Dutilh Novaes, C. (2020). The Dialogical Roots of Deduction: Historical, Cognitive, and Philosophical Perspectives on Reasoning. Cambridge University Press. https://doi.org/10.1017/9781108800792
Atomic notes
- Three main features of deductive arguments, after Dutilh Novaes
- Deductive dialogues are Prover-Skeptic games, after Dutilh Novaes
- Informal proofs and proof practices are linguistic entities, after Dutilh Novaes
Key terms
- Necessary truth-preservation = if the premises are true, then the conclusion is necessarily true.
- Monotonicity = “if an inference from A and B to C is valid, then adding any arbitrary premise D will not block the inference to the conclusion C from A, B, and D” (5).
- Bracketing belief = the requirement for a deductive reasoner to consider only the connections between premises and conclusions, taking the actual content of both at face value.
- Adversarial collaboration = where parties in the dialogue collaborate to arrive at a comprehensive proof by disagreeing with each other and/or remining skeptical of their interlocutor’s position.
- Transferable proof = from Easwaran (2009), “a proof must be such that a relevant expert will become convinced of the truth of the conclusion of the proof just by consideration of each of the steps in the proof.”
Selected concepts and passages
What is a deductive argument?
- Deductive arguments have three necessary aspects, in order of greatest to least importance: necessary truth-preservation, step-wise structure, and bracketing belief.
- Distinguishing between deduction and other inference strategies: “Inductive, abductive, and more generally defeasible inferences, which lack the property of necessary truth-preservation, also lack the property of monotonicity” (5).
- Sufficient perspicuity depends on the context: “The level of granularity required for a deductive argument to be considered adequate will vary according to context; for example, a mathematical proof presented in a journal for professional mathematicians will typically be more ‘dense,’ i.e. less detailed, than a proof presented in an introductory textbook for students” (7).
- From Descartes: “A deductive argument is a way of expounding to others what one already knows” (51).
The Prover-Skeptic dialogues
- The Dawkins question
- “An account of logic and deduction…will only have true explanatory value insofar as plausible stories for such [accounts] can be told – that is, if these stories rely on practices and activities that the agents in question (most plausibly, but not necessarily, human agents) would have good reasons to engage in” (46).
- “In other words, are Prover and Skeptic trying to do something intelligible, and is doing it the same thing as winning in the game?” (59).
- Lakatos’ (Hegelian) dialectical philosophy of mathematics: “The general idea seems to be that a clash of contraries – proofs and refutations – gives rise to a third ‘thing’ which represents a synthesis of the two contraries” (48).
- “Then not only do refutations act as fermenting agents for proof-analysis, but proof analysis may act as a fermenting agent for refutations! What an unholy alliance between seeming enemies! (Lakatos, 1976, p. 48)”
- Deductive dialogues should be understood as Prover-Skeptic games, not Prover-Refuter: “When presented with sufficiently persuasive argumentation, a skeptic may well become convinced of a conclusion, but this will not constitute any kind of ‘loss’ for him. In other words, a skeptic is not actively trying to disprove the arguer, but he16 will only become convinced of the conclusion if the argument is strongly persuasive” (50).
A dialogical perspective of mathematical practice
- Proofs for professional mathematicians have inferential gaps: “From a dialogical perspective, these gaps are unproblematic, and in fact anticipated: a proof is typically formulated with the right level of detail, the right granularity and inferential decomposition, for the specific audience to whom it is directed. In fact, too much detail would be counterproductive for a proof understood as a communicative device, as it would make the argument tedious” (208).
- Prover-Skeptic games in large scale, online collaboration: “Once a candidate proof or argument is put forward, those involved in the interaction will presumably behave as Skeptics, looking for possible flaws or steps in the proof which remain obscure. (Naturally, this is an empirical hypothesis that could be studied systematically with the large databases of interactions from Polymath and MathOverflow.)” (214).