Is modus ponens valid for the indicative conditional? McGee [1985] famously presents several alleged counterexamples to this inference rule. More recently, Kolodny and MacFarlane [2010] and Willer [2010] argue that modus ponens is unreliable in certain hypothetical contexts. However, none of these attacks undermines an informational conception of logic on which modus ponens is valid.

After some motivating remarks in Section 1, in Section 2 we show how to replace an axiomatic basis for any one of a broad range of sentential logics having finitely many axiom schemes and Modus Ponens as the sole proper rule, by a basis with the same axiom schemes and finitely many one-premiss rules. Section 3 mentions some questions arising from this replacement procedure , explores another such procedure, and discusses some aspects of the consequence relations associated (...) with the different axiomatizations in play. Several open problems are mentioned. An appendix briefly treats the issue of a similar ‘at most one-premiss rules’ reformulation of proof systems with sequent-to-sequent rules. (shrink)

The paper is concerned with a logical difficulty which Lionel Shapiro’s deflationist theory of logical consequence (as well as the author’s favoured, non-deflationist theory) gives rise to. It is argued that Shapiro’s non-contractive approach to solving the difficulty, although correct in its broad outlines, is nevertheless extremely problematic in some of its specifics, in particular in its failure to validate certain intuitive rules and laws associated with the principle of modus ponens. An alternative non-contractive theory is offered which (...) does not suffer from the same problem. (shrink)

Considering the instability of nonlinear dynamics, the deductive inference rule Modus ponens itself is not enough to guarantee the validity of reasoning sequences in the real physical world, and similar results cannot necessarily be obtained from similar causes. Some kind of stability hypothesis should be added in order to draw meaningful conclusions. Hence, the uncertainty of deductive inference appears to be like that of inductive inference, and the asymmetry between deduction and induction becomes unrecognizable such as to (...) undermine the basis for the fundamental cleavage between analytic truth and synthetic truth, as W. V. O. Quine pointed out. Induction is not inferior to deduction from a pragmatic point of view. (shrink)

Lattice BCK logic is the expansion of the well known Meredith implicational logic BCK expanded with lattice conjunction and disjunction. Although its natural axiomatization has three rules named modus ponens, ∨‐rule and ∧‐rule, we show that we can give an equivalent presentation with just modus ponens and ∧‐rule, however it is impossible to obtain an equivalent presentation with modus ponens as unique rule. In this paper we study and characterize all (...) axiomatic extensions of lattice BCK logic with modus ponens as unique rule. We obtain an infinite chain of proper axiomatic extensions with this property. Moreover, we prove that there is no weakest axiomatic extension of Lattice BCK‐logic admitting modus ponens as unique rule. (shrink)

“Weak relevant model structures” (wr-ms) are defined on “weak relevant matrices” by generalizing Brady’s model structure ${\mathcal{M}_{\rm CL}}$ built upon Meyer’s Crystal matrix CL. It is shown how to falsify in any wr-ms the Generalized Modus Ponens axiom and similar schemes used to derive Curry’s Paradox. In the last section of the paper we discuss how to extend this method of falsification to more general schemes that could also be used in deriving Curry’s Paradox.

Modus ponens (from A and “if A then C” infer C) is one of the most basic inference rules. The probabilistic modus ponens allows for managing uncertainty by transmitting assigned uncertainties from the premises to the conclusion (i.e., from P(A) and P(C|A) infer P(C)). In this paper, we generalize the probabilistic modus ponens by replacing A by the conditional event A|H. The resulting inference rule involves iterated conditionals (formalized by conditional random quantities) and (...) propagates previsions from the premises to the conclusion. Interestingly, the propagation rules for the lower and the upper bounds on the conclusion of the generalized probabilistic modus ponens coincide with the respective bounds on the conclusion for the (non-nested) probabilistic modus ponens. (shrink)

The paper deals with the epistemological problems posed by the application of the modus ponens rule in cognitive processing of distributed information. Distinguishing the deductive correctness of a rule of inference from its normativity for rational belief, we consider ‘counterexamples’ of modus ponens proposed in modern epistemology of logic that endanger not its model-theoretic validity, but epistemic rationality for boundedly rational cognitive agents. We explain the epistemic lacunas in the closure of knowledge according to (...) modus ponens revealed by Dretske’s ‘counterexamples’ by a variety of standards of validity that go beyond ‘common sense’. However, the epistemic ‘counterexamples’ of modus ponens proposed by McGee, Kolodny and MacFarlane require, in our opinion, an appeal to less obvious epistemic phenomena, which can be discovered by reasoning for interpretation. We demonstrate that the supposed ‘counterexamples’ to modus ponens do not conflict with either the epistemic neutrality of logic or its epistemic relevance to boundedly rational agents. On the contrary, the appeal to reasoning for interpretation, which is needed to resolve the difficulties caused by these ‘counterexamples’, helps to achieve a balance between the epistemic constraints of reasoning agents and the logical requirements for the correctness of their interpretive and deductive cognitive operations. (shrink)

In a recent pair of publications, Richard Bradley has offered two novel no-go theorems involving the principle of Preservation for conditionals, which guarantees that one’s prior conditional beliefs will exhibit a certain degree of inertia in the face of a change in one’s non-conditional beliefs. We first note that Bradley’s original discussions of these results—in which he finds motivation for rejecting Preservation, first in a principle of Commutativity, then in a doxastic analogue of the rule of modus (...) class='Hi'>ponens —are problematic in a significant number of respects. We then turn to a recent U-turn on his part, in which he winds up rescinding his commitment to modus ponens, on the grounds of a tension with the rule of Import-Export for conditionals. Here we offer an important positive contribution to the literature, settling the following crucial question that Bradley leaves unanswered: assuming that one gives up on full-blown modus ponens on the grounds of its incompatibility with Import-Export, what weakened version of the principle should one be settling for instead? Our discussion of the issue turns out to unearth an interesting connection between epistemic undermining and the apparent failures of modus ponens in McGee’s famous counterexamples. (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 compositional structure of language might have led one to expect that a proper analysis of simple conditionals would have been adequate to determine the analysis of iterated conditionals. But McGee has presented an interesting group of examples that shows that this is not so for indicative conditionals. The examples are particularly arresting since they appear to show that modus ponens does not hold as a generally valid rule of inference for conditionals in natural language.

This paper presents an adaptive logic enhancement of conditional logics of normality that allows for defeasible applications of Modus Ponens to conditionals. In addition to the possibilities these logics already offer in terms of reasoning about conditionals, this way they are enriched by the ability to perform default inferencing. The idea is to apply Modus Ponens defeasibly to a conditional and a fact on the condition that it is ‘safe' to do so concerning the factual and (...) conditional knowledge at hand. It is for instance not safe if the given information describes exceptional circumstances: although birds usually fly, penguins are exceptional to this rule. The two adaptive standard strategies are shown to correspond to different intuitions, a skeptical and a credulous reasoning type, which manifest themselves in the handling of so-called floating conclusions. (shrink)

En este trabajo se presenta una explicación para el problema de las versiones abstractas de la tarea de selección de las cuatro tarjetas de Peter Wason: los discretos resultados de los participantes cuando se enfrentan a ellas. Nuestra explicación apunta a que los sujetos no comprenden la regla de la tarea como un condicional, sino como un bicondicional, provocando tal circunstancia que no elijan las tarjetas correctas. Santamaría planteó una objeción para todo enfoque que defienda que en la tarea de (...) selección se produce una interpretación bicondicional de la regla, pero nosotros también respondemos aquí a dicha objeción atendiendo a la primacía que en el razonamiento humano puede tener la regla lógica del modus ponens. Igualmente, exponemos un estudio, basado en diálogos de seminario tutoriales al modo socrático, realizado por nosotros con el propósito de comprobar si nuestra explicación puede ser viable y comentamos los resultados obtenidos. In this paper, we present an explanation for the problem of the abstract versions of Peter Wason’s four-card selection task: modest subjects’ results when they are confronted with these tasks. Our explanation poses that individuals do not comprehend the task rule as a conditional sentence, but as a biconditional one, causing that subjects do not choose the correct cards. Santamaría presented an objection for any approach that suggests a biconditional interpretation in the task selection. In this work, we discuss this objection based on the apparent primacy of a logical rule in human reasoning: modus ponens. Additionally, we present a study based on Socratic dialogue seminars with a tutorial style, developed by us, to check whether our explanation is viable or not, and discuss the results achieved. (shrink)

This paper studies, with techniques ofAlgebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential logics have been isolated. These logics are weaker than the positive implicative logic. We have (...) determined their algebraic models and the relationships between them, and have classified them according to several standard criteria of Abstract Algebraic Logic. One of the logics is protoalgebraic but neither equivalential nor weakly algebraizable, a rare situation where very few natural examples were hitherto known. In passing we have found new, alternative presentations of positive implicative logic, both in Hilbert style and in Gentzen style, and have characterized it in terms of the restricted Deduction Theorem: it is the weakest logic satisfying Modus Ponens and the Deduction Theorem restricted to at most 2 side formulas. The algebraic part of the work has lead to the class of quasi-Hilbert algebras, a quasi-variety of implicative algebras introduced by Pla and Verdú in 1980, which is larger than the variety of Hilbert algebras. Its algebraic properties reflect those of the corresponding logics and Gentzen systems. (shrink)

In virtue of what are we justified in employing the rule of inference Modus Ponens? One tempting approach to answering this question is to claim that we are justified in employing Modus Ponens purely in virtue of facts concerning meaning or concept-possession. In this paper, we argue that such meaning-based accounts cannot be accepted as the fundamental account of our justification.

A recent article by Jeff Kochan contains a discussion of modus ponens that among other thing alleges that the paradox of the heap is a counterexample to it. In this note I show that it is the conditional major premise of a modus ponens inference, rather than the rule itself, that is impugned. This premise is the contrapositive of the inductive step in the principle of mathematical induction, confirming the widely accepted view that it is (...) the vagueness of natural language predicates, not modus ponens , that is challenged by Sorites. (shrink)

The logic of single-conclusion proofs () is introduced. It combines the verification property of proofs with the single valuedness of proof predicate and describes the operations on proofs induced by modus ponens rule and proof checking. It is proved that is decidable, sound and complete with respect to arithmetical proof interpretations based on single-valued proof predicates. The application to arithmetical inference rules specification and -admissibility testing is considered. We show that the provability in gives the complete admissibility (...) test for the rules which can be specified by schemes in -language. The test is supplied with the ground proof extraction algorithm which eliminates the admissible rules from a -proof by utilizing the information from the corresponding -proofs. (shrink)

Colin Howson argues that (1) my sociologistic reliabilism sheds no light on the objectivity of epistemic content, and that (2) sorites does not threaten the reliability of modus ponens . I reply that argument (1) misrepresents my position, and that argument (2) is beside the point.

A number of theoretical positions in psychology—including variants of case‐based reasoning, instance‐based analogy, and connectionist models—maintain that abstract rules are not involved in human reasoning, or at best play a minor role. Other views hold that the use of abstract rules is a core aspect of human reasoning. We propose eight criteria for determining whether or not people use abstract rules in reasoning, and examine evidence relevant to each criterion for several rule systems. We argue that there is substantial (...) evidence that several different inferential rules, including modus ponens, contractual rules, causal rules, and the law of large numbers, are used in solving everyday problems. We discuss the implications for various theoretical positions and consider hybrid mechanisms that combine aspects of instance and rule models. (shrink)

Recently Ekman gave a derivation in natural deduction such that it either contains a substantial redundant part or else is not normal. It is shown that this problem is caused by a non-normality inherent in the usual modus ponens rule.

In contemporary discussions of the Ramsey Test for conditionals, it is commonly held that (i) supposing the antecedent of a conditional is adopting a potential state of full belief, and (ii) Modus Ponens is a valid rule of inference. I argue on the basis of Thomason Conditionals (such as ' If Sally is deceiving, I do not believe it') and Moore's Paradox that both claims are wrong. I then develop a double-indexed Update Semantics for conditionals which takes (...) these two results into account while doing justice to the key intuitions underlying the Ramsey Test. The semantics is extended to cover some further phenomena, including the recent observation that epistemic modal operators give rise to something very like, but also very unlike, Moore's Paradox. (shrink)

This paper surveys the various forms of Deduction Theorem for a broad range of relevant logics. The logics range from the basic system B of Routley-Meyer through to the system R of relevant implication, and the forms of Deduction Theorem are characterized by the various formula representations of rules that are either unrestricted or restricted in certain ways. The formula representations cover the iterated form,A 1 .A 2 . ... .A n B, the conjunctive form,A 1&A 2 & ...A n (...) B, the combined conjunctive and iterated form, enthymematic version of these three forms, and the classical implicational form,A 1&A 2& ...A n B. The concept of general enthymeme is introduced and the Deduction Theorem is shown to apply for rules essentially derived using Modus Ponens and Adjunction only, with logics containing either (A B)&(B C) .A C orA B .B C .A C. (shrink)

In our thought, we employ rules of inference and belief-forming methods more generally. For instance, we (plausibly) employ deductive rules such as Modus Ponens, ampliative rules such as Inference to the Best Explanation, and perceptual methods that tell us to believe what perceptually appears to be the case. What explains our entitlement to employ these rules and methods? This chapter considers the motivations for broadly internalist answers to this question. It considers three such motivations—one based on simple cases, (...) one based on a general conception of epistemic responsibility, and one based on skeptical scenarios. The chapter argues that none of these motivations is successful. The first two motivations lead to forms of internalism—Extreme Method Internalism and Defense Internalism—that are too strong to be tenable. The third motivation motivates Mental Internalism (Mentalism), which does not fit with plausible accounts of entitlement. (shrink)

In [15], [16] G. Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated ε-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals Φ A of order type 0 which realize the Herbrand normal form A H of A. Subsequently more perspicuous proofs of this fact via functional interpretation (combined with normalization) and cut-elimination were found. These proofs however do (...) not carry out the no-counterexample interpretation as a local proof interpretation and don't respect the modus ponens on the level of the no-counterexample interpretation of formulas A and A → B. Closely related to this phenomenon is the fact that both proofs do not establish the condition (δ) and--at least not constructively-- (γ) which are part of the definition of an 'interpretation of a formal system' as formulated in [15]. In this paper we determine the complexity of the no-counterexample interpretation of the modus ponens rule for (i) PA-provable sentences, (ii) for arbitrary sentences A, B ∈ L(PA) uniformly in functionals satisfying the no-counterexample interpretation of (prenex normal forms of) A and A → B, and (iii) for arbitrary A, B ∈ L(PA) pointwise in given α( 0 ) -recursive functionals satisfying the no-counterexample interpretation of A and A → B. This yields in particular perspicuous proofs of new uniform versions of the conditions (γ), (δ). Finally we discuss a variant of the concept of an interpretation presented in [17] and show that it is incomparable with the concept studied in [15], [16]. In particular we show that the no-counterexample interpretation of PA n by α( n (ω))-recursive functionals (n ≥ 1) is an interpretation in the sense of [17] but not in the sense of [15] since it violates the condition (δ). (shrink)

The sentential calculiR, under discussion, are axiomatizable and implication is among their primitive terms. The modus ponens and the rule of substitution are their primitive rules. ByS r is denoted the set of sentences obtained from the formulae of the calculusR by substituting sentences of a given language for all variables. The variablesx, y, z ... represent the elements of the setS r , the variablesX, Y, Z ... represent the subsets ofS R . The formulacxy designates (...) an implication withx as its antecedent andy as its consequent,cxy is always an element ofS R δ(X) means, thatX is closed with respect to the modus ponens rule.A R designates the class of allS R -substitutions of the axioms of the sentential calculusR. (shrink)

1. There is only one rule of inference, modus ponens. This is true both in the presentations of Begriffsschrift and Grundgesetze. There are other ways of making transitions between propositions in proofs, but these are never labeled by Frege “rules of inference.” These pertain to scope of quantification, parsing of formulas, introduction of definitions, conventions for the use and replacement of the various letters, and certain structural reorganizations, ; cf. the list in Gg §48.

We are justified in employing the rule of inference Modus Ponens (or one much like it) as basic in our reasoning. By contrast, we are not justified in employing a rule of inference that permits inferring to some difficult mathematical theorem from the relevant axioms in a single step. Such an inferential step is intuitively “too large” to count as justified. What accounts for this difference? In this paper, I canvass several possible explanations. I argue that (...) the most promising approach is to appeal to features like usefulness or indispensability to important or required cognitive projects. On the resulting view, whether an inferential step counts as large or small depends on the importance of the relevant rule of inference in our thought. (shrink)

There is a renewed debate about modus ponens. Strikingly, the recent counterexamples in Cantwell, Dreier and MacFarlane and Kolodny are generated by restricted readings of the ‘if’-clause. Moreover, it can be argued on general grounds that the restrictor view of conditionals developed in Kratzer and Lewis leads to counterexamples to modus ponens. This paper provides a careful analysis of modus ponens within the framework of the restrictor view. Despite appearances to the contrary, there is (...) a robust sense in which modus ponens is valid, owing to the fact that conditionals do not only allow for restricted readings but have bare interpretations, too. (shrink)

As it is well known, in the forties of the past century, Curry proved that in any logic S closed under Modus Ponens, uniform substitution of propositional variables and the Contraction Law, the naïve Comprehension axiom trivializes S in the sense that all propositions are derivable in S plus CA. Not less known is the fact that, ever since Curry published his proof, theses and rules weaker than W have been shown to cause the same effect as W (...) causes. Among these, the Contraction rule or the Modus Ponens axiom, for example, are to be noted. But, moreover, as Brady has proved, even the Generalized Modus Ponens axiom or the Generalized Contraction rule give rise to “Curry’s Paradox” under the same circumstances as W does. In some previous work by us, “weak relevant model structures” are defined on “weak relevant matrices” by generalizing Brady’s model structure MCL built upon Meyer’s Crystal matrix CL. We have proved that wr-ms only verify logics with the “depth relevance condition”. The aim of this paper is to show how to falsify gMPa and gRW in certain wr-ms. In particular, it will be shown that gMPa is falisfied in any wr-ms and gRW in any wr-ms verifying Routley and Meyer’s basic positive logic B+. (shrink)

People accept conclusions of valid conditional inferences (e.g., if p then q, p therefore q) less, the more disablers (circumstances that prevent q to happen although p is true) exist. We investigated whether rules that through their phrasing exclude disablers evoke higher acceptance ratings than rules that do not exclude disablers. In three experiments we re-phrased content-rich conditionals from the literature as either universal or existential rules and embedded these rules in Modus Ponens and Modus Tollens inferences. (...) In Experiments 2 and 3, we also used abstract rules. The acceptance of conclusions increased when the rule was phrased with “all” instead of “some” and the number of disablers had a higher impact on existential rules than on universal rules. Further, the effect of quantifier was more pronounced for abstract rules and when tested within subjects. We discuss the relevance of phrasing, quantifiers and knowledge on reasoning. (shrink)

Chancy modus ponens is the following inference scheme: ‘probably φ’, ‘if φ, then ψ’, therefore, ‘probably ψ’. I argue that Chancy modus ponens is invalid in general. I further argue that the invalidity of Chancy modus ponens sheds new light on the alleged counterexample to modus ponens presented by McGee. I close by observing that, although Chancy modus ponens is invalid in general, we can recover a restricted sense in which (...) this scheme of inference is valid. (shrink)

We explore the problems that confront any attempt to explain or explicate exactly what a primitive logical rule of inferenceis, orconsists in. We arrive at a proposed solution that places a surprisingly heavy load on the prospect of being able to understand and deal with specifications of rules that are essentiallyself-referring. That is, any rule$\rho $is to be understood via a specification that involves, embedded within it, reference to rule$\rho $itself. Just how we arrive at this position (...) is explained by reference to familiar rules as well as less familiar ones with unusual features. An inquiry of this kind is surprisingly absent from the foundations of inferentialism—the view that meanings of expressions (especially logical ones) are to be characterized by the rules of inference that govern them. (shrink)

If modus ponens is valid, then you should take up smoking.

In the recent paper “Naive modus ponens”, Zardini presents some brief considerations against an approach to semantic paradoxes that rejects the transitivity of entailment. The problem with the approach is, according to Zardini, that the failure of a meta-inference closely resembling modus ponens clashes both with the logical idea of modus ponens as a valid inference and the semantic idea of the conditional as requiring that a true conditional cannot have true antecedent and false (...) consequent. I respond on behalf of the non-transitive approach. I argue that the meta-inference in question is independent from the logical idea of modus ponens, and that the semantic idea of the conditional as formulated by Zardini is inadequate for his purposes because it is spelled out in a vocabulary not suitable for evaluating the adequacy of the conditional in semantics for non-transitive entailment. I proceed to generalize the semantic idea of the conditional and show that the most popular semantics for non-transitive entailment satisfies the new formulation. (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)

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)

What accounts for how we know that certain rules of reasoning, such as reasoning by Modus Ponens, are valid? If our knowledge of validity must be based on some reasoning, then we seem to be committed to the legitimacy of rule-circular arguments for validity. This paper raises a new difficulty for the rule-circular account of our knowledge of validity. The source of the problem is that, contrary to traditional wisdom, a universal generalization cannot be inferred just (...) on the basis of reasoning about an arbitrary object. I argue in favor of a more sophisticated constraint on reasoning by universal generalization, one which undermines a rule-circular account of our knowledge of validity. (shrink)

There is only one physically possible process that builds and operates purposive systems in nature: natural selection. What it does is build and operate systems that look to us purposive, goal directed, teleological. There really are not any purposes in nature and no purposive processes ether. It is just one vast network of linked causal chains. Darwinian natural selection is the only process that could produce the appearance of purpose. That is why natural selection must have built and must continually (...) shape the intentional causes of purposive behavior. Fodor’s argument against Darwinian theory involves a biologist’s modus tollens which is a cognitive scientist’s modus ponens. Assuming his argument is valid, the right conclusion is not that Darwin’s theory is mistaken but that Fodor’s and any other non-Darwinian approaches to the mind are wrong. It shows how getting things wrong in the philosophy of biology leads to mistaken conclusions with the potential to damage the acceptance of a theory with harmful consequences for human well-being. Fodor has shown that the real consequence of rejecting a Darwinian approach to the mind is to reject a Darwinian theory of phylogenetic evolution. This forces us to take seriously a notion that otherwise would not have much of a chance: that when it comes to the nature of mental states, indeterminacy rules. This is an insight that should have the most beneficial impact on freeing cognitive neuroscience from demands on the adequacy of its theories that it could never meet. (shrink)

This paper deals with two issues in the field of reasoning by analogy in the law. The one issue is whether there exists such a thing as analogous rule application, or whether there is only the ‘normal’ application of a broadened rule. It is argued that if rules, as the entities made by a legislator, are distinguished from generalised solutions for cases, the idea of analogous application of rules makes sense. It is also shown how the so-called ‘reason-based (...) model of rule application’, in contrast to the traditional modus ponens or subsumption model, makes it easy to give a logical account of analogous rule application. The other issue is how to argue logically about whether two cases are sufficiently similar to adopt the outcome of the one case for the other case. Section 3 provides a general logical model to establish this. The model is based on the comparison of the reasons for and against a particular solution in the two cases. (shrink)

In this paper I argue that pluralism at the level of logical systems requires a certain monism at the meta-logical level, and so, in a sense, there cannot be pluralism all the way down. The adequate alternative logical systems bottom out in a shared basic meta-logic, and as such, logical pluralism is limited. I argue that the content of this basic meta-logic must include the analogue of logical rules Modus Ponens and Universal Instantiation. I show this through a (...) detailed analysis of the ‘adoption problem’, which manifests something special about MP and UI. It appears that MP and UI underwrite the very nature of a logical rule of inference, due to all rules of inference being conditional and universal in their structure. As such, all logical rules presuppose MP and UI, making MP and UI self-governing, basic, unadoptable, and required in the meta-logic for the adequacy of any logical system. (shrink)

Tableau formulations are given for the relevance logics E (Entailment), R (Relevant implication) and RM (Mingle). Proofs of equivalence to modus-ponens-based formulations are vialeft-handed Gentzen sequenzen-kalküle. The tableau formulations depend on a detailed analysis of the structure of tableau rules, leading to certain global requirements. Relevance is caught by the requirement that each node must be used; modality is caught by the requirement that only certain rules can cross a barrier. Open problems are discussed.

This article concerns the structure of defeasible arguments like: 'If Bob has red spots, Bob has the measles; Bob has red spots; therefore Bob has the measles.' The issue is whether such arguments have the form of modus ponens or not. Either way there is a problem. If they don't have the form of modus ponens, the common opinion to the contrary taught in leading logic textbooks is wrong. But if they do have the form of (...) modus ponens, doubts are raised about the conventional dogma that all arguments having the form of modus ponens are deductively valid. By carefully examining arguments on both sides of the issue, reasonable doubts are raised about the view that all arguments having a modus ponens form are valid. (shrink)

In this article the argumentation structure of the court’s decision in the Popov v. Hayashi case is formalised in Prakken’s (Argument Comput 1:93–124; 2010) abstract framework for argument-based inference with structured arguments. In this framework, arguments are inference trees formed by applying two kinds of inference rules, strict and defeasible rules. Arguments can be attacked in three ways: attacking a premise, attacking a conclusion and attacking an inference. To resolve such conflicts, preferences may be used, which leads to three corresponding (...) kinds of defeat, after which Dung’s (Artif Intell 77:321–357; 1995) abstract acceptability semantics can be used to evaluate the arguments. In the present paper the abstract framework is instantiated with strict inference rules corresponding to first-order logic and with defeasible inference rules for defeasible modus ponens and various argument schemes. The main techniques used in the formal reconstruction of the case are rule-exception structures and arguments about rule validity. Arguments about socio-legal values and the use of precedent cases are reduced to arguments about rule validity. The tree structure of arguments, with explicit subargument relations between arguments, is used to capture the dependency relations between the elements of the court’s decision. (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)

We give a negative solution to the problem, posed by A. Chagrov and M. Zakharyaschev, of whether every quasi-normal propositional modal logic can be axiomatized by an independent set of axioms, with the inference rules of Substitution and Modus Ponens.

A well-known result, going back to the twenties, states that, under some reasonable assumptions, any logic can be characterized as the set of formulas satisfied by a matrix 〈,F〉, whereis an algebra of the appropriate type, andFa subset of the domain of, called the set of designated elements. In particular, every quasi-classical modal logic—a set of modal formulas, containing the smallest classical modal logicE, which is closed under the inference rules of substitution and modus ponens—is characterized by such (...) a matrix, wherenow is a modal algebra, andFis a filter of. If the modal logic is in fact normal, then we can do away with the filter; we can study normal modal logics in the setting of varieties of modal algebras. This point of view was adopted already quite explicitly in McKinsey and Tarski [8]. The observation that the lattice of normal modal logics is dually isomorphic to the lattice of subvarieties of a variety of modal algebras paved the road for an algebraic study of normal modal logics. The algebraic approach made available some general results from Universal Algebra, notably those obtained by Jónsson [6], and thereby was able to contribute new insights in the realm of normal modal logics [2], [3], [4], [10].The requirement that a modal logic be normal is rather a severe one, however, and many of the systems which have been considered in the literature do not meet it. For instance, of the five celebrated modal systems, S1–S5, introduced by Lewis, S4 and S5 are the only normal ones, while only SI fails to be quasi-classical. The purpose of this paper is to generalize the algebraic approach so as to be applicable not just to normal modal logics, but to quasi-classical modal logics in general. (shrink)

The subsystem S of Parry's AI [10] (obtained by omitting modus ponens for the material conditional) is axiomatized and shown to be strongly complete for a class of three valued Kripke style models. It is proved that S is weakly complete for the class of consistent models, and therefore that Ackermann's rule is admissible in S. It also happens that S is decidable and contains the Lewis system S4 on translation — though these results are not presented (...) here. S is arguably the most relevant relevant logic known at this time to be decidable. (shrink)