Abstract
We present a dynamic approach to Peirce’s original construal of abductive logic as a logic of conjecture making, and provide a new decidable, contraction-free and cut-free proof system for the dynamic logic of abductive inferences with neighborhood semantics. Our formulation of the dynamic logic of abduction follows the philosophical and scientific track that led Peirce to his late, post-1903 characterization of abductive conclusions as investigands, namely invitations to investigate propositions conjectured at the level of pre-beliefs.
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Notes
Reference R is to the Peirce Papers deposited at the Harvard University Houghton Library, as cataloged by Richard Robin [25], by the manuscript number, page number (if available) and year (if available).
All reasoning has, according to Peirce, a diagrammatic form in thought. Ideally, one ought to use the graphical logic of Peirce’s existential graphs in order to represent the logical forms of propositions and the inferential relationships between them, even in the case of non-deductive logic and reasoning. We do not discuss the topic of graphical logic and its relations to abduction in the present paper. Let us note, however, that Peirce himself presented at least one logical diagram which he apparently meant to represent abductive rather than deductive reasoning (Syllabus, R 478s, 1903, a draft page).
July 16, 1905, R L 493. This letter has escaped earlier notice because it was not included in the 1966 microfilm edition (R) and not published in the collection of letters between Peirce and Welby, Semiotics and Significs (SS, 1977).
This formulation of KD45 differs from the standard one in [11, p. 39]. The difference is that we use the axiom \((N_{B_{a}})\) instead of the rule of necessitation, the latter being that from φ that has been already derived infer Baφ. One can easily prove that the resulting system is equivalent to KD45.
There are some counterexamples to these properties debated among philosophers over the years, but for our purposes of contrasting them with the principles of conjecture-making such complications are not crucial.
Semantically, this corresponds to the property of density in possible-worlds semantics.
Peirce’s recommendation was to take the scientific attitude toward “buried secrets”: that we should resist making the fallacy of pre-empting what the future science might bring before us, even when we would right now be faced with meager or no evidence whatsoever concerning the status of such secrets, or even no prospects of ever gaining one.
To further clarify: we assume bivalence and that propositions are non-vague in the meta-theory. We could entertain tautologies being assumed, in the somewhat unwarranted sense in which from holding them something unwelcome might follow, such as an intuitionist mathematician arguing against a classical mathematician. In such examples we conjecture the hypothetical conditional, namely we conjecture that, supposing that the LEM holds in infinite domains, then non-constructive proofs would have to be allowed in such domains (and thus something unwelcome might have to be accepted, such as existence proofs without producing the objects that the indirect proof talks about, etc.). In these kinds of arguments, we assume a tautology, or a logical or necessary truth, but what we conjecture is the entire conditional formula. We thank the reviewer for suggesting possible counterexamples to our argument that conjecturing tautologies (or dually, logical or necessary falsities) is meaningless and illicit in the context of scientific reasoning. A conjecture is not the same thing as an assumption or supposition. A conjecture is a conclusion of an abductive argument. The conclusion of an abductive argument can well present itself in the form of a hypothetical conditional, such as “If LEM, then something unwelcome may occur for a classical mathematician”. But then it is this entire conditional form that will be the investigand of abductive reasoning, subject to further inquiry in order to extract its various consequences and hence its real meaning in the future proceedings of the investigation.
Beliefs, as a wealth of research has confirmed, tend to be strongly irrational and biased types of attitudes, but this based on different, empirical and psychological kinds of observations.
Textual evidence for setting up such a hierarchy in this manner is abundant in Peirce’s writings: “An increase of information by induction, hypothesis [abduction], or analogy, is a presumption” ([24] CP 2.430). “Animated by that [great and cheerful] hope [for true guesses], we are to proceed to the construction of a hypothesis” (EP 2:107). “The entire fabric of science has to be built up out of surmises at truth” ([24] CP 7.87). “Retroduction … depends on our hope, sooner or later, to guess at the conditions under which a given kind of phenomena will present itself” (L 477).
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This first author is supported by Chinese National Funding of Social Sciences (grant no. 16CZX049). The second author is supported by the Academy of Finland (project 1270335, Diagrammatic Mind (DiaMind): Logical and Cognitive Aspects of Iconicity, 2013–2017) and the Estonian Research Council (project PUT 1305, (Abduction in the Age of Fundamental Uncertainty, 2016–2018), Principle Investigator A.-V. Pietarinen).
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Ma, M., Pietarinen, AV. Let Us Investigate! Dynamic Conjecture-Making as the Formal Logic of Abduction. J Philos Logic 47, 913–945 (2018). https://doi.org/10.1007/s10992-017-9454-x
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DOI: https://doi.org/10.1007/s10992-017-9454-x