A consistent finding in research on conditional reasoning is that individuals are more likely to endorse the valid modus ponens (MP) inference than the equally valid modus tollens (MT) inference. This pattern holds for both abstract task and probabilistic task. The existing explanation for this phenomenon within a Bayesian framework (e.g., Oaksford & Chater, 2008) accounts for this asymmetry by assuming separate probability distributions for both MP and MT. We propose a novel explanation within a computational-level Bayesian (...) account of reasoning according to which “argumentation is learning”. We show that the asymmetry must appear for certain prior probability distributions, under the assumption that the conditional inference provides the agent with new information that is integrated into the existing knowledge by minimizing the Kullback-Leibler divergence between the posterior and prior probability distribution. We also show under which conditions we would expect the opposite pattern, an MT-MP asymmetry. (shrink)

This paper defends a counterexample to Modus Tollens, and uses it to draw some conclusions about the logic and semantics of indicative conditionals and probability operators in natural language. Along the way we investigate some of the interactions of these expressions with 'knows', and we call into question the thesis that all knowledge ascriptions have truth-conditions.

We establish a probabilized version of modus tollens, deriving from p(E|H)=a and p()=b the best possible bounds on p(). In particular, we show that p() 1 as a, b 1, and also as a, b 0. Introduction Probabilities of conditionals Conditional probabilities 3.1 Adams' thesis 3.2 Modus ponens for conditional probabilities 3.3 Modus tollens for conditional probabilities.

Medical diagnosis has been traditionally recognized as a privileged field of application for so called probabilistic induction. Consequently, the Bayesian theorem, which mathematically formalizes this form of inference, has been seen as the most adequate tool for quantifying the uncertainty surrounding the diagnosis by providing probabilities of different diagnostic hypotheses, given symptomatic or laboratory data. On the other side, it has also been remarked that differential diagnosis rather works by exclusion, e.g. by modus tollens, i.e. deductively. By drawing (...) on a case history, this paper aims at clarifying some points on the issue. Namely: 1) Medical diagnosis does not represent, strictly speaking, a form of induction, but a type, of what in Peircean terms should be called ‘abduction’ (identifying a case as the token of a specific type); 2) in performing the single diagnostic steps, however, different inferential methods are used for both inductive and deductive nature: modus tollens, hypothetical-deductive method, abduction; 3) Bayes’ theorem is a probabilized form of abduction which uses mathematics in order to justify the degree of confidence which can be entertained on a hypothesis given the available evidence; 4) although theoretically irreconcilable, in practice, both the hypothetical- deductive method and the Bayesian one, are used in the same diagnosis with no serious compromise for its correctness; 5) Medical diagnosis, especially differential diagnosis, also uses a kind of “probabilistic modus tollens”, in that, signs (symptoms or laboratory data) are taken as strong evidence for a given hypothesis not to be true: the focus is not on hypothesis confirmation, but instead on its refutation [Pr (¬ H/E1, E2, …, En)]. Especially at the beginning of a complicated case, odds are between the hypothesis that is potentially being excluded and a vague “other”. This procedure has the advantage of providing a clue of what evidence to look for and to eventually reduce the set of candidate hypotheses if conclusive negative evidence is found. 6) Bayes’ theorem in the hypothesis-confirmation form can more faithfully, although idealistically, represent the medical diagnosis when the diagnostic itinerary has come to a reduced set of plausible hypotheses after a process of progressive elimination of candidate hypotheses; 7) Bayes’ theorem is however indispensable in the case of litigation in order to assess doctor’s responsibility for medical error by taking into account the weight of the evidence at his disposal. (shrink)

Suppose that some contradictions are true – for example, that as I walk through the door, I’m inside and I’m not inside. Then we argue 'if I'm walking through the door, I'm inside; I'm not inside; therefore, I'm not walking through the door' is an invalid instance of modus tollens.

Epistemic modals in consequent place of indicative conditionals give rise to apparent counterexamples to Modus Tollens. Familiar assumptions behind familiar truth conditional theories of embedded modality facilitate a prima facie explanation—viz., that the target cases harbor epistemic modal equivocations. However, this sort of explanation goes too far. It fosters other predictions of equivocation in places where in fact there are none. It is argued that the solution is to drop the credo that modal claims are inherently relational in (...) favor of a view that treats them as inherently quantificational. In particular it is suggested that epistemic modals express covert mass noun descriptions of information of the form “the actionable information stands in logical relation L to prejacent p”. We demonstrate how this approach unlocks the equivocation problem. (shrink)

Modus Tollens is the following valid deductive argument form: “If P, then Q. Not Q. Therefore not P.” I show how this structure plays an important part in everyday argument and in everyday non-argument; I show how the argument form fits into non-argument cases. The structure is common as argument, as rhetorical emphasis, and as explanation. Students can see how this pattern is rooted in everyday thought, when elements of the structure are unspoken but nonetheless relied upon, what (...) pictures the structure evokes, and how these pictures and this pattern fit into everyday thought and discourse. Many examples are provided. A homework handout is presented which encourages the student to find and explicate sample cases from current media, world literature, movies, proverbs, etc. (shrink)

There are narrowest bounds for P(h) when P(e) = y and P(h/e) = x, which bounds collapse to x as y goes to 1. A theorem for these bounds -- bounds for probable modus ponens -- entails a principle for updating on possibly uncertain evidence subject to these bounds that is a generalization of the principle for updating by conditioning on certain evidence. This way of updating on possibly uncertain evidence is appropriate when updating by ’probability kinematics’ or ’Jeffrey-conditioning’ (...) is, and apparently in countless other cases as well. A more complicated theorem due to Karl Wagner -- bounds for probable modus tollens -- registers narrowest bounds for P(not h) when P(not e) = y and P(e/h) = x. This theorem serves another principle for updating on possibly uncertain evidence that might be termed ’contraditioning’, though it is for a way of updating that seems in practice to be frequently not appropriate. It is definitely not a way of putting down a theory -- for example, a random-chanc. (shrink)

I argue that vagueness produces counterexamples to modus tollens. I begin by outlining cases where indicative and counterfactual conditionals seem intuitively to be determinate even when their antecedents are borderline and their consequents are determinately false. Accepting these intuitions has some revisionary implications; however, rejecting them leads to unacceptable consequences for our knowledge of conditionals. I thus take it that we should accept that our intuitions are reliable. I show it follows that modus tollens fails. I (...) conclude by defending this consequence from some objections. (shrink)

The precedent studies on the validity of Modus ponens and Modus tollens have been carried out with most regard to a major type of conditionals in which the conditional clause is a sufficient condition for the main clause. But we sometimes, in natural language arguments, find other types of conditionals in which the conditional clause is a necessary or necessary and sufficient condition for the main clause. In this paper I reappraise, on the basis of new definitions (...) of Modus ponens and Modus tollens, their validity/invalidity in natural language arguments in consideration of all types of conditionals. (shrink)

We present a meta-analytic review on the processing of negations in conditional reasoning about affirmation problems (Modus Ponens: "MP", Affirmation of the Consequent "AC") and denial problems (Denial of the Antecedent "DA", and Modus Tollens "MT"). Findings correct previous generalisations about the phenomena. First, the effects of negation in the part of the conditional about which an inference is made, are not constrained to denial problems. These inferential-negation effects are also observed on AC. Second, there generally are (...) reliable effects of a negation in the clause referred to by the categorical premise, and these referred-negation effects are constrained to the logically fallacious AC and DA inferences. All findings are presented and discussed in relation to contemporary mental model (MM) and mental logic (ML) theories. It is argued that a double-negation elimination hypothesis provides a sufficient explanation of inferential-negation effects within both MM theory and ML theory, if the latter is extended by a validating search for counter examples. Both MM and ML theories adhere to a processing scheme that allows them to incorporate an account of referred-negation effects based on the thesis that counter-example frequency is modulated by the scope of a contrast class delineated by a false affirmative. We conclude that MM and ML theories provide adequate processing schemes to accommodate for the explanatory hypotheses, at least in principle. In practice, both approaches remain equivocal as regards the connectivity and interactivity with long-term memory knowledge invoked in generating, manipulating, and testing the mental representations of negative state of affairs. (shrink)

That one person's modus ponens is another's modus tollens is the bane of philosophy because it strips many philosophical arguments of their persuasive force. The problem is that philosophical arguments become mere pantomemes: arguments that are reasonable to resist simply by denying the conclusion. Appeals to proof, intuition, evidence, and truth fail to alleviate the problem. Two broad strategies, however, do help in certain circumstances: an appeal to normal informal standards of what is reasonable (nisowir) and argument (...) by interpretation. The method of explication features prominently in both strategies, and the paper extends this method to apply to nisowir, introducing the concept of canonical explication. The paper illustrates the two strategies with examples of arguments from formal epistemology and suggests that an appeal to nisowir might help to defend against philosophical scepticism by shifting the burden of proof to the sceptic. (shrink)

We present a meta-analytic review on the processing of negations in conditional reasoning about affirmation problems (Modus Ponens: “MP”, Affirmation of the Consequent “AC”) and denial problems (Denial of the Antecedent “DA”, and Modus Tollens “MT”). Findings correct previous generalisations about the phenomena. First, the effects of negation in the part of the conditional about which an inference is made, are not constrained to denial problems. These inferential-negation effects are also observed on AC. Second, there generally are (...) reliable effects of a negation in the clause referred to by the categorical premise, and these referred-negation effects are constrained to the logically fallacious AC and DA inferences. All findings are presented and discussed in relation to contemporary mental model (MM) and mental logic (ML) theories. It is argued that a double-negation elimination hypothesis provides a sufficient explanation of inferential-negation effects within both MM theory and ML theory, if the latter is extended by a validating search for counter examples. Both MM and ML theories adhere to a processing scheme that allows them to incorporate an account of referred-negation effects based on the thesis that counter-example frequency is modulated by the scope of a contrast class delineated by a false affirmative. We conclude that MM and ML theories provide adequate processing schemes to accommodate for the explanatory hypotheses, at least in principle. In practice, both approaches remain equivocal as regards the connectivity and interactivity with long-term memory knowledge invoked in generating, manipulating, and testing the mental representations of negative state of affairs. (shrink)

Broadly speaking, the contemporary scientific realist is concerned to justify belief in what we might call theoretical truth, which includes truth based on ampliative inference and truth about unobservables. Many, if not most, contemporary realists say scientific realism should be treated as ‘an overarching scientific hypothesis’ (Putnam 1978, p. 18). In its most basic form, the realist hypothesis states that theories enjoying general predictive success are true. This hypothesis becomes a hypothesis to be tested. To justify our belief in the (...) realist hypothesis, realists commonly put forward an argument known as the ‘no-miracles argument’. With respect to the basic hypothesis this argument can be stated as follows: it would be a miracle were our theories as successful as they are, were they not true; the only possible explanation for the general predictive success of our scientific theories is that they are true. (shrink)

Relatively little is known about those who consistently produce the valid response to Modus Tollens (MT) problems. In two studies, people who responded correctly to MT problems indicated how “convinced” they were by proofs of conditional reasoning conclusions. The first experiment showed that MT competent reasoners found accurate proofs of MT reasoning more convincing than similar “proofs” of invalid reasoning. Similarly, there was a tendency for MT competent reasoners to find an initial counterfactual supposition more convincing than did (...) people who were less competent in MT. The second experiment showed that when individuals produced the correct MT response, and found correct MT proofs to be more convincing than “bogus” proofs, they were also less likely to find the conclusions to Denying the Antecedent, or Affirming the Consequent problems valid, compared to individuals who could not discriminate between valid and bogus MT proofs. These findings are discussed in terms of both their implications for the mental logic and mental models positions, and individual differences in System 1 and System 2 reasoning. (shrink)

Epistemic modals in consequent place of indicative conditionals give rise to apparent counterexamples to Modus Ponens and Modus Tollens. Familiar assumptions of fa- miliar truth conditional theories of modality facilitate a prima facie explanation—viz., that the target cases harbor epistemic modal equivocations. However, these explana- tions go too far. For they foster other predictions of equivocation in places where in fact there are no equivocations. It is argued here that the key to the solution is to drop (...) the assumption that modal claims are inherently relational (i.e., that they ex- press a logical relation between a prejacent and a premise-set) in favor of a view that treats them as inherently quantificational. In particular it is suggested that modals are mass noun descriptions of information. We demonstrate how this approach unlocks the equivocation problem. (shrink)

Two experiments examined the role of conditional reasoning in the logical deduction game, Mastermind . An analysis suggested that Modus Tollens (MT) reasoning could be used to determine the code structure, for example, in determining if any of the colours in the code are repeated. Consistent with this analysis, Experiment 1 showed that only MT errors are correlated with the number of hypotheses advanced in Mastermind . A subsequent analysis showed that conditional reasoning such as Affirming the Consequent (...) (AC) and Denying the Antecedent (DA) could lead to particularly damaging inferences only when the code was four different colours. When that was known before play, Experiment 2 showed that AC errors, but not MT errors, were significantly correlated with Mastermind hypotheses advanced. A stepwise multiple regression analysis supported these findings: When the solvers knew they were playing a four-colour code, there was a slight diminution in the variance explained by MT errors, and a significant increase in the variance explained by AC errors. An analysis of the number of different possible codes that could be consistent with hypotheses actually played showed that the number of such codes is far fewer when the code consists of four different colours than when its structure is not known. This analysis suggests that reasoners are therefore unlikely to discover many alternative causes for the feedback given when the code consists of four different colours, and it is under these conditions that humans are most likely to engage in AC reasoning. (shrink)

There has long been a history of studies investigating how people (“ordinary people”) perform on tasks that involve deductive reasoning. The upshot of these studies is that people characteristically perform some deductive tasks well but others badly. For instance, studies show that people will typically perform MP (“modus ponens”: from ‘If A then B’ and ‘A’, infer ‘B’) and bi-conditional MP (from: ‘A if and only if B’ and ‘A’, infer ‘B’) correctly when invited to make the inference and (...) additionally can discover of their own accord when such inferences are appropriate. On the other hand, the same studies show that people typically perform MT (“modus tollens”: from ‘If A then B’ and ‘not-B’, infer ‘not-A’) and biconditional MT badly. They not only do not recognize when it is appropriate to draw such inferences, but also they will balk at doing them even when they are told that they can make it. Related to these shortcomings seems to be the inability of people to understand that contrapositives are equivalent (that ‘If A then B’ is equivalent to ‘If not-B then not-A’). [Studies of people’s deductive inference-drawing abilities have a long history, involving many studies in the early 20th century concerning Aristotelian syllogisms. But the current spate of studies draws much of its impetus from Wason (Wason, 1968; see also Wason & Johnson-Laird, 1972). ] The general conclusion seems to be that there are specific areas where “ordinary people” do not perform very logically. This conclusion will not come as a surprise to teachers of elementary logic, who have long thought that the majority of “ordinary people” are inherently illogical and need deep and forceful schooling in order to overcome this flaw. (shrink)

ABSTRACT: This paper traces the earliest development of the most basic principle of deduction, i.e. modus ponens (or Law of Detachment). ‘Aristotelian logic’, as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classified as ‘hypothetical syllogisms’. However, Aristotle did not discuss (...) such arguments, nor did he call any arguments ‘hypothetical syllogisms’. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them ‘hypothetical syllogisms’; nor did they describe them as ponendo ponens, etc. The tradition of the four argument forms and the classification of the arguments as hypothetical syllogisms hence need some explaining. In this paper, I offer some explanations by tracing the development of certain elements of Aristotle’s logic via the early Peripatetics to the logic of later antiquity. I consider the questions: How did the four argument forms arise? Why were there four of them? Why were arguments of these forms called ‘hypothetical syllogisms’? On what grounds were they considered valid? I argue that such arguments were neither part of Aristotle’s dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle’s logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some 20th century theories. (shrink)

Recently, there has been a shift away from traditional truth-conditional accounts of meaning towards non-truth-conditional ones, e.g., expressivism, relativism and certain forms of dynamic semantics. Fueling this trend is some puzzling behavior of modal discourse. One particularly surprising manifestation of such behavior is the alleged failure of some of the most entrenched classical rules of inference; viz., modus ponens and modus tollens. These revisionary, non-truth-conditional accounts tout these failures, and the alleged tension between the behavior of modal (...) vocabulary and classical logic, as data in support of their departure from tradition, since the revisionary semantics invalidate some of these patterns. I, instead, offer a semantics for modality with the resources to accommodate the puzzling data while preserving classical logic, thus affirming the tradition that modals express ordinary truth-conditional content. My account shows that the real lesson of the apparent counterexamples is not the one the critics draw, but rather one they missed: namely, that there are linguistic mechanisms, reflected in the logical form, that affect the interpretation of modal language in a context in a systematic and precise way, which have to be captured by any adequate semantic account of the interaction between discourse context and modal vocabulary. The semantic theory I develop specifies these mechanisms and captures precisely how they affect the interpretation of modals in a context, and do so in a way that both explains the appearance of the putative counterexamples and preserves classical logic. (shrink)

The modus ponens (A -> B, A :. B) is, along with modus tollens and the two logically not valid counterparts denying the antecedent (A -> B, ¬A :. ¬B) and affirming the consequent, the argument form that was most often investigated in the psychology of human reasoning. The present contribution reports the results of three experiments on the probabilistic versions of modus ponens and denying the antecedent. In probability logic these arguments lead to conclusions with (...) imprecise probabilities. In the modus ponens tasks the participants inferred probabilities that agreed much better with the coherent normative values than in the denying the antecedent tasks, a result that mirrors results found with the classical argument versions. For modus ponens a surprisingly high number of lower and upper probabilities agreed perfectly with the conjugacy property (upper probabilities equal one complements of the lower probabilities). When the probabilities of the premises are imprecise the participants do not ignore irrelevant (“silent”) boundary probabilities. The results show that human mental probability logic is close to predictions derived from probability logic for the most elementary argument form, but has considerable difficulties with the more complex forms involving negations. (shrink)

The book provides an in-depth discussion on the human nature concept from different perspectives and from different disciplines, analyzing its use in the doping debate and researching its normative overtones. The relation between natural talent and enhanced abilities is scrutinized within a proper conceptual and theoretical framework: is doping to be seen as a factor of the athlete’s dehumanization or is it a tool to fulfill his/her aspirations to go faster, higher and stronger? Which characteristics make sports such a peculiar (...) subject of ethical discussion and what are the, both intrinsic and extrinsic, moral dangers and opportunities involved in athletic enhancement? This volume combines fundamental philosophical anthropological reflection with applied ethics and socio-cultural and empirical approaches. Furthermore it presents guidelines to decision- and policy-makers on local, national and international levels. (shrink)

There is an increasing awareness of the importance of food security, of which the UN’s Millennium Development Goals are the best measure. Although some progress has been made in some regions, much progress still needs to be made in Sub-Saharan Africa, where agriculture largely remains subsistence and personal savings extremely low, and where population growth outstrips economic growth.Thus, there has been a renewed effort to bring these problems back on the development agenda. Food insecurity is a major manifestation of poverty, (...) and hence agricultural development is the most effective way out of rural poverty.The ethical imperative is to bring about the political will to effect such development. Institutions must be built that focus on smallholder development as an engine of economic growth. Along with classic issues of development , the rural poor, who respond well to market-based development and are thus good micro-credit risks, must be given access to markets and finances.The drive to bring the rural poor into the market is frustrated, however, by the subsidies that the Northern nations use to prop up their own markets and farmers. One of the most pressing ethical questions, then, remains how the North can continue to support its subsidy system, while, at the same time, contributing to development work. (shrink)