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.

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)

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)

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)

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)

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)

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)

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)

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)

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)

Harm-reduction approaches are used to reduce the burden of risky human behaviour without necessarily aiming to stop the behaviour. We discuss what an introduction of harm reduction for doping in sports would mean in parallel with a relaxation of the antidoping rule. We analyse what is ethically at stake in the following five levels: (1) What would it mean for the athlete (the self)? (2) How would it impact other athletes (the other)? (3) How would it affect the phenomenon of (...) sport as a game and its fair play basis (the play)? (4) What would be the consequences for the spectator and the role of sports in society (the display)? and (5) What would it mean for what some consider as essential to being human (humanity)? For each level, we present arguments for and against doping and then discuss what a harm-reduction approach, within a dynamic regime of a partially relaxed antidoping rule, could imply. We find that a harm-reduction approach is morally defensible and potentially provides a viable escape out of the impasse resulting from the impossibility of attaining the eradication of doping. The following question remains to be answered: Would a more relaxed position, when combined with harm-reduction measures, indeed have less negative consequences for society than today9s all-out antidoping efforts that aim for abstinence? We provide an outline of an alternative policy, allowing a cautious step-wise change to answer this question and then discuss the ethical aspects of such a policy change. (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)

There are two possible realist defense strategies against the pessimistic meta-induction and Laudan’s meta-modus tollens: the selective strategy, claiming that discarded theories are partially true, and the discontinuity strategy, denying that pessimism about past theories can be extended to current ones. A radical version of discontinuity realism is proposed by Gerald Doppelt: rather than discriminating between true and false components within theories, he holds that superseded theories cannot be shown to be even partially true, while present best theories (...) are demonstrably completely true. I argue that this position, running counter both the cumulativity of science and fallibilism, is untenable; it cannot account for the success of past theories, nor for the failures of current theories, and rather than shutting the door to the pessimistic historical objections it opens it wide. The best strategy, instead, joins the selective idea there was both some truth and some falsity in discarded theories, like in current ones, with the moderate discontinuity idea that the truth rate in present best theories is much greater than in past ones. (shrink)

This chapter focuses on one of the common fallacies in Western philosophy, 'denying the antecedent'. Like affirming the consequent, denying the antecedent is also a fallacious form of reasoning in formal logic. This time the problem occurs when the minor premise of a propositional syllogism denies the antecedent of a conditional statement. Denying the antecedent makes the mistake of assuming that if the antecedent is denied, then the consequent must also be denied. Like modus ponens, modus tollens (...) is a valid argument form because the truth of the premises guarantees the truth of the conclusion; however, like affirming the consequent, denying the antecedent is an invalid argument form because the truth of the premises does not guarantee the truth of the conclusion. The invalidity of denying the antecedent is confirmed by a truth table presented in the chapter. (shrink)

This thesis is about indicative conditionals and apparent counterexamples to classically valid argument forms. Specifically, it applies the following four theories: - material (inspired by Grice (1961, 1975 and 1989)); - possible-worlds (inspired by Stalnaker (1981); Lewis (1976); and Kratzer (2012)), - suppositional (inspired by Adams (1975) and Edgington (1995 and 2014)); and - hybrid (inspired by Jackson (1987)) to try and solve the following two counterexamples: - Vann McGee’s to modus ponens (1985); and - Lewis Carroll’s to (...) class='Hi'>modus tollens (1894). I argue that none of the theories I consider can explain – without facing any problems – the three individually plausible but jointly inconsistent theses that give rise to the apparent counterexamples. The three theses of the trilemma are the following: (1) the argument is invalid; (2) the argument is an instance of modus ponens or modus tollens; and (3) modus ponens or modus tollens is valid. (shrink)

This paper assesses the prospects of a pragmatist theory of content. I begin by criticising the theory presented in D.H. Mellor’s essay ‘Successful Semantics’. I then identify problems and lacunae in the pragmatist theory of meaning sketched in Chapter 13 of Dummett’s The Logical Basis of Metaphysics. The prospects are brighter, I contend, for a tempered pragmatism, in which the theory of content is permitted to draw upon irreducible notions of truth and falsity. I sketch the shape of such a (...) theory and illustrate the role of its pragmatist elements by showing how they point towards a promising account of the truth conditions of indicative conditionals. A feature of the account is that it validates Modus Ponens whilst invalidating Modus Tollens. (shrink)

A modus tollens against zero-dimensional material objects is presented from the premises (i) that if there are zero-dimensional material objects then there are bare particulars, and (ii) that there are no bare particulars. The argument for the first premise proceeds by elimination. First, bare particular theory and bundle theory are motivated as the most appealing theories of property exemplification. It is then argued that the bundle theorist’s Ockhamism ought to lead her to reject spatiotemporally located zero-dimensional property instances. (...) Finally, it is argued that since she must accept such instances if she accepts zero-dimensional material object bundles, she ought to avoid the latter. This leaves bare particular theory as the default view of zero-dimensional material objects. The argument for the second premise invokes the thesis that the exemplification of at least one sparse property is a prerequisite for the existence of any particular. It is argued from Humean considerations that bare particulars fail this prerequisite. (shrink)

As the final component of a chain of reasoning intended to take us all the way to logical nihilism, Russell (2018) presents the atomic sentence ‘prem’ which is supposed to be true when featuring as premise in an argument and false when featuring as conclusion in an argument. Such a sentence requires a non-reflexive logic and an endnote by Russell (2018) could easily leave the reader with the impression that going non-reflexive suffices for logical nihilism. This paper shows how one (...) can obtain non-reflexive logics in which ‘prem’ behaves as stipulated by Russell (2018) but which nonetheless has valid inferences supporting uniform substitution of any formula for propositional variables such as modus tollens and modus ponens. (shrink)

Kai kurios šiuolaikinio kognityvinio mokslo teorijos teigia, kad žmogaus prote veikia dvi sistemos: sistema, vykdanti greitą intuityvų mąstymą, bei sistema, vadovaujanti lėtam logiškam mąstymui. Būtų galima manyti, kad šių sistemų egzistavimą patikrinti sudėtinga. Šiame straipsnyje pateikiamas būdas palaipsniui patvirtinti šių dviejų sistemų egzistavimą. Pasitelkiamas dviejų sistemų, pasireiškiančių per mentalinių modelių teoriją, principas. Be to, laikantis Carnapo redukcijos idėjos, straipsnyje aprašomos dvi procedūros, kuriomis hipotezė patvirtinama palaipsniui. Viena iš jų tyrinėja, kaip žmogaus protas nagrinėja žmogaus proto veiklą darant išvedimus pagal (...) class='Hi'>modus tollendo tollens. Kita hipotezė tyrinėja išvedimus iš griežtosios disjunkcijos su trimis disjunktais. (shrink)