Abstract
We answer an old challenge by Susan Haack to the idea that deduction can be justified in a deductive way. Haack's challenge is that, just as modus ponens needs to be used in the standard justification of modus ponens from the truth table of the conditional, a deviant rule that she calls “modus morons” can apparently be justified in a similar rule-circular way. Should modus morons really be on a par with modus ponens, this would cast grave doubts on the standard justification of modus ponens. We use methods from natural deduction to describe the situation in a more comprehensive way than Haack, and go on to show that modus morons is not on a par with modus ponens, thus undermining Haack’s argument. Specifically, we use ideas from philosophical proof-theory to show that modus morons does not harmonize with the standard introduction rule for the conditional, for the following four reasons: The conditional defined by the combination of the two rules fails to satisfy the inversion principle due to Dag Prawitz, it fails to satisfy certain principles of the inferential truth theory due to Neil Tennant, there can be no truth table for it, and last but not least, it leads to inconsistency in the strongest sense of trivializing every system it is added to. In sum this helps rehabilitate modus ponens, and shows the value of philosophical proof-theory.
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Notes
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In the words of Thomas Nagel: “Certain forms of thought can’t be intelligibly doubted because they force themselves into every attempt to think about anything.” (Nagel 1997, 61).
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In our presentation of Haack’s main argument, we follow (Haack 1996, 186 f.).
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To anticipate, these are the arguments concerning reduction, inferential truth theory, and truth tables. The fourth argument, which concerns inconsistency, is our own.
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Here is the original quote: „Zu jedem der logischen Zeichen […] gehört genau eine Schlußfigur, die das Zeichen […] ‘einführt’, und eine die es ‘beseitigt’ […]. Die Einführungen stellen sozusagen die ‘Definitionen’ der betreffenden Zeichen dar, und die Beseitigungen sind letzten Endes nur Konsequenzen hiervon […]“ (Gentzen 1934, 189).
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Note that there are several contenders for explicating harmony! This leads us to a side remark about the hope for precision and the plurality of acceptable logical systems discussed today. Gentzen had hoped that we need only make the idea that an elimination rule must fit the respective introduction rule precise, and we will find that the elimination rule is a function of the introduction rule. (“By making these ideas more precise it should be possible to display the [elimination rules] as single-valued functions of their corresponding [introduction rules], on the basis of certain requirements.” (Gentzen 1964, 195). Here is the original quote: “Durch Präzisierung dieser Gedanken dürfte es möglich sein, die B-Schlüsse [Beseitigungs-Regeln] auf Grund gewisser Anforderungen als eindeutige Funktionen der zugehörigen E-Schlüsse [Einführungs-Regeln] nachzuweisen.” (Gentzen 1934, 189)) As there are different serious contenders for an explication of harmony today, we obviously have not yet reached a point where we can be quite sure which function it is that determines elimination rules once the introduction rules are given. And probably it is a good thing for the community of logicians that the one and only best explication of harmony has not been found yet, because such a function would privilege one understanding of each logical connective over all others, thus severely reducing the number of acceptable logical systems and cutting off many interesting discussions. E.g., the discussion about classical versus intuitionistic negation would be decided once and for all. And that would be a result that is just too strong, because there are many considerations, often in conflict with each other, that are relevant to the question of which logical system provides the best explication of the intuitive notion of inference.
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Here is the original quote: “Ein Beispiel möge verdeutlichen, wie das gemeint ist: Die Formel A → B durfte eingeführt werden, wenn eine Herleitung von B aus der Annahmeformel A vorlag. Will man sie nun mit Beseitigung des Zeichens → wieder verwenden […], so kann man das gerade in der Weise tun, daß man aus einem bewiesenen A sofort B schließt, denn A → B dokumentiert ja das Bestehen einer Herleitung von B aus A. Wohlgemerkt: Es braucht hierbei nicht auf einen ‘inhaltlichen Sinn’ des Zeichens → Bezug genommen zu werden.” (Gentzen 1934, 189; notation altered)
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Take for instance modus ponens: The major premiss is ‘A → B’ for it contains ‘→’. The only minor premiss needed to deduce ‘B’ is ‘A’.
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Strictly speaking, if ‘Τ (…)’ is to be understood as a truth-predicate, we would also need a nameforming operator for a formula like this to be well-formed. But we will suppress this point here.
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In this case and the others, we only display the induction step of a proof by induction.
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In addition, the rules of Semantic Ascent and Disquotation are applied to atomic formulas or to formulas of the level immediately before the respective induction step.
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‘\( T\left(\mathrm{A}\wedge \mathrm{B}\right)\to\ \left[T\left(\mathrm{A}\right)\wedge T\left(\mathrm{B}\right)\right] \)’ in the case of conjunction.
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‘\( T\left(\mathrm{A}\wedge \mathrm{B}\right)\to\ \left[T\left(\mathrm{A}\right)\wedge T\left(\mathrm{B}\right)\right] \)’ and ‘\( \left[T\left(\mathrm{A}\right)\wedge T\left(\mathrm{B}\right)\right]\to T\left(\mathrm{A}\wedge \mathrm{B}\right) \)’ in the case of conjunction.
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To see this more clearly consider the following. If conditional proof is a valid form of argument, then it is truth-preserving. Everyone who wants to use it as an introduction rule for an operator would presuppose this much. So, if conditional proof is truth-preserving, then each instantiation for a certain distribution of truth values over ‘A’ and ‘B’ leads to the same truth value for the conditional ‘A \( \to \) B’. Now take e.g. a conditional proof that starts with a derivation of `A \( \vee \neg \)A’ from the assumption ‘A’. (We get this by replacing ‘B’ with ‘A \( \vee \neg \)A’ in the general form of the rule of conditional proof.)
$$ \frac{\begin{array}{c}\left[\mathrm{A}\right]\\ {}\vdots \\ {}\mathrm{A}\vee \neg \mathrm{A}\end{array}}{\mathrm{A}\to \mathrm{A}\vee \neg \mathrm{A}}\to \hbox{-} \mathrm{intro} $$In this example ‘\( \mathrm{A} \vee \neg \mathrm{A} \)’ is derived from the assumption ‘A’, and then ‘\( \mathrm{A}\ \to\ \mathrm{A} \vee \neg \mathrm{A} \)’ is deduced. Now ‘A’ could well be false. But even then ‘\( \mathrm{A} \vee \neg \mathrm{A} \)’ would be true. So we have the very distribution of truth values for the antecedent and the consequent of a conditional that we are interested in. Now the conclusion ‘\( \mathrm{A}\ \to\ \mathrm{A} \vee \neg \mathrm{A} \)’ is derivable according to the rule of conditional proof and hence is a theorem. So it must be true. But if the conclusion of this example is true and conditional proof is truth-preserving in general, then the conclusion of such a conditional proof—a conditional with a false antecedent and a true consequent—must be true in general.
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We are, of course, not alone in this assessment. Boghossian writes: “The satisfying intermediate position concerning warrant transfer, I therefore want to propose, is that in the case of fundamental inference the implicated rule must be meaning-constituting. Unlike the purely external requirement of truth-preservation, this view explains why the thinker is entitled to the rule; and yet unlike the impossible internalism, it does so without requiring that the thinker know that the rule is truth-preserving.” (Boghossian 2001, 29) And here, Tennant agrees with Boghossian: “Boghossian’s suggested solution to his problem is that the introduction and elimination rules for a (well behaved) logical connective are meaning-constituting. They are therefore not in need of any justification—or at least are none the poorer for having only rule-circular justifications. This is a view already widely held within the above-mentioned community [of philosophical proof-theorists]. So I am sure that I shall not be the only proof-theoretically inclined philosopher of logic who would regard Boghossian as having confirmed us in our views.” (Tennant 2005, 648)
References
Belnap, Nuel. 1962. Tonk, plonk and plink. Analysis 22(6): 130–134.
Boghossian, Paul. 2001. How are objective epistemic reasons possible? Philosophical Studies 106(1/2): 1–40.
Carroll, Lewis. 1895. What the tortoise said to Achilles. Mind 4(14): 278–280.
Dummett, Michael. 1973. Frege: Philosophy of language. London: Duckworth.
Dummett, Michael. 1978 [1973]. The justification of deduction. In Truth and other enigmas, 290–318. Cambridge, MA: Harvard University Press.
Dummett, Michael. 1991. The logical basis of metaphysics. Cambridge, MA: Harvard University Press.
Haack, Susan. 1996. Deviant logic, fuzzy logic: Beyond the formalism. Chicago: University of Chicago Press.
Gentzen, Gerhard. 1934. Untersuchungen über das logische Schließen I. Mathematische Zeitschrift 39(2): 175–210.
Gentzen, Gerhard. 1964. Investigations into logical deduction. American Philosophical Quarterly 1(4): 288–306.
Hjortland, Ole T. 2009. The structure of logical consequence: Proof-theoretic conceptions. PhD thesis, University of St Andrews. http://hdl.handle.net/10023/892. Retrieved 14 May 2014
Humberstone, Lloyd. 2011. The connectives. Cambridges, MA/London: MIT Press.
Nagel, Thomas. 1997. The last word. New York: Oxford University Press.
Prawitz, Dag. 2006 [1965]. Natural deduction. A proof-theoretical study. Mineola: Dover.
Prior, Arthur. 1960. The runabout inference-ticket. Analysis 21(2): 38–39.
Tennant, Neil. 2005. Rule-circularity and the justification of deduction. Philosophical Quarterly 55(221): 625–648.
Varzi, Achille C. 1998. Review of S. Haack, deviant logic, fuzzy logic: Beyond the formalism. Philosophical Review 107(3): 468–471.
Wright, Crispin. 2001. On basic logical knowledge. Reflections on Paul Boghossian’s ‘how are objective epistemic reasons possible?’. Philosophical Studies 106(1–2): 41–85.
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Bloch, S., Pleitz, M., Pohlmann, M., Wrobel, J. (2016). Deviant Rules: On Susan Haack’s “The Justification of Deduction”. In: Göhner, J., Jung, EM. (eds) Susan Haack: Reintegrating Philosophy. Münster Lectures in Philosophy, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-24969-8_5
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