The relation between ex falso and disjunctive syllogism, or even the justification of ex falso based on disjunctive syllogism, is an old topic in the history of logic. This old topic reappears in contemporary logic since the introduction of minimal logic by Johansson. The disjunctive syllogism seems to be part of our general non-problematic inferential practices and superficially it does not seem to be related to or to depend on our acceptance of the frequently (...) disputable ex falso rule. We know that the acceptance of the ex falso is a sufficient condition for the acceptance of the disjunctive syllogism, but the interesting question is: is the ex falso a necessary condition for the acceptance of the disjunctive syllogism? The aim of the present paper is to discuss some possible ways to define systems that combines the preservation of the disjunctive syllogism with the rejection of the ex falso. (shrink)

Disjunctive Syllogism, that is, the inference from 'not-A or B' and 'A', to 'B' can lead from true premises to a false conclusion if each of the sentences 'A' and 'not-A' is a statement of a partial truth such that affirming one of them amounts to denying the other, without each being the contradictory of the other. Such sentences inevitably occur whenever a situation which for its proper precise description needs the use of expressions such as 'most probably (...) true' and so forth, is described (less precisely) by sentences not containing such expressions. (shrink)

In the present paper, we prove the normalization theorem and the consistency of the first-order classical logic with disjunctive syllogism. First, we propose the natural deduction system SCD for classical propositional logic having rules for conjunction, implication, negation, and disjunction. The rules for disjunctive syllogism are regarded as the rules for disjunction. After we prove the normalization theorem and the consistency of SCD, we extend SCD to the system SPCD for the first-order classical logic with (...) class='Hi'>disjunctive syllogism. It can be shown that SPCD is conservative extension to SCD. Then, the normalization theorem and the consistency of SPCD are given. (shrink)

The validity of argument by disjunctive syllogism has been denied by proponents of relevant and paraconsistent logic. DS is stigmatised for its role in inferences — most notably C.I. Lewis's derivation of that fallacy of irrelevance ex falso quodlibet — that involve both it and other rules of inference governing disjunction, or, to speak more precisely, other rules of inference taken to apply to the very same disjunction that obeys DS. In avoiding these inferences the road less travelled (...) is to deny the identity rather than to deny DS: what follows is, then, an exercise in disjoining disjunctions. (shrink)

Routley-Meyer type ternary relational semantics are defined for relevant logics including Routley and Meyer’s basic logic B plus the reductio rule and the disjunctive syllogism. Standard relevant logics such as E and R (plus γ ) and Ackermann’s logics of ‘strenge Implikation’ Π and Π ′ are among the logics considered.

Since a clear account of the fallacy of false disjunctive syllogism is missing in the literature, the fallacy is defined and its three types are differentiated after some preliminaries. Section 4 further elaborates the differentia specifica for each of the three types by analyzing relevant argument criticism of each, as well as the related profiles of dialogue. After defining false disjunctive syllogisms, it becomes possible to distinguish between a false dilemma and a false disjunctive syllogism: (...) section 5 analyzes their similarities and section 6 explains their differences. (shrink)

We provide Routley-Meyer type semantics for relevant logics including Contractionless Ticket Entailment TW (without the truth constant t and o) plus reductio R and Ackermann’s rule γ (i.e., disjunctive syllogism). These logics have the following properties. (i) All have the variable sharing property; some of them have, in addition, the Ackermann Property. (ii) They are stable. (iii) Inconsistent theories built upon these logics are not necessarily trivial.

Relevantism, as a matter of definition, rejects classical logic as incorrect and adopts instead a relevant logic as encapsulating correct inference. It rejects classical logic on the grounds that the rule of Material Detachment, from A and not A or B to infer B,, sometimes leads from truth to falsity. Relevantism — although promoted by some relevant logicians, and an integral part of ultralogic — has recently encountered heavy, but interesting, criticism from relevance logicians themselves.

This chapter focuses on one of the common fallacies in Western philosophy called 'affirming a disjunct' (AAD). It presents a few examples of fallacies in arguments caused by an ambiguity in the English word or. Because context makes the meaning clear in everyday usage, we might never have thought about this ambiguity before, but we actually use the word or in two very distinct ways. One type of or is known as “inclusive”, and the other type is known as “exclusive”. (...) AAD occurs when someone mistakenly believes that an inclusive or is an exclusive or. Importantly, not every or statement can fall prey to AAD. The argument known as a disjunctive syllogism sets out two possibilities that are mutually exclusive. When someone is dead, she/he is not alive, and when someone is alive, she/he is not dead. (shrink)

Mody and Carey (2016) investigated children's capacity to reason by the disjunctive syllogism by hiding stickers within two pairs of cups (i.e., there is one sticker in cup A or B, and one in cup C or D) and then showing one cup to be empty. They found that children as young as 3 years of age chose the most likely cup (i.e., not A, therefore choose B; and disregard C and D) and suggested that these children were (...) representing the dependent relationship between A and B by applying the logical operator “or”. However, it is possible that children succeeded using simpler strategies, such as avoiding the empty cup and choosing within the manipulated pair. We devised a new version of the task in which a sticker was visibly removed from one of the four cups so that 2.5- to 5-year-old children (N = 100) would fail if they relied on such strategies. We also included a conceptual replication of Mody and Carey's (2016) original condition. Our results replicated their findings and showed that even younger children, 2.5 years of age, could pass above chance levels. Yet, 2.5-, 3- and 4-year-olds failed the new condition. Only 5-year-old children performed above chance in both conditions and so provided compelling evidence of deductive reasoning from the premise “A or B", where “or” is exclusive. We propose that younger children may instead conceive of the relationship between A and B as inclusive “or” across both versions of the task. (shrink)

The form of the conditional syllogism resembles that of the categorical syllogism, while its subject matter is at least a conditional premise, but its conclusion is always conditional conjunctive or disjunctive. This mixed structure to which we apply the rules of the categorical syllogism, is a structure of which Aristotle did not have an idea, and which the Stoics did not conceive, and which the non-Arabian logicians did not know until in modern times. But what we (...) have to notice here is the putting of a conditional matter in the form of the categorical syllogism, and it is this kind of hybridization, if we dare to say, which generated this mixed structure which appeared for the first time in the history of logic in the treatise on the logic of Ibn Sina and which can be considered a discovery by this author until proof to the contrary, and that the ancient Arabian logicians have taken the habit of exhibiting in their treatises. (shrink)

Aristotle's modal syllogistic has been an object of study ever since the time of Theophrastus; but these studies have been somewhat desultory. Remarkably, in the 1990s several new lines of research have appeared, with series of original publications by Fred Johnson, Richard Patterson and Ulrich Nortmann. Johnson presented for the first time a formal semantics adequate to a de re reading of the apodeictic syllogistic; this was based on a simple intuition linking the modal syllogistic to Aristotelian metaphysics. Nortmann developed (...) an ingenious de dicto analysis. Patterson articulated the links between the modal syllogistic and the metaphysics, using an analysis which strictly speaking is neither de re nor de dicto. My own studies in this field date from 1976, when my colleague Peter Roeper and I jointly wrote a paper "Aristotle's apodeictic syllogisms" for the XXIInd History of Logic Conference in Krakow. This paper contained the disjunctive reading of particular affirmative apodeictic propositions, which I still favour. Nonetheless, I did not consider that paper's results decisive or comprehensive enough to publish, and my 1981 book The Syllogism contained no treatment of the modal syllogism. The paper's ideas lay dormant till 1989, when I read Johnson's and Patterson's initial articles. I began publishing on the topic in 1991. Gradually my thoughts acquired a certain comprehensiveness and systematicity, till in 1993 I was able to take a semester's sabbatical to write up a draft of this book. (shrink)

Boethius' treatise De Hypotheticis Syllogismis provided twelfth-century philosophers with an introduction to the logic of conditional and disjunctive sentences but this work is the only part of the logica vetus which is no longer studied in the twelfth century. In this paper I investigate why interest in Boethius acount of hypothetical syllogisms fell off so quickly. I argue that Boethius' account of compound sentences is not an account of propositions and once a proper notion of propositionality is available the (...) argument forms accepted by Boethius are seen to be incoherent. It was Peter Abaelard who first understood the nature of propositionality and propositional connectives and used this to criticise Boethius' claims in De Hypothetics Syllogismis. In place Boethius' confusion Abaelard offered a simple and correct account of the hypothetical syllogism. (shrink)

As is known, a logic S is paraconsistent if the rule ECQ (E contradictione quodlibet) is not a rule of S. Not less well known is the fact that Lewis’ modal logics are not paraconsistent. Actually, Lewis vindicates the validity of ECQ in a famous proof currently known as the “Lewis’ proof” or “Lewis’ argument.” This proof essentially leans on the Disjunctive Syllogism as a rule of inference. The aim of this paper is to define a series of (...) paraconsistent logics included in S4 where the Disjunctive Syllogism is valid only as a rule of proof. (shrink)

This paper employs the metaphor of hunting to discuss intellectual investigation. Drawing on the example of Chrysippus’ dog, an animal whose behaviour supposedly reflects disjunctive syllogistic reasoning, the article traces the history of thought. It concludes by summarizing the contribution of Chrysippus’ dog to the fields of literature, philosophy and the visual arts. -/- .

Contents \t\t\t\t\t \tTRANSLATOR'S INTRODUCTION \t\t1 \t \tNOTE ON THE TRANSLATION \t\t39 \t OBSERVATIONS ON THE FEELING OF THE BEAUTIFUL AND SUBLIME \t\t\t\t\t \tSECTION ONE: \t\t\t\t \t\tOf the Distinct Objects of the Feeling of the Beautiful and Sublime \t\t45 \tSECTION TWO: \t\t\t\t \t\tOf the Attributes of the Beautiful and Sublime.

Relevant logics have traditionally been viewed as paraconsistent. This paper shows that this view of relevant logics is wrong. It does so by showing forth a logic which extends classical logic, yet satisfies the Entailment Theorem as well as the variable sharing property. In addition it has the same S4-type modal feature as the original relevant logic E as well as the same enthymematical deduction theorem. The variable sharing property was only ever regarded as a necessary property for a logic (...) to have in order for it to not validate the so-called paradoxes of implication. The Entailment Theorem on the other hand was regarded as both necessary and sufficient. This paper shows that the latter theorem also holds for classical logic, and so cannot be regarded as a sufficient property for blocking the paradoxes. The concept of suppression is taken up, but shown to be properly weaker than that of variable sharing. (shrink)

This chapter contains sections titled: Non‐Sequiturs are Bad The Real Use of Premises Implication From Proof Theory to Semantics Adding Conjunction The Problem of Disjunction Routley and Meyer's Ternary Relation Rules for Disjunction The Semantics of Negation Rules for Negation Disjunctive Syllogism Logics Stronger than R Logics Weaker than R Relevant Logics and Natural Language Conditionals Theory of Properties Summary.

Imagine a dog tracing a scent to a crossroads, sniffing all but one of the exits, and then proceeding down the last without further examination. According to Sextus Empiricus, Chrysippus argued that the dog effectively employs disjunctive syllogism, concluding that since the quarry left no trace on the other paths, it must have taken the last. The story has been retold many times, with at least four different morals: (1) dogs use logic, so they are as clever as (...) humans; (2) dogs use logic, so using logic is nothing special; (3) dogs reason well enough without logic; (4) dogs reason better for not having logic. This paper traces the history of Chrysippus's dog, from antiquity up to its discussion by relevance logicians in the twentieth century. (shrink)

Take a correct sequent of formal logic, perhaps a simple logical truth, like the law of excluded middle, or something with premises, like disjunctive syllogism, but basically a claim of the form \.Γ can be empty. If you don’t like my examples, feel free to choose your own, everything I have to say should apply to those as well. Such a sequent attributes the properties of logical truth or logical consequence to a schematic sentence or argument. This paper (...) aims to answer the question of how beliefs in such attributions are justified, on both its descriptive and normative interpretations; I aim to say when we generally take ourselves to be justified in forming such beliefs, and to make it plausible that beliefs formed this way really are justified.We can ask such questions about many domains but there are special difficulties for answering them in logic. Some of the difficulties stem from the fact that logic is thought t.. (shrink)

We first consider the entailment logic MC, based on meaning containment, which contains neither the Law of Excluded Middle (LEM) nor the Disjunctive Syllogism (DS). We then argue that the DS may be assumed at least on a similar basis as the assumption of the LEM, which is then justified over a finite domain or for a recursive property over an infinite domain. In the latter case, use is made of Mathematical Induction. We then show that an instance (...) of the LEM is intrumental in the proof of Cantor's Theorem, and we then argue that this is based on a more general form than can be reasonably justified. We briefly consider the impact of our approach on arithmetic and naive set theory, and compare it with intuitionist mathematics and briefly with recursive mathematics. Our "Four Basic Logical Issues" paper would provide useful background, the current paper being an application of the some of the ideas in it. (shrink)

Epistemic modals have peculiar logical features that are challenging to account for in a broadly classical framework. For instance, while a sentence of the form ‘p, but it might be that not p’ appears to be a contradiction, 'might not p' does not entail 'not p', which would follow in classical logic. Likewise, the classical laws of distributivity and disjunctive syllogism fail for epistemic modals. Existing attempts to account for these facts generally either under- or over-correct. Some theories (...) predict that 'p and might not p', a so-called epistemic contradiction, is a contradiction only in an etiolated sense, under a notion of entailment that does not allow substitution of logical equivalents; these theories underpredict the infelicity of embedded epistemic contradictions. Other theories savage classical logic, eliminating not just rules that intuitively fail, like distributivity and disjunctive syllogism, but also rules like non-contradiction, excluded middle, De Morgan’s laws, and disjunction introduction, which intuitively remain valid for epistemic modals. In this paper, we aim for a middle ground, developing a semantics and logic for epistemic modals that makes epistemic contradictions genuine contradictions and that invalidates distributivity and disjunctive syllogism but that otherwise preserves classical laws that intuitively remain valid. We start with an algebraic semantics, based on ortholattices instead of Boolean algebras, and then propose a more concrete possibility semantics, based on partial possibilities related by compatibility. Both semantics yield the same consequence relation, which we axiomatize. Then we show how to extend our semantics to explain parallel phenomena involving probabilities and conditionals. The goal throughout is to retain what is desirable about classical logic while accounting for the non-classicality of epistemic vocabulary. (shrink)

I critique an ancient argument for the possibility of non-linguistic deductive inference. The argument, attributed to Chrysippus, describes a dog whose behavior supposedly reflects disjunctive syllogistic reasoning. Drawing on contemporary robotics, I urge that we can equally well explain the dog's behavior by citing probabilistic reasoning over cognitive maps. I then critique various experimentally-based arguments from scientific psychology that echo Chrysippus's anecdotal presentation.

One of the main controversies of the Logic Schools of the 12th century centered on the question: What follows from the impossible? In this paper arguments for two diametrically opposed positions are examined. The author of the ‘Avranches Text’ who probably belonged to the school of the Parvipontani defended the view that from an impossible proposition everything follows (‘Ex impossibili quodlibet’). In particular he developed a proof to show that by means of so-called ‘disjunctive syllogism’ any arbitrary proposition (...) B can be logically derived from a pair of contradictory propositions A and Not-A. The author of the Ars Meliduna instead argued that nothing follows from an impossible proposition (‘ex falso nihil sequitur’). This view is supported by various counterexamples which aimed to show that the admission of impossible premises would give rise to inconsistent conclusions. Upon closer analysis these inconsistencies do not, however, have the formal structure of a real contradiction like A and Not-A, but rather the structure of two rivalling conditionals like ‘If B then A’ and ‘If B then Not-A’. Hence these counterexamples rather have to be considered as refutations of the basic principles of ‘connexive logic’. (shrink)

Because we humans speak a public language, there has always been a special reason to suppose that we have a language of thought. For nonhuman animals, this special reason is missing, and the issue is less straightforward. On the one hand, there is evidence of various types of nonlinguistic representations, such as analog magnitude representations, which can explain many types of intelligent behavior. But on the other hand, the mere fact that some aspects of animal cognition can be explained by (...) nonlinguistic representations hardly suffices to show that animal minds are bereft of any sentence-like representations. This paper explores these complexities and argues that empirical research into the logical abilities of nonhuman animals provides a more direct way of evaluating whether they have a language of thought. Along the way, I offer a novel suggestion of how the hypothesis that animals draw disjunctive syllogisms can be empirically distinguished from the hypothesis, defended by Michael Rescorla, that animals reason probabilistically in accordance with Bayes’ Law. (shrink)

This paper argues that some widely used laws of implication are false, and arguments based upon them invalid. These laws are Exportation, Commutation, (as well as various restricted forms of these), Exported Syllogism and Disjunctive Syllogism. All these laws are false for the same reason – that they license the suppression or replacement in some position of some class of propositions which cannot legitimately be suppressed or replaced. These laws fail to preserve the property of sufficiency of (...) premiss set for conclusion. They are false, and can be seen to be false, independently of their respon- sibility for the paradoxes. Hence the main ‘independent’ argument for the paradoxes – that they follow from an allegedly immaculate set of laws – is undermined. Counterexamples to all these laws are produced. (shrink)

There is a strong case to be made for thinking that an obscure logic, KR, is better than classical logic and better than any relevant logic. The argument for KR over relevant logics is that KR counts disjunctive syllogism valid, and this is the biggest complaint about relevant logics. The argument for KR over classical logic depends on the normativity of logic and the paradoxes of implication. The paradoxes of implication are taken by relevant logicians to justify relevant (...) logic, but considerations on the normativity of logic show that only some of the paradoxes of implication are genuine. KR avoids all the genuine paradoxes of implication, unlike classical logic. Overall, KR avoids the genuine paradoxes of implication and avoids the major objection to relevant logics. This combination of features provides strong reason to give KR a place in the conversation about the right logic. (shrink)

This paper gives an account of Anderson and Belnap’s selection criteria for an adequate theory of entailment. The criteria are grouped into three categories: criteria pertaining to modality, those pertaining to relevance, and those related to expressive strength. The leitmotif of both this paper and its prequel is the relevant legitimacy of disjunctive syllogism. Relevant logics are commonly held to be paraconsistent logics. It is shown in this paper, however, that both E and R can be extended to (...) explosive logics which satisfy all of Anderson and Belnap’s selection criteria, provided the truth-constant known as the Ackermann constant is available. One of the selection criteria related to expressive strength is having an “enthymematic” conditional for which a deduction theorem holds. I argue that this allows for a new interpretation of Anderson and Belnap’s take on logical consequence, namely as committing them to pluralism about logical consequence. (shrink)

This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Restall, J Philos Logic 22(5):481–511, 1993 ) concerning the modelling conditions for the axioms of assertion A → (( A → B ) → B ) (there called c 6) and permutation ( A → ( B → C )) → ( B → ( A → C )) (there called c 7). We show that the modelling conditions for assertion and permutation proposed (...) in ‘Simplified Semantics’ overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the “base point” is inconsistent. This problem is not restricted to ‘Simplified Semantics.’ The techniques of that paper are used in Graham Priest’s textbook An Introduction to Non-Classical Logic (Priest, 2001 ), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose two different corrections. (shrink)

This paper offers general support for what Valerie Plumwood’s paper is trying to achieve by supporting the rejection of each of her four “false laws of logic”: exportation, illegitimate replacement, commutation (aka. permutation) and disjunctive syllogism. We start by considering her general characterizations of entailment, beginning with her stated definition of entailment as the converse of deducibility. However, this applies to a wide range of relevant logics and so is not able to be used as a criterion for (...) deciding what laws to include in a logic. In this context, we examine the two key differences between deduction from premises to conclusion and entailment from antecedent to consequent. We also consider her use of sufficiency as a general characterizing feature. We then discuss Plumwood’s syntactic criteria used to reject the first three of her false laws of logic and add the Relevance Condition in this context. We next consider semantic characterizing criteria for a logic. After making a case against using truth, we introduce Brady’s logic MC of meaning containment. We then examine the content semantics for MC and use it to reject all of Plumwood’s false laws of logic together with some others. We follow with the related Depth Relevance Condition, which is a syntactic cri- terion satisfied by MC. This clearly rejects the first three of these laws and many others, but not the fourth law. We conclude by giving our overall support for her general enterprise. (shrink)

Some nonlinguistic systems of representation display some of the six features of a language-of-thought (LoT) delineated by Quilty-Dunn et al. But they conjecture something stronger: That all six features cooccur homeostatically in nonlinguistic thought. Here I argue that there is no good evidence for nonlinguistic deductive reasoning involving the disjunctive syllogism. Animals and prelinguistic children probably do not make logical inferences.

In this paper we present a logic that determines when implications in a classical logic context express a relevant connection between antecedent and consequent. In contrast with logics in the relevance logic literature, we leave classical negation intact—in the sense that the law of non-contradiction can be used to obtain relevant implications, as long as there is a connection between antecedent and consequent. On the other hand, we give up the requirement that our theory of relevance should be able to (...) define a new standard of deduction. We present and argue for a list of requirements such a logical theory of classical relevance needs to meet and go on to formulate a system that respects each of these requirements. The presented system is a Tarski logic that extends the relevance logic R with a new relevant implication which allows for Disjunctive Syllogism and similar rules. This is achieved by interpreting the logical symbols in the antecedents in a stronger way than the logical symbols in consequents. A proof theory and an algebraic semantics are formulated and interesting metatheorems are proven. Finally we give a philosophical motivation for our non-standard relevant implication and the asymmetric interpretation of antecedents and consequents. (shrink)

In this paper we shall try to understand what it is to beg the question, and since begging the question is generally believed to be linked with circularity, we shall also explore this relationship. Finally, we shall consider whether certain forms of valid argument can go through smoothly in anepistemio context without begging the question. We shall consider, especially, the claims of the disjunctive syllogism in this regard.

In this paper we first develop a logic independent account of relevant implication. We propose a stipulative denition of what it means for a multiset of premises to relevantly L-imply a multiset of conclusions, where L is a Tarskian consequence relation: the premises relevantly imply the conclusions iff there is an abstraction of the pair such that the abstracted premises L-imply the abstracted conclusions and none of the abstracted premises or the abstracted conclusions can be omitted while still maintaining valid (...) L-consequence. Subsequently we apply this denition to the classical logic consequence relation to obtain NTR-consequence, i.e. the relevant CL-consequence relation in our sense, and develop a sequent calculus that is sound and complete with regard to relevant CL-consequence. We present a sound and complete sequent calculus for NTR. In a next step we add rules for an object language relevant implication to the sequent calculus. The object language implication reflects exactly the NTR-consequence relation. One can see the resulting logic NTR-> as a relevant logic in the traditional sense of the word. By means of a translation to the relevant logic R, we show that the presented logic NTR is very close to relevance logics in the Anderson-Belnap-Dunn-Routley-Meyer tradition. However, unlike usual relevant logics, NTR is decidable for the full language, Disjunctive Syllogism and Adjunction are valid, and neither Modus Ponens nor the Cut rule are admissible. (shrink)

According to John A Barker, whether an argument begs the question is purely a matter of logical form. According to me, it is also a matter of epistemic conditions; some arguments which beg the question in some contexts need not beg the question in every context. I point out difficulties in Barker's treatment and defend my own views against some of his criticisms. In the concluding section, "Alleged difficulties with disjunctive syllogism," I defend the validity of disjunctive (...) syllogism against the views of Alan Ross Anderson and Nuel D Belnap, Jr., ask pointed questions about the notion of intensional disjunction, and suggest how my treatment of begging the question can be extended to deal with the so-called paradoxes of strict implication. (shrink)