialog Interaktif di Kanal YouTube Metro TV News (Editorial Media Indonesia)

The focus of this study is to analyze and to explain about arguments in interactive dialogue in television programs. The use of arguments is critically important. It is because without knowing and understanding them, it can trigger misunderstanding and chaos in the dialogue. The specific focus of this study consists of reasoning in interactive dialogue arguments, evidence in interactive dialogue arguments, along with inference and implication in interactive dialogue arguments. Qualitative approach was used in this study. The data of this study were in the form of verbal data from the dialogue of Editorial Media Indonesia Metro TV video. The verbal data were in the form of phrases, words, prepositions, and sentences. The data sources were taken from YouTube. The data gathering techniques that was used in this study was documentation, by collecting videos. The results of the study showed reasoning, evidence, inference, and implication were found in interactive dialogue. In the inductive reasoning, generalization in casual and analogical relationship was found. In inductive reasoning, categorical and hypothetical syllogism were found. Evidence in arguments tested with data, information, and facts. The data and information testing used observation and expert opinions. The facts testing used consistency. In inference and implication, causal and assumption factors were found. Keywords: arguments, interactive dialogue, televisions programs Abstrak: Fokus penelitian ini menganalisis dan menjelaskan argumentasi dalam dialog interaktif di kanal YouTube Metro TV News. Penggunaan argumentasi sangat penting karena jika tidak mengetahui dan memahaminya dengan jelas dapat memicu kesalahpahaman dan kegaduhan dalam berdialog. Fokus khusus penelitian meliputi penalaran dalam argumentasi dialog interaktif, evidensi dalam argumentasi dialog interaktif, serta inferensi dan implikasi dalam argumentasi dialog interaktif. Pendekatan penelitian ini adalah pendekatan kualitatif. Data penelitian ini berupa data verbal dari tuturan dialog interaktif yang ada di dalam video Editorial Media Indonesia Metro TV. Data verbal tersebut berupa frasa, kata, proposisi, atau kalimat. Sumber data penelitian diambil melalui platform YouTube. Teknik pengumpulan data yang digunakan adalah studi dokumen, dengan cara mengumpulkan video-video. Hasil penelitian menunjukkan argumentasi dalam dialog interaktif, ditemukan penalaran, evidensi, inferensi dan implikasi. Penalaran dalam argumentasi terdapat penalaran induktif dan deduktif. Penalaran induktif ditemukan proses generalisasi, hubungan kausal, dan analogi. Penalaran deduktif ditemukan silogisme kategorial dan silogisme hipotetis. Evidensi dalam argumentasi menguji dengan data, informasi, dan fakta. Menguji data dan informasi menggunakan observasi dan meminta pendapat autoritas. Menguji fakta menggunakan segi konsistensi. Inferensi dan implikasi dalam argumentasi ditemukan faktor dugaan, faktor akibat, dan asumsi-asumsi. Kata kunci: argumentasi, dialog interaktif, kanal YouTube Metro TV News

On Basic Probability Logic Inequalities

We give some simple examples of applying some of the well-known elementary probability theory inequalities and properties in the field of logical argumentation. A probabilistic version of the hypothetical syllogism inference rule is as follows: if propositions A, B, C, A→B, and B→C have probabilities a, b, c, r, and s, respectively, then for probability p of A→C, we have f(a,b,c,r,s)≤p≤g(a,b,c,r,s), for some functions f and g of given parameters. In this paper, after a short overview of known rules related to conjunction and disjunction, we proposed some probabilized forms of the hypothetical syllogism inference rule, with the best possible bounds for the probability of conclusion, covering simultaneously the probabilistic versions of both modus ponens and modus tollens rules, as already considered by Suppes, Hailperin, and Wagner.

Perspectives of Fuzzy Logic and Their Applications

Fuzzy logic is a highly suitable and applicable basis for developing knowledge-based systems in engineering and applied sciences. The concepts of a fuzzy number plays a fundamental role in formulating quantitative fuzzy variable. These are variable whose states are fuzzy numbers. When in addition, the fuzzy numbers represent linguistic concepts, such as very small, small, medium, and so on, as interpreted in a particular contest, the resulting constructs are usually called linguistic variables. Each linguistic variable the states of which are expressed by linguistic terms interpreted as specific fuzzy numbers is defined in terms of a base variable, the value of which are real numbers within a specific range. A base variable is variable in the classical sense, exemplified by the physical variable (e.g., temperature, pressure, speed, voltage, humidity, etc.) as well as any other numerical variable (e.g., age, interest rate, performance, salary, blood count, probability, reliability, etc.). Logic is the science of reasoning. Symbolic or mathematical logic is a powerful computational paradigm. Just as crisp sets survive on a 2-state membership (0/1) and fuzzy sets on a multistage membership [0 - 1], crisp logic is built on a 2-state truth-value (true or false) and fuzzy logic on a multistage truth-value (true, false, very true, partly false and so on). The author now briefly discusses the crisp logic and fuzzy logic. The aim of this paper is to explain the concept of classical logic, fuzzy logic, fuzzy connectives, fuzzy inference, fuzzy predicate, modifier inference from conditional fuzzy propositions, generalized modus ponens, generalization of hypothetical syllogism, conditional, and qualified propositions. Suitable examples are given to understand the topics in brief.

Der Kettenschluss – eine Verteidigung

Proponents of conditional logics such as David Lewis and Robert Stalnaker reject inferences containing counterfactuals from "if A, B" and "if B, C" to "if A, C" due to ordinary language counterexamples. Contextualists defend this inference rule called "hypothetical syllogism" or "transitivity" on the basis of a possible word semantics, which, however, assigns implausible truth values to certain counterfactuals. My defence of hypothetical syllogism avoids this problem, as it rests on Nelson Goodman's uncontroversial, metaphysically parsimonious assumption that we accept counterfactuals as true only under certain conditions from which, in conjunction with the antecedent, the consequent can be inferred. The counterexamples to hypothetical syllogism can be rebutted because their premises are doxastically noncotenable; that is, there is no set of conditions under which the premises can be accepted as jointly true.

THE PERIPATETIC PROGRAM IN CATEGORICAL LOGIC: LEIBNIZ ON PROPOSITIONAL TERMS

Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$.