Argumentation Without Arguments

A well-known ambiguity in the term ‘argument’ is that of argument as an inferential structure and argument as a kind of dialogue. In the first sense, an argument is a structure with a conclusion supported by one or more grounds, which may or may not be supported by further grounds. Rules for the construction and criteria for the quality of arguments in this sense are a matter of logic. In the second sense, arguments have been studied as a form of dialogical interaction, in which human or artificial agents aim to resolve a conflict of opinion by verbal means. Rules for conducting such dialogues and criteria for their quality are part of dialogue theory. Usually, formal accounts of argumentation dialogues in logic and artificial intelligence presuppose an argument-based logic. That is, the ways in which dialogue participants support and attack claims are modelled as the construction of explicit arguments and counterarguments (in the inferential sense). However, in this paper formal models of argumentation dialogues are discussed that do not presuppose arguments as inferential structures. The motivation for such models is that there are forms of inference that are not most naturally cast in the form of arguments (such as abduction, statistical reasoning and coherence-based reasoning) but that can still be the subject of argumentative dialogue. Some recent work in artificial intelligence is discussed which embeds non-argumentative inference in an argumentative dialogue system, and some general observations are drawn from this discussion.     

Illusory inferences with quantifiers

The psychological study of reasoning with quantifiers has predominantly focused on inference patterns studied by Aristotle about two millennia ago. Modern logic has shown a wealth of inference patterns involving quantifiers that are far beyond the expressive power of Aristotelian syllogisms, and whose psychology should be explored. We bring to light a novel class of fallacious inference patterns, some of which are so attractive that they are tantamount to cognitive illusions. In tandem with recent insights from linguistics that quantifiers like “some” are treated as wh-questions, these illusory inferences are predicted by the erotetic theory of reasoning, which postulates that a process akin to question asking and answering is behind human inference making.     

A Logic For Inductive Probabilistic Reasoning

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from “70% of As are Bs” and “a is an A” infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey’s rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation ⊨ that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone.     

Substitution Logic: An Extension of Syllogism

I examine the theoretical difficulties of Aristotle’s syllogism and the traditional syllogism. I propose a more unified ordinary thinking logic different from the syllogism. I show that the new logic based on the substitution of thinking elements can be used to describe the reasoning process of human minds more properly, bypassing rigid figures, moods and cumbersome rules of the syllogism. I also show that the new logic combines the categorical inference with relation and modal inferences, expanding the scope of the syllogism so that more complex quantification inferences can be captured. I conclude that the substitution of thinking elements is the basic characteristics of human thinking, and the substitutions can be further applied not only to all research fields of abstract and image thinking but also to practical fields of action methodology.     

Peter Winch on ‘Aristotelian’ and ‘Socratic’ Reasoning

Peter Winch often returned to questions about the nature of logic. In the context of his work on Wittgenstein and political philosophy in the 1990s, Winch described a contrast between ‘Aristotelian’ and ‘Socratic’ reasoning. Aristotelian conceptions of reasoning, attributed to Frege and Russell, would see logic as a formal science and rationality as consistency with pre‐existent rules of inference. The Socratic conception, attributed to Wittgenstein, understands rational argument as a form of socially embedded dialogue that involves moral relationships and a dimension of depth. Rational persuasion may also involve use of persuasive images and examples.     

Fallacies of Accident

In this paper I will attempt a unified analysis of the various examples of the fallacy of accident given by Aristotle in the Sophistical Refutations. In many cases the examples underdetermine the fallacy and it is not trivial to identify the fallacy committed. To make this identification we have to find some error common to all the examples and to show that this error would still be committed even if those other fallacies that the examples exemplify were not. Aristotle says that there is only one solution “against the argument” as opposed to “against the man”, and it is this solution the paper attempts to find. It is a characteristic mark of my analysis that some arguments that we might normally be inclined to say are fallacious turn out to be valid and that some arguments that we would normally be inclined to say are valid turn out to be fallacious. This is (in part) because what we call validity in modern logic is not the same as the apodicticity that Aristotelian syllogisms require in order to be used in science. The fallacies of accident, uniquely among the fallacies, are failures of apodicticity rather than failures of, in particular, semantic entailment. This makes sense in a tensed and token-based logic such as Aristotle’s. I conclude that the closest analogue to the fallacy of accident that we can point to is a fallacy in modal logic, viz., the fallacy of necessity.     

Aristotle's Prior Analytics and Boole's Laws of Thought

Prior Analytics by the Greek philosopher Aristotle (384?–?322 BCE) and Laws of Thought by the English mathematician George Boole (1815?–?1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle's system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss many other historically and philosophically important aspects of Boole's book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole's contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology.

… using mathematical methods … has led to more knowledge about logic in one century than had been obtained from the death of Aristotle up to … when Boole's masterpiece was published.     

Mathematical statistics and metastatistical analysis

This paper deals with meta-statistical questions concerning frequentist statistics. In Sections 2 to 4 I analyse the dispute between Fisher and Neyman on the so called logic of statistical inference, a polemic that has been concomitant of the development of mathematical statistics. My conclusion is that, whenever mathematical statistics makes it possible to draw inferences, it only uses deductive reasoning. Therefore I reject Fisher's inductive approach to the statistical estimation theory and adhere to Neyman's deductive one. On the other hand, I assert that Neyman-Pearson's testing theory, as well as Fisher's tests of significance, properly belong to decision theory, not to logic, neither deductive nor inductive. I then also disagree with Costantini's view of Fisher's testing model as a theory of hypothetico-deductive inferences.In Section 5 I disapprove Hacking₁'s evidentialists criticisms of the Neyman-Pearson's theory of statistics (NPT), as well as Hacking₂'s interpretation of NPT as a theory of probable inference. In both cases Hacking misses the point. I conclude, by claiming that Mayo's conception of the Neyman-Pearson's testing theory, as a model of learning from experience, does not purport any advantages over Neyman's behavioristic model.     

The eco-cognitive model of abduction: Ἀπαγωγή now: Naturalizing the logic of abduction

Abduction (ἀπαγωγή, in ancient Greek, often translated as “leading away” or “reduction”) is a procedure in which something that lacks classical explanatory epistemic virtue can be accepted because it has virtue of another kind: Gabbay and Woods [15] contend (GW-schema) that abduction presents an ignorance-preserving or (ignorance-mitigating) character. From this perspective abductive reasoning is a response to an ignorance-problem; through abduction the basic ignorance – that does not have to be considered a total “ignorance” – is neither solved nor left intact. Abductive reasoning is an ignorance-preserving accommodation of the problem at hand. Is abduction really ignorance-preserving? To better answer this question I will introduce (and take advantage of) an eco-cognitive model (EC-Model) of abduction. It will be illustrated that through abduction, knowledge can be enhanced, even when abduction is not considered an inference to the best explanation in the classical sense of the expression, that is an inference necessarily characterized by an empirical evaluation phase, or an inductive phase, as Peirce called it. To further deepen the eco-cognitive character of abduction a simple genealogy of logic is provided: Aristotle clearly states that in syllogistic theory local/environmental cognitive factors – external to that peculiar inferential process, for example regarding users/reasoners, are given up. Indeed, to define syllogism Aristotle first of all insists that all syllogisms are valid and contends that the necessity of this kind of reasoning is related to the circumstance that “no further term from outside (ἔξωθɛν) is needed”, in sum syllogism is the fruit of a kind of eco-cognitive immunization. At the same time Aristotle presents a seminal perspective on abduction: the second part of the article considers the famous passage in Chapter B25 of Prior Analytics concerning ἀπαγωγή (“leading away”), also studied by Peirce. I contend that some of the current well-known distinctive characters of abductive cognition are already expressed, which are in tune with the EC-Model. By providing an illustration of the role of the method of analysis and of the middle terms in Plato's dialectic argumentation, considered as related to the diorismic/poristic process in ancient geometry – also, later on, emphasized by Proclus – I maintain that it is just this intellectual heritage which informs Aristotle' Chapter B25 on ἀπαγωγή. Even if, in general, Aristotle seems to sterilize, thanks to the invention of syllogistic theory, every “dialectic” background of reasoning, nevertheless in Chapter B25 he is still pointing to the fundamental inferential role in reasoning of those externalities that substantiate the process of “leading away” (ἀπαγωγή). Hence, we can gain a new positive perspective about the “constitutive” eco-cognitive character of abduction, just thanks to Aristotle himself. Finally, the paper presents an excursus on Aristotle's enthymemes from signs, disregarded by Peirce, but extremely important to stress the Aristotelian treatment of what I have called selective abduction. A forthcoming companion paper [35] will further deepen the EC-Model of abduction stressing stricter logical aspects: the first result will be that, contrarily to the classical logical view, relevance and plausibility in abductive reasoning have to be relativized and so the epistemologically embarrassing concepts of irrelevance and implausibility exculpated: they are not always offensive to reason.     

Theories of the syllogism: A meta-analysis

Syllogisms are arguments about the properties of entities. They consist of 2 premises and a conclusion, which can each be in 1 of 4 "moods": All A are B, Some A are B, No A are B, and Some A are not B. Their logical analysis began with Aristotle, and their psychological investigation began over 100 years ago. This article outlines the logic of inferences about syllogisms, which includes the evaluation of the consistency of sets of assertions. It also describes the main phenomena of reasoning about properties. There are 12 extant theories of such inferences, and the article outlines each of them and describes their strengths and weaknesses. The theories are of 3 main sorts: heuristic theories that capture principles that could underlie intuitive responses, theories of deliberative reasoning based on formal rules of inference akin to those of logic, and theories of deliberative reasoning based on set-theoretic diagrams or models. The article presents a meta-analysis of these extant theories of syllogisms using data from 6 studies. None of the 12 theories provides an adequate account, and so the article concludes with a guide-based on its qualitative and quantitative analyses-of how best to make progress toward a satisfactory theory.     

How to use probabilities in reasoning

Conclusions Probabilities are important in belief updating, but probabilistic reasoning does not subsume everything else (as the Bayesian would have it). On the contrary, Bayesian reasoning presupposes knowledge that cannot itself be obtained by Bayesian reasoning, making generic Bayesianism an incoherent theory of belief updating. Instead, it is indefinite probabilities that are of principal importance in belief updating. Knowledge of such indefinite probabilities is obtained by some form of statistical induction, and inferences to non-probabilistic conclusions are carried out in accordance with the statistical syllogism. Such inferences have been the focus of much attention in the nonmonotonic reasoning literature, but the logical complexity of such inference has not been adequately appreciated.     

古典对当方阵和广义量词

古典对当方阵可同溯到亚里士多德逻辑,并且自此后就一直被广泛地讨论,特别是在中世纪和现代。它刻画了所有、没有、并非所有和某些这四个量词之间的特定逻辑关系,即对当关系。亚里十多德和传统逻辑学家,以及人多数当代语言学家,都把“所有”看作具有存在预设,也即“所有A是B”可以推山“存在A”,而现代逻辑则放弃了这一假定。用现代逻辑对“所有”的解释来代替亚里十多德的解释（对“并非所有”也可以作类似处理）,就产生了现代版本的对当方阵。近年来有许多争论,探讨这两个方阵中哪一个是正确的。本文中我的主要观点是,这个问题不是,或者不应该主要是关于存在预设的,毋宁说它是关于否定的模式的。我认为现代方阵表述了自然语言中否定的一般模式,而传统方阵则没有做到这一点。明乎此,不仅需要把对当方阵应用于四个亚里士多德量词,还需要把它应用到这一类型的广义量词上。现代方阵上的任一量词所展示的否定的模式,常常不是在传统方阵中发现的对当关系。本文提供了一些技术性结果和工具,阐述了解释各种英语限定词的量词方阵的若干例子。本文最后一个例子引入了否定的第二模式。它伴随特定复杂量词出现,也能够在方阵中被表达。     

Complete Virtue and the Definition of Happiness in Aristotle

In this paper, I challenge the standard reading of complete virtue (ἀρετή τελεία) in those disputed passages of Nicomachean Ethics and Eudemian Ethics. I argue that, for Aristotle, complete virtue is neither (i) wisdom nor (ii) a whole set of all virtues. Rather, it is a term used by Aristotle to denote any virtue that is in its complete or perfect form. In light of this reading, I offer a pluralist interpretation of Aristotelian happiness. I argue that for Aristotle, the life-long exercise of a predominant virtue—as long as it is exercised in its complete or perfect form—will suffice for human happiness. The so-called inclusivist and intellectualist notions of Aristotelian happiness, thus understood, are merely two forms (viz. the composite and the non-composite form) of the pluralist notion of Aristotelian happiness. And if I am right, my pluralist interpretation provides an alternative, if not better, solution to the long-standing problem of “dual happiness” in Aristotle.     

Pragmatism Regained

This article views the confrontation between pragmatism and Kant’s Critical undertaking as very possibly the single most consequential agon of contemporary philosophy, given the utter irreconcilability of their respective ways of addressing the concerns of First Philosophy, with regard to the enabling conditions of cognitive realism. Pragmatism favors an informal, fluxive, “instrumentalist” form of empiricism, impossible to complete, opposed to any and all the ontic and epistemic fixities of Kant’s Rationalism. Reason (Vernunft) cannot be more than a fiction. Kant has no supporting criterion of realism. The article proposes an empirical criterion of the distinctive (given) “duality” of sensory “appearings” and “appeareds” (that is, objects) as yielding a plausible “cognitive faculty” (involving “reasoning”—inference, for instance—but not Vernunft) that, in accord with the drift of evolutionism and Aristotle, readily extends to languageless animals as well as humans. It serves as the linchpin of the paper.     

Applied Logic without Psychologism

Logic is a celebrated representation language because of its formal generality. But there are two senses in which a logic may be considered general, one that concerns a technical ability to discriminate between different types of individuals, and another that concerns constitutive norms for reasoning as such. This essay embraces the former, permutation-invariance conception of logic and rejects the latter, Fregean conception of logic. The question of how to apply logic under this pure invariantist view is addressed, and a methodology is given. The pure invariantist view is contrasted with logical pluralism, and a methodology for applied logic is demonstrated in remarks on a variety of issues concerning non-monotonic logic and non-monotonic inference, including Charles Morgan’s impossibility results for non-monotonic logic, David Makinson’s normative constraints for non-monotonic inference, and Igor Douven and Timothy Williamson’s proposed formal constraints on rational acceptance.     

Syllogistic: Old Wine in New Bottles

In the late nineteenth century there were two very active lines of research in the field of formal logic. First, logicians (mostly in English-speaking countries) were engaged in formulating a generally traditional logic as an algebra, a part of mathematics; second, logicians (mostly on the continent) were busy building a non-traditional logic that could serve, not as a part of, but as the foundation of, mathematics. By the end of the First World War the former line had been pretty well abandoned while the second continued to expand. However, that old abandoned line, stretching from Aristotle, through the Scholastics and then Leibniz to the nineteenth century algebraists, had not been completely forgotten. One of those logicians who has recently worked on the restoration (and, importantly, the extension) of that line is Fred Sommers. His Term Logic preserves a number of traditional insights (especially involving the theory of logical syntax), while also enjoying a power to account for formal inference at least comparable to that of the standard logic now in place.     

Probabilities on Sentences in an Expressive Logic

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter)examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic.     

Reasoning with quantifiers

In the semantics of natural language, quantification may have received more attention than any other subject, and one of the main topics in psychological studies on deductive reasoning is syllogistic inference, which is just a restricted form of reasoning with quantifiers. But thus far the semantical and psychological enterprises have remained disconnected. This paper aims to show how our understanding of syllogistic reasoning may benefit from semantical research on quantification. I present a very simple logic that pivots on the monotonicity properties of quantified statements--properties that are known to be crucial not only to quantification but to a much wider range of semantical phenomena. This logic is shown to account for the experimental evidence available in the literature as well as for the data from a new experiment with cardinal quantifiers ("at least n" and "at most n"), which cannot be explained by any other theory of syllogistic reasoning.     

Completion,reduction and analysis: three proof-theoretic processes in aristotle’s prior analytics

Three distinctly different interpretations of Aristotle’s notion of a sullogismos in Prior Analytics can be traced: (1) a valid or invalid premise-conclusion argument (2) a single, logically true conditional proposition and (3) a cogent argumentation or deduction. Remarkably the three interpretations hold similar notions about the logical relationships among the sullogismoi. This is most apparent in their conflating three processes that Aristotle especially distinguishes: completion (A4-6)reduction(A7) and analysis (A45). Interpretive problems result from not sufficiently recognizing Aristotle’s remarkable degree of metalogical sophistication to distinguish logical syntax from semantics and, thus, also from not grasping him to refine the deduction system of his underlying logic. While it is obvious that Aristotle most often uses ‘sullogimos’ to denote a valid argument of a certain kind, we show that at Prior Analytics A4-6, 7, 45 Aristotle specifically treats a sullogismos as an elemental argument pattern having only valid instances and that such a pattern then serves as a rule of deduction in his syllogistic logic. By extracting Aristotle’s understanding of three proof-theoretic processes, this paper provides new insight into what Aristotle thinks reasoning syllogistically is and, moreover, it resolves three problems in the most recent interpretation that takes a sullogismos to be a deduction     

Aristotle on the Unity of Composite Substance

Aristotle claims that a sensible substance is composed of form and matter, while he insists that it is a unity in a strict sense. So there is the question—in what sense can a composite thing be a unity? Aristotle’s key solution lies in his account of matter as potentiality and form as actuality. Many scholars are bewildered by his laconic solutions, and there are mainly two approaches undertaken in interpretations. One is called “projective”; the other is called “explanatory.” But neither interpretation is satisfying. The main tasks of this paper are to reexamine the problem and the two interpretations, then to argue that the composition of a sensible substance should be understood in light of its coming-to-be; that its unity refers to its being a functional unity.