The medieval theory of consequence

The recovery of Aristotle??s logic during the twelfth century was a great stimulus to medieval thinkers. Among their own theories developed to explain Aristotle??s theories of valid and invalid reasoning was a theory of consequence, of what arguments were valid, and why. By the fourteenth century, two main lines of thought had developed, one at Oxford, the other at Paris. Both schools distinguished formal from material consequence, but in very different ways. In Buridan and his followers in Paris, formal consequence was that preserved under uniform substitution. In Oxford, in contrast, formal consequence included analytic consequences such as ??If it??s a man, then it??s an animal??. Aristotle??s notion of syllogistic consequence was subsumed under the treatment of formal consequence. Buridan developed a general theory embracing the assertoric syllogism, the modal syllogism and syllogisms with oblique terms. The result was a thoroughly systematic and extensive treatment of logical theory and logical consequence which repays investigation.     

Term Kinds and the Formality of Aristotelian Modal Logic

Recent formalizations of Aristotle's modal syllogistic have made use of an interpretative assumption with precedent in traditional commentary: That Aristotle implicitly relies on a distinction between two classes of terms. I argue that the way Rini (2011. Aristotle's Modal Proofs: Prior Analytics A8–22 in Predicate Logic, Dordrecht: Springer) employs this distinction undermines her attempt to show that Aristotle gives valid proofs of his modal syllogisms. Rini does not establish that Aristotle gives valid proofs of the arguments which she takes to best represent Aristotle's modal syllogisms, nor that Aristotle's modal syllogisms are instances of any other system of schemata that could be used to define an alternative notion of validity. On the other hand, I argue, Robert Kilwardby's ca. 1240 commentary on the Prior Analytics makes use of a term-kind distinction so as to provide truth conditions for Aristotle's necessity propositions which render Aristotle's conversion rules and first figure modal syllogisms formally valid. I reconstruct a suppositio semantics for syllogistic necessity propositions based on Kilwardby's text, and yield a consequence relation which validates key results in the assertoric, pure necessity and mixed necessity-assertoric syllogistics.     

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.     

Vollkommene Syllogismen und reine Vernunftschlüsse: Aristoteles und Kant. Eine Stellungnahme zu Theodor Eberts Gegeneinwänden. Teil 1

In an earlier article (s. J Gen Philos Sci 40:341–355, 2009), I have rejected an interpretation of Aristotle’s syllogistic which (since Patzig) is predominant in the literature on Aristotle, but wrong in my view. According to this interpretation, the distinguishing feature of perfect syllogisms is their being evident. Theodor Ebert has attempted to defend this interpretation by means of objections (s. J Gen Philos Sci 40:357–365, 2009) which I will try to refute in part [1] of the following article. I want to show that (1) according to Aristotle’s Prior Analytics perfect and imperfect syllogisms do not differ by their being evident, but by the reason for their being evident, (2) Aristotle uses the same words to denote proofs of the validity of perfect and imperfect syllogisms („apodeixis“, “deiknusthai” etc.), (3) accordingly, Aristotle defines perfect syllogisms not as being evident, but as “requiring nothing beyond the things taken in order to make the necessity evident“, i.e. as not “requiring one or more things that are necessary because of the terms assumed, but that have not been taken among the propositions” (APr. I. 1), (4) the proofs by which the validity of perfect assertoric syllogisms can be shown according to APr. I. 4 are based on the Dictum de omni et nullo, (5) the fact that Aristotle describes these proofs only in rough outlines corresponds to the fact that his proofs of the validity of other fundamental rules are likewise produced in rough outlines, e.g. his proof of the validity of conversio simplex in APr. I. 2, which usually has been misunderstood (also by Ebert): (6) Aristotle does not prove the convertibility of E-sentences by presupposing the convertibility of I-sentences; only the reverse is true.     

Syllogistic reasoning and belief-bias inhibition in school children: evidence from a negative priming paradigm

Research on deductive reasoning in adolescents and adults has shown that errors in deductive logic are not necessarily due to a lack of logical ability but can stem from an executive failure to inhibit biases. Few studies have examined this dissociation in children. Here, we used a negative priming paradigm with 64 children (8-10 years old) to test the role of cognitive inhibition in syllogisms with belief-bias effects. On trials where negative priming was predicted, results were as follows: For the first syllogism (A), the strategy 'unbelievable-equals-invalid' had to be inhibited. The logic of the syllogism led to affirming a conclusion inconsistent with one's knowledge of the world, such as 'All elephants are light.' For the second syllogism (B), one's real-world knowledge and the syllogism's logic were congruent but the latter required affirming exactly what had been inhibited for A (i.e. that elephants are heavy). A negative priming effect on the A-B sequence was reflected in a significant drop in reasoning performance on B. This supports the idea that during cognitive development, inhibitory control is required for success on syllogisms where beliefs and logic interfere.     

Content and strategy in syllogistic reasoning.

Syllogistic reasoning has been investigated as a general deductive process (Johnson-Laird & Byrne, 1991; Revlis, 1975; Rips, 1994). However, several studies have demonstrated the role of cognitive strategies in this type of reasoning. These strategies focus on the method used by the participants (Ford, 1995; Gilhooly, Logie, Wetherick, & Wynn, 1993) and strategies related to different interpretations of the quantified premises (Roberts, Newstead, & Griggs, 2001). In this paper, we propose that content (as well as individual cognitive differences) is an important factor in inducing a certain strategy or method for syllogistic resolution. Specifically, we suggest that syllogisms with a causal conditional premise that can be extended by an agency premise induce the use of a conditional method. To demonstrate this, we carried out two experiments. Experiment 1 provided evidence that this type of syllogism leads participants to draw the predicted conditional conclusions, in contrast with control content syllogisms. In Experiment 2, we demonstrated that the drawing of conditional conclusions is based on a causal conditional to an agent representation of the syllogism premises. These results support the role of content as inducing a particular strategy for syllogistic resolution. The implications of these results are discussed.     

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.     

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.     

Framing deductive reasoning with emotional content: An fMRI study

In the literature concerning the study of emotional effect on cognition, several researches highlight the mechanisms of reasoning ability and the influence of emotions on this ability. However, up to now, no neuroimaging study was specifically devised to directly compare the influence on reasoning performance of visual task-unrelated with semantic task-related emotional information.In the present functional fMRI study, we devised a novel paradigm in which emotionally negative vs. neutral visual stimuli (context) were used as primes, followed by syllogisms composed of propositions with emotionally negative vs. neutral contents respectively. Participants, in the MR scanner, were asked to assess the logical validity of the syllogisms. We have therefore manipulated the emotional state and arousal induced by the visual prime as well as the emotional interference exerted by the syllogism content.fMRI data indicated a medial prefrontal cortex deactivation and lateral/dorsolateral prefrontal cortex activation in conditions with negative context. Furthermore, a lateral/dorsolateral prefrontal cortex modulation caused by syllogism content was observed. Finally, behavioral data confirmed the influence of emotional task-related stimuli on reasoning ability, since the performance was worse in conditions with syllogisms involving negative emotions. Therefore, on the basis of these data, we conclude that emotional states can impair the performance in reasoning tasks by means of the delayed general reactivity, whereas the emotional content of the target may require a larger amount of top-down resources to be processed.     

逻辑训练对不同理性思维方式大学生三段论推理的影响

采用理性—经验思维方式量表对105名大学生进行测量,选出高、低理性思维方式各35名大学生进行实验,探讨了逻辑训练对不同理性思维方式大学生三段论推理的影响。结果表明：(1)高理性思维方式个体三段论推理的平均反应时快于低理性思维方式个体;非冲突类型题目推理的平均正确率高于冲突类型题目;逻辑训练可以提高个体三段论推理的平均反应时和正确率;(2)逻辑训练后非冲突任务类型题目的平均正确率高于冲突任务类型题目;(3)逻辑训练对低理性思维方式个体冲突任务类型题目平均正确率的提高效果更明显。这意味着逻辑训练可以显著提升个体的推理成绩,但不能完全消除信念偏差对推理的影响,而且逻辑训练对低理性思维方式个体推理成绩的提升效果更加明显。     

Phonological and visual distinctiveness effects in syllogistic reasoning: Implications for mental models theory

Two experiments are reported in which the representational distinctiveness of terms within categorical syllogisms was manipulated in order to examine the assumption of mental models theory that abstract, spatially based representations underpin deduction. In Experiment 1, participants evaluated conclusion validity for syllogisms containing either phonologically distinctive terms (e.g., harks, paps, and fids) or phonologically nondistinctive terms (e.g., fuds, fods, and feds). Logical performance was enhanced with the distinctive contents, suggesting that the phonological properties of syllogism terms can play an important role in deduction. In Experiment 2, participants received either the phonological materials from Experiment 1 or syllogisms involving distinctive or nondistinctive visual contents. Logical inference was again enhanced for the distinctive contents, whether phonological or visual in nature. Our findings suggest a broad involvement of multimodal information in syllogistic reasoning and question the assumed primacy of abstract, spatially organized representations in deduction, as is claimed by mental models theorists.     

Introduction to the Special Issue “Phenomenological Studies in Multidimensional Views”

How should Scheler’s critique of Kant’s ethics be interpreted? This paper focuses on two aspects of Scheler’s critique of Kant’s ethics: 1) the problem of “formalism” in Kant’s ethics, and 2) the problem of the “ethics of autonomy” and “ethics of heteronomy.” Generally speaking, Scheler’s project has a “modern” starting point; that is to say, his work starts with the rejection or critique of Kant and Aristotle. Most essentially, Scheler’s “material ethics of values” (ethics of person) must stay autonomous. Following Kant, Scheler takes Aristotle’s theory as an “ethics of heteronomy,” and then competes with Kant within the “ethics of autonomy” and further develops his own “ethics of personal autonomy.”     

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.     

Divergent Thinking Styles of the Hemispheres: How Syllogisms Are Solved during Transitory Hemisphere Suppression

Psychiatric patients solved syllogisms while recovering from transitory ictal suppression of one hemisphere by electroconvulsive therapy (ECT). The premises were familiar or unfamiliar, true or false. While the right hemisphere was suppressed, syllogisms were usually solved by theoretical, deductive reasoning even when the factual answer was known a priori, the premises were obviously false and the conclusions were absurd. While their left hemisphere was suppressed, the same subjects applied their prior knowledge; if the syllogism content was unfamiliar or false, they refused to answer. We postulate a left-hemisphere mechanism capable of decontextualized mental operations and a right-hemisphere mechanism, the operation of which is context-bound and incapable of abstraction. We show that each hemisphere tends to overextend its perspective on the problem and that in the intact brain they both contribute to an extent that depends on the characteristics of the problem at hand.     

Neo-Aristotelian Social Justice: An Unanswered Question

In this paper I assess the possibility of advancing a modern conception of social justice under neo-Aristotelian lights, focussing primarily on conceptions that assert a fundamental connection between social justice and eudaimonia. After some preliminary remarks on the extent to which a neo-Aristotelian account must stay close to Aristotle’s own, I focus on Martha Nussbaum’s sophisticated neo-Aristotelian approach, which I argue implausibly overworks the aspects of Aristotle’s thought it appeals to. I then outline the shape of a deeper and more general, and as yet unanswered, problem facing neo-Aristotelian accounts: how to justify the claim that the point of a just society is to assist or enable its members to flourish.     

Why the view of intellect in De Anima I 4 is not Aristotle’s own

In De Anima I 4, Aristotle describes the intellect (nous) as a sort of substance, separate and incorruptible. Myles Burnyeat and Lloyd Gerson take this as proof that, for Aristotle, the intellect is a separate eternal entity, not a power belonging to individual humans. Against this reading, I show that this passage does not express Aristotle’s own views, but dialectically examines a reputable position (endoxon) about the intellect that seems to show that it can be subject to change. The passage’s argument for the indestructibility of intellect via an analogy to perception does not fit with Aristotle’s own views. Aristotle thinks that perception operates via bodily organs, but denies this of understanding. He also requires separability from the body for indestructibility, something this analogy rules out. However, Aristotle’s Platonist interlocutors may well endorse such an argument. My dialectical interpretation best resolves the interpretative difficulties and explains its place in the larger context, Aristotle’s discussion of Platonist views on the soul. Aristotle presents a challenge to his insistence that the soul is subject to change, dialectically resolves that challenge, and then ends by reserving the right to give a different account of the intellect.     

Belief bias and relational reasoning

When people evaluate categorical syllogisms, they tend to reject unbelievable conclusions and accept believable ones irrespective of their validity. Typically, this effect is particularly marked for invalid conclusions that are possible, but do not necessarily follow, given the premises. However, smaller believability effects can also be detected for other types of conclusion. Three experiments are reported here, in which an attempt was made to determine whether belief bias effects can manifest themselves on the relational inference task. Subjects evaluated the validity of conclusions such as William the Conqueror was king after the Pyramids were built (temporal task) or Manchester is north of Bournemouth (spatial task) with respect to their premises. All of the majorfindings for equivalent categorical syllogism tasks were replicated. However, the overall size of the main effect of believability appears to be related to task presentation, a phenomenon not previously identified for categorical syllogisms and which current theories of belief bias have difficulty explaining.     

Statistical inference and Aristotle's Rhetoric

Formal logic operates in a closed system where all the information relevant to any conclusion is present, whereas this is not the case when one reasons about events and states of the world. Pollard and Richardson drew attention to the fact that the reasoning behind statistical tests does not lead to logically justifiable conclusions. In this paper statistical inferences are defended not by logic but by the standards of everyday reasoning. Aristotle invented formal logic, but argued that people mostly get at the truth with the aid of enthymemes—incomplete syllogisms which include arguing from examples, analogies and signs. It is proposed that statistical tests work in the same way—in that they are based on examples, invoke the analogy of a model and use the size of the effect under test as a sign that the chance hypothesis is unlikely. Of existing theories of statistical inference only a weak version of Fisher's takes this into account. Aristotle anticipated Fisher by producing an argument of the form that there were too many cases in which an outcome went in a particular direction for that direction to be plausibly attributed to chance. We can therefore conclude that Aristotle would have approved of statistical inference and there is a good reason for calling this form of statistical inference classical.     

Evolutionary Biology and Classical Teleological Arguments for God's Existence

Much has been made of how Darwinian thinking destroyed proofs for the existence of God from ‘design’ in the universe. I challenge that prevailing view by looking closely at classical ‘teleological’ arguments for the existence of God. One version championed by Aristotle and Thomas Aquinas stems from how chance is not a sufficient kind of ultimate explanation of the universe. In the course of constructing this argument, I argue that the classical understanding of teleology is no less necessary in modern Darwinian biology than it was in Aristotle's time. In fact, modern biology strengthens the claims that teleological arguments make by vindicating many of their key features. As a consequence, I show how Aristotle and Aquinas' teleological argument for an intelligent First Cause remains valid.