Results for 'Counterpossibles'

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  1. Counterpossibles in Science: The Case of Relative Computability.Matthias Jenny - 2018 - Noûs 52 (3):530-560.
    I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as 'If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable,' which is true, and 'If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable,' which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What’s more, (...)
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  2. On Counterpossibles.Jens Christian Bjerring - 2013 - Philosophical Studies 168 (2):327-353.
    The traditional Lewis–Stalnaker semantics treats all counterfactuals with an impossible antecedent as trivially or vacuously true. Many have regarded this as a serious defect of the semantics. For intuitively, it seems, counterfactuals with impossible antecedents—counterpossibles—can be non-trivially true and non-trivially false. Whereas the counterpossible "If Hobbes had squared the circle, then the mathematical community at the time would have been surprised" seems true, "If Hobbes had squared the circle, then sick children in the mountains of Afghanistan at the time (...)
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  3.  37
    Counterpossibles for Modal Normativists.Theodore D. Locke - 2019 - Synthese 198 (2):1235-1257.
    Counterpossibles are counterfactuals that involve some metaphysical impossibility. Modal normativism is a non-descriptivist account of metaphysical necessity and possibility according to which modal claims, e.g. ‘necessarily, all bachelors are unmarried’, do not function as descriptive claims about the modal nature of reality but function as normative illustrations of constitutive rules and permissions that govern the use of ordinary non-modal vocabulary, e.g. ‘bachelor’. In this paper, I assume modal normativism and develop a novel account of counterpossibles and claims about (...)
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  4. Counterpossible Non-Vacuity in Scientific Practice.Peter Tan - 2019 - Journal of Philosophy 116 (1):32-60.
    The longstanding philosophical orthodoxy on counterfactuals holds, in part, that counterfactuals with metaphysically impossible antecedents are indiscriminately vacuously true. Drawing on a number of examples from across scientific practice, I argue that science routinely treats counterpossibles as non-vacuously true and also routinely treats other counterpossibles as false. In fact, the success of many central scientific endeavors requires that counterpossibles can be non-vacuously true or false. So the philosophical orthodoxy that counterpossibles are indiscriminately vacuously true is inconsistent (...)
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  5. Counterpossibles and Similarity.David Vander Laan - 2004 - In Frank Jackson & Graham Priest (eds.), Lewisian Themes: The Philosophy of David K. Lewis. Oxford, UK: Clarendon Press. pp. 258-275.
    Several themes of David Lewis's theory of counterfactuals, especially their sensitivity to context, pave the way for a viable theory of non-trivial counterpossibles. If Lewis was successful in defending his account against the early objections, a semantics of counterpossibles can be defended from similar objections in the same way. The resulting theory will be extended to address 'might' counterfactuals and questions about the relative "nearness" of impossible worlds.
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  6. Counterpossibles (Not Only) for Dispositionalists.Barbara Vetter - 2016 - Philosophical Studies 173 (10):2681-2700.
    Dispositionalists try to provide an account of modality—possibility, necessity, and the counterfactual conditional—in terms of dispositions. But there may be a tension between dispositionalist accounts of possibility on the one hand, and of counterfactuals on the other. Dispositionalists about possibility must hold that there are no impossible dispositions, i.e., dispositions with metaphysically impossible stimulus and/or manifestation conditions; dispositionalist accounts of counterfactuals, if they allow for non-vacuous counterpossibles, require that there are such impossible dispositions. I argue, first, that there are (...)
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  7.  40
    Classical Counterpossibles.Rohan French, Patrick Girard & David Ripley - 2022 - Review of Symbolic Logic 15 (1):259-275.
    We present four classical theories of counterpossibles that combine modalities and counterfactuals. Two theories are anti-vacuist and forbid vacuously true counterfactuals, two are quasi-vacuist and allow counterfactuals to be vacuously true when their antecedent is not only impossible, but also inconceivable. The theories vary on how they restrict the interaction of modalities and counterfactuals. We provide a logical cartography with precise acceptable boundaries, illustrating to what extent nonvacuism about counterpossibles can be reconciled with classical logic.
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  8.  46
    Are Counterpossibles Epistemic?Daniel Dohrn - 2021 - Pacific Philosophical Quarterly 102 (1):51-72.
    It has been suggested that intuitions supporting the nonvacuity of counterpossibles can be explained by distinguishing an epistemic and a metaphysical reading of counterfactuals. Such an explanation must answer why we tend to neglect the distinction of the two readings. By way of an answer, I offer a generalized pattern for explaining nonvacuity intuitions by a stand-and-fall relationship to certain indicative conditionals. Then, I present reasons for doubting the proposal: nonvacuists can use the epistemic reading to turn the table (...)
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  9. Williamson on Counterpossibles.Berto Francesco, David Ripley, Graham Priest & Rohan French - 2018 - Journal of Philosophical Logic 47 (4):693-713.
    A counterpossible conditional is a counterfactual with an impossible antecedent. Common sense delivers the view that some such conditionals are true, and some are false. In recent publications, Timothy Williamson has defended the view that all are true. In this paper we defend the common sense view against Williamson’s objections.
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  10. Counterpossibles.Timothy Williamson - 2018 - Topoi 37 (3):357-368.
    The paper clarifies and defends the orthodox view that counterfactual conditionals with impossible antecedents are vacuously true against recent criticisms. It argues that apparent counterexamples to orthodoxy result from uncritical reliance on a fallible heuristic used in the processing of conditionals. A comparison is developed between such counterpossibles and vacuously true universal generalizations.
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  11. Counterpossibles.Barak Krakauer - 2012 - Dissertation, University of Massachusetts
    Counterpossibles are counterfactuals with necessarily false antecedents. The problem of counterpossibles is easiest to state within the "nearest possible world" framework for counterfactuals: on this approach, a counterfactual is true when the consequent is true in the "nearest" possible world where the antecedent is true. Since counterpossibles have necessarily false antecedents, there is no possible world where the antecedent is true. On the approach favored by Lewis, Stalnaker, Williamson, and others, counterpossibles are all trivially true. I (...)
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  12. Remarks on Counterpossibles.Berit Brogaard & Joe Salerno - 2013 - Synthese 190 (4):639-660.
    Since the publication of David Lewis’ Counterfactuals, the standard line on subjunctive conditionals with impossible antecedents (or counterpossibles) has been that they are vacuously true. That is, a conditional of the form ‘If p were the case, q would be the case’ is trivially true whenever the antecedent, p, is impossible. The primary justification is that Lewis’ semantics best approximates the English subjunctive conditional, and that a vacuous treatment of counterpossibles is a consequence of that very elegant theory. (...)
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  13. Counterpossible Dependence and the Efficacy of the Divine Will.Kenneth L. Pearce - 2017 - Faith and Philosophy 34 (1):3-16.
    The will of an omnipotent being would be perfectly efficacious. Alexander Pruss and I have provided an analysis of perfect efficacy that relies on non-trivial counterpossible conditionals. Scott Hill has objected that not all of the required counterpossibles are true of God. Sarah Adams has objected that perfect efficacy of will (on any analysis) would be an extrinsic property and so is not suitable as a divine attribute. I argue that both of these objections can be answered if the (...)
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  14. Counterpossibles.Alexander W. Kocurek - 2021 - Philosophy Compass 16 (11):e12787.
  15.  16
    Counterpossibles and Normal Defaults in the Filioque Controversy.Jacob Archambault - 2019 - Logica Universalis 13 (4):443-455.
    A counterpossible conditional, or counterpossible for short, is a conditional proposition whose antecedent is impossible. The filioque doctrine is a dogma of western Christian Trinitarian theology according to which the Holy Spirit proceeds from the Father and the Son. The filioque doctrine was the principal theological reason for the Great Schism, the split between Eastern Orthodoxy and western Christianity, which continues today. In the paper, I review one of the earliest medieval defenses of the doctrine in Anselm of Canterbury, and (...)
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  16. Why Counterpossibles Are Non-Trivial.Berit Brogaard & Joe Salerno - forthcoming - In Vincent Hendricks (ed.), Synthese volume.
    I. Non-Trivial Counterpossibles On Lewis’ account, a subjunctive of the form ‘if it were the case that p, it would be the case that q’ (represented as ‘p → q’) is to be given the following rough meta-linguistic truth-conditions1.
     
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  17.  43
    Counterpossible Reasoning in Physics.Alastair Wilson - 2021 - Philosophy of Science 88 (5):1113-1124.
    This article explores three ways in which physics may involve counterpossible reasoning. The first way arises when evaluating false theories: to say what the world would be like if the theory were true, we need to evaluate counterfactuals with physically impossible antecedents. The second way relates to the role of counterfactuals in characterizing causal structure: to say what causes what in physics, we need to make reference to physically impossible scenarios. The third way is novel: to model metaphysical dependence in (...)
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  18. Counterpossibles and the ‘Terrible’ Divine Command Deity.Richard Brian Davis & W. Paul Franks - 2015 - Religious Studies 51 (1):1-19.
    In a series of articles, Wes Morriston has launched what can only be considered a full-scale assault on the divine command theory (DCT) of morality. According to Morriston, proponents of this theory are committed to an alarming counterpossible: that if God did command an annual human sacrifice, it would be morally obligatory. Since only a ‘terrible’ deity would do such a ‘terrible’ thing, we should reject DCT. Indeed, if there were such a deity, the world would be a terrible place—certainly (...)
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  19. Grounding Entails Counterpossible Non‐Triviality.Alastair Wilson - 2018 - Philosophy and Phenomenological Research 92 (3):716-728.
    This paper outlines a non-reductive counterfactual account of grounding along interventionist lines, and uses the account to argue that taking grounding seriously requires ascribing non-trivial truth-conditions to a range of counterpossible counterfactuals. This result allows for a diagnosis of a route to scepticism about grounding, as deriving at least in part from scepticism about non-trivial counterpossible truth and falsity.
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  20. Wierenga on Theism and Counterpossibles.Fabio Lampert - 2019 - Philosophical Studies 176 (3):693-707.
    Several theists, including Linda Zagzebski, have claimed that theism is somehow committed to nonvacuism about counterpossibles. Even though Zagzebski herself has rejected vacuism, she has offered an argument in favour of it, which Edward Wierenga has defended as providing strong support for vacuism that is independent of the orthodox semantics for counterfactuals, mainly developed by David Lewis and Robert Stalnaker. In this paper I show that argument to be sound only relative to the orthodox semantics, which entails vacuism, and (...)
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  21.  39
    The Science of Counterpossibles Vs. The Counterpossibles of Science.Daniel Dohrn - forthcoming - British Journal for the Philosophy of Science.
    Orthodoxy has it that all counterpossibles are vacuously true. Yet there are strong arguments both for and against the use of non-vacuous counterpossibles in metaphysics. Even more compelling evidence may be expected from science. Arguably philosophy should defer to best scientific practice. If scientific practice comes with a commitment to non-vacuous counterpossibles, this may be the decisive reason to reject semantic orthodoxy and accept non-vacuity. I critically examine various examples of the purported scientific use of non-vacuous (...) and argue that they are not convincing. They neither establish that scientific practice comes with a commitment to the non-vacuity of counterpossibles, nor that incurring such a commitment would be useful in scientific practice. I illustrate a variety of counterstrategies on behalf of orthodoxy. (shrink)
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  22. Game Counterpossibles.Felipe Morales Carbonell - 2020 - Argumenta 6 (1):117-133.
    Counterpossibles, counterfactuals conditional with impossible antecedents, are notoriously contested; while the standard view makes them trivially true, some authors argue that they can be non-trivially true. In this paper, I examine the use of counterfactuals in the context of games, and argue that there is a case to be made for their non-triviality in a restricted sense. In particular, I examine the case of retro problems in chess, where it can happen that one is tasked with evaluating counterfactuals about (...)
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  23. Sceptical Counterpossibilities.Barbara Winters - 1981 - Pacific Philosophical Quarterly 62 (1):30.
     
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  24. Counterpossibles and the Nature of Impossible Worlds.Mattias Skipper Rasmussen - 2016 - SATS 17 (2):145-158.
    One well-known objection to the traditional Lewis-Stalnaker semantics of counterfactuals is that it delivers counterintuitive semantic verdicts for many counterpossibles (counterfactuals with necessarily false antecedents). To remedy this problem, several authors have proposed extending the set of possible worlds by impossible worlds at which necessary falsehoods may be true. Linguistic ersatz theorists often construe impossible worlds as maximal, inconsistent sets of sentences in some sufficiently expressive language. However, in a recent paper, Bjerring (2014) argues that the “extended” Lewis-Stalnaker semantics (...)
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  25.  45
    Semantics for Counterpossibles.Yale Weiss - 2017 - Australasian Journal of Logic 14 (4):383-407.
    The object of this paper is to examine two approaches to giving non-vacuous truth conditions for counterpossibles, counterfactuals with impossible antecedents. I first develop modifications of a Lewis-style sphere semantics with impossible worlds. I argue that this approach sanctions intuitively invalid inferences and is supported by philosophically problematic foundations. I then develop modifications of certain ceteris paribus conditional logics with impossible worlds. Tableaux are given for each of these in an appendix and soundness and completeness results are proved. While (...)
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  26.  7
    Counterpossibles, Story Prefix and Trivialism.Maciej Sendłak - 2021 - Synthese 199 (3-4):7283-7301.
    The aim of this paper is to argue in favor of the view that some counterpossibles are false. This is done indirectly by showing that accepting the opposite view, i.e., one that ascribes truth to each and every counterpossible, results in the claim that every necessarily false theory has exactly the same consequences. Accordingly, it is shown that taking every counterpossible to be true not only undermines the value of debates over various alternative theories and their consequences, but also (...)
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  27. Williamson on Counterpossibles.Joe Salerno & Berit Brogaard - 2007 - The Reasoner.
    Lewis/Stalnaker semantics has it that all counterpossibles (i.e., counterfactual conditionals with impossible antecedents) are vacuously true. Non-vacuism, by contrast, says the truth-values of counterpossibles are affected by the truth-values of the consequents. Some counterpossibles are true, some false. Williamson objects to non-vacuism. He asks us to consider someone who answered ‘11’ to ‘What is 5 + 7?’ but who mistakenly believes that he answered ‘13’. For the non-vacuist, (1) is false, (2) true: (1) If 5 + 7 (...)
     
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  28. Conceivability, Rigidity and Counterpossibles.Jesper Kallestrup - 2009 - Synthese 171 (3):377 - 386.
    Wright (In Gendler and Hawthorne (Eds.), Conceivability and possibility, 2002) rejects some dominant responses to Kripke’s modal argument against the mind-body identity theory, and instead he proposes a new response that draws on a certain understanding of counterpossibles. This paper offers some defensive remarks on behalf of Lewis’ objection to that argument, and it argues that Wright’s proposal fails to fully accommodate the conceivability intuitions, and that it is dialectically ineffective.
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  29.  64
    Alternative Frameworks and Counterpossibles.Maciej Sendłak - 2016 - Grazer Philosophische Studien 93 (1):24-41.
    The aim of this paper is to show why the theories of impossible worlds do not fully solve the problem of counterpossibles, but merely shift it. Moreover, by making a distinction between two types of languages, we will show that some expectations about proper theory of counterfactuals might be too great.
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  30. How Close Are Impossible Worlds? A Critique of Brogaard and Salerno’s Account of Counterpossibles.Dan Baras - 2019 - Dialectica 73 (3):315-329.
    Several theorists have been attracted to the idea that in order to account for counterpossibles, i.e. counterfactuals with impossible antecedents, we must appeal to impossible worlds. However, few have attempted to provide a detailed impossible worlds account of counterpossibles. Berit Brogaard and Joe Salerno’s ‘Remarks on Counterpossibles’ is one of the few attempts to fill in this theoretical gap. In this article, I critically examine their account. I prove a number of unanticipated implications of their account that (...)
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  31. An Extended Lewis-Stalnaker Semantics and The New Problem of Counterpossibles.Jeffrey Goodman - 2004 - Philosophical Papers 33 (1):35-66.
    Closest-possible-world analyses of counterfactuals suffer from what has been called the ‘problem of counterpossibles’: some counterfactuals with metaphysically impossible antecedents seem plainly false, but the proposed analyses imply that they are all (vacuously) true. One alleged solution to this problem is the addition of impossible worlds. In this paper, I argue that the closest possible or impossible world analyses that have recently been suggested suffer from the ‘new problem of counterpossibles’: the proposed analyses imply that some plainly true (...)
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  32. Counterpossibles, Impossible Worlds, and the Notion of Similarity.Maciej Sendłak - 2017 - In Gillman Payette & Rafal Urbaniak (eds.), Applications of Formal Philosophy. The Road Less Travelled. Springer Verlag.
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  33.  36
    The Quietest Challenge to the Axiology of God: A Cognitive Approach to Counterpossibles.Joshua Mugg - 2016 - Faith and Philosophy 33 (4):441-460.
    Guy Kahane asks an axiological question: what value would (or does) God’s existence bestow on the world? Supposing God’s existence is a matter of necessity, this axiological question faces a problem because answering it will require assessing the truth-value of counterpossibles. I argue that Kahane, Paul Moser, and Richard Davis and Paul Franks fail in their attempts to render the axiological question substantive. I then offer my own solution by bringing work in cognitive psychology and philosophy of mind to (...)
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  34. Spinoza and Counterpossible Inferences.Galen Barry - 2021 - Pacific Philosophical Quarterly 102.
    Spinoza reasons about impossibilities on a regular basis. But he also says they're unthinkable and that reasoning is a mental process. How can he do this? The paper defends a linguistic account of counterpossible inferences in Spinoza's geometrical method.
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  35. God and Counterpossibles.Richard Brian Davis - 2006 - Religious Studies 42 (4):371.
    In this paper I critically examine Brian Leftow's attempt to construct a theistic semantics for counterpossibles, one that can be used to make sense of the fact that propositions, which exist necessarily, nevertheless depend on God as their cause. I argue that the impressive theoretical framework erected by Leftow cannot guarantee an asymmetrical dependence of propositions on God, and ultimately leads to a semantic collapse in which every counterpossible comes out false. I end by defending an alternative account of (...)
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  36. Theism and Counterpossibles.Edward Wierenga - 1998 - Philosophical Studies 89 (1):87-103.
  37.  16
    Calculus and Counterpossibles in Science.Brian McLoone - 2020 - Synthese 198 (12):12153-12174.
    A mathematical model in science can be formulated as a counterfactual conditional, with the model’s assumptions in the antecedent and its predictions in the consequent. Interestingly, some of these models appear to have assumptions that are metaphysically impossible. Consider models in ecology that use differential equations to track the dynamics of some population of organisms. For the math to work, the model must assume that population size is a continuous quantity, despite that many organisms are necessarily discrete. This means our (...)
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  38.  43
    The Problem of Counterpossibles.Daniel H. Cohen - 1987 - Notre Dame Journal of Formal Logic 29 (1):91-101.
  39. The Dependence Problem: Theism, Counterpossibles, and Necessity.Richard Brian Davis - 1998 - Dissertation, University of Toronto
    This dissertation explores various attempts to solve the Dependence Problem problem posed by the following question: How can necessary truths stand to God in a one-way relation of dependence, given that neither they nor God could have failed to exist?
     
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  40.  51
    On the Pragmatic Approach to Counterpossibles.Maciej Sendłak - 2019 - Philosophia 47 (2):523-532.
    Nina Emery and Christopher Hill proposed a pragmatic approach toward the debate about counterpossibles—i.e., counterfactuals with impossible antecedents. The core of this approach is to move the burden of the problem from the notion of truth value into the notion of assertion. This is meant to explain our pre-theoretical intuitions about counterpossibles while claiming that each and every counterpossible is vacuously true. The aim of this paper is to indicate a problematic aspect of this view.
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  41. Fictionalism, the Safety Result and Counterpossibles.Lukas Skiba - 2019 - Analysis 79 (4):647-658.
    Fictionalists maintain that possible worlds, numbers or composite objects exist only according to theories which are useful but false. Hale, Divers and Woodward have provided arguments which threaten to show that fictionalists must be prepared to regard the theories in question as contingently, rather than necessarily, false. If warranted, this conclusion would significantly limit the appeal of the fictionalist strategy rendering it unavailable to anyone antecedently convinced that mathematics and metaphysics concern non-contingent matters. I try to show that their arguments (...)
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  42.  46
    Zagzebski and Interesting Counterpossibles.Richard Brian Davis - 1997 - Proceedings of the American Catholic Philosophical Association 71:125-136.
  43.  7
    Zagzebski and Interesting Counterpossibles.Richard Brian Davis - 1997 - Proceedings of the American Catholic Philosophical Association 71:125-136.
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  44. Realizm modalny i okresy warunkowe z niemożliwymi poprzednikami [Modal Realism and Counterpossibles].Maciej Sendłak - 2014 - Filozofia Nauki 22 (4).
    To solve the problem of counterpossibles, many philosophers have been arguing that one needs to invoke impossible worlds. This extension of the ontology of modality should save the analysis of counterfactuals from being insensitive to the problem of counterpossibles. Since theories of impossible worlds are extensions of original accounts of modalities, it is worth stressing that proper analyses of counterpossibles should not weaken the latter.In this paper I argue that these theories of impossible wolrds, which are based (...)
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  45.  10
    About the Basis for the Debate of Counterpossibles.Maciej Sendłak - 2019 - Studia Semiotyczne—English Supplement 30:43-59.
    According to the most popular theories, counterfactuals with impossible antecedents are vacuously true. Critiques of this view argue that contrary to this, we tend to consider only some of them true and others to be false. In his recent paper Timothy Williamson has ingeniously explained the motivations for the orthodox view and argued that although there are some heuristic reasons that may suggest the plausibility of the unorthodox view, they are fallible. The most important of Williamson’s arguments is that the (...)
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  46. Sensitivity, Safety, and Impossible Worlds.Guido Melchior - 2021 - Philosophical Studies 178 (3):713-729.
    Modal knowledge accounts that are based on standards possible-worlds semantics face well-known problems when it comes to knowledge of necessities. Beliefs in necessities are trivially sensitive and safe and, therefore, trivially constitute knowledge according to these accounts. In this paper, I will first argue that existing solutions to this necessity problem, which accept standard possible-worlds semantics, are unsatisfactory. In order to solve the necessity problem, I will utilize an unorthodox account of counterfactuals, as proposed by Nolan, on which we also (...)
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  47. Omission Impossible.Sara Bernstein - 2016 - Philosophical Studies 173 (10):2575-2589.
    This paper gives a framework for understanding causal counterpossibles, counterfactuals imbued with causal content whose antecedents appeal to metaphysically impossible worlds. Such statements are generated by omissive causal claims that appeal to metaphysically impossible events, such as “If the mathematician had not failed to prove that 2+2=5, the math textbooks would not have remained intact.” After providing an account of impossible omissions, the paper argues for three claims: (i) impossible omissions play a causal role in the actual world, (ii) (...)
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  48. Counterlogicals as Counterconventionals.Alexander W. Kocurek & Ethan J. Jerzak - 2021 - Journal of Philosophical Logic 50 (4):673-704.
    We develop and defend a new approach to counterlogicals. Non-vacuous counterlogicals, we argue, fall within a broader class of counterfactuals known as counterconventionals. Existing semantics for counterconventionals, 459–482 ) and, 1–27 ) allow counterfactuals to shift the interpretation of predicates and relations. We extend these theories to counterlogicals by allowing counterfactuals to shift the interpretation of logical vocabulary. This yields an elegant semantics for counterlogicals that avoids problems with the usual impossible worlds semantics. We conclude by showing how this approach (...)
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  49. Counteridenticals.Alexander W. Kocurek - 2018 - The Philosophical Review 127 (3):323-369.
    A counteridentical is a counterfactual with an identity statement in the antecedent. While counteridenticals generally seem non-trivial, most semantic theories for counterfactuals, when combined with the necessity of identity and distinctness, attribute vacuous truth conditions to such counterfactuals. In light of this, one could try to save the orthodox theories either by appealing to pragmatics or by denying that the antecedents of alleged counteridenticals really contain identity claims. Or one could reject the orthodox theory of counterfactuals in favor of a (...)
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  50.  80
    The Prospects for a Monist Theory of Non-Causal Explanation in Science and Mathematics.Alexander Reutlinger, Mark Colyvan & Karolina Krzyżanowska - forthcoming - Erkenntnis:1-21.
    We explore the prospects of a monist account of explanation for both non-causal explanations in science and pure mathematics. Our starting point is the counterfactual theory of explanation (CTE) for explanations in science, as advocated in the recent literature on explanation. We argue that, despite the obvious differences between mathematical and scientific explanation, the CTE can be extended to cover both non-causal explanations in science and mathematical explanations. In particular, a successful application of the CTE to mathematical explanations requires us (...)
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