Abstract
Using the proof of Peirce’s Law [{(x → y) → x} → x] as an example, I show how bilateral tableau systems (or “2-sided trees”) are not only more economical than rival systems of logical proof, they also better reflect the reasoning Peirce actually gives for securing the law’s acceptance as an axiom. Moreover, bilateral proof trees are readily adapted to Peirce’s own graphical notation, producing a proof system in that notation that is even more efficient and easier to learn than Peirce’s system of permissions. This is in part due to the fact that Peirce’s graphical notation is similarly bilateral. In effect bilateral proof trees in Peirce’s notation can be understood as representing the space of outcomes for a game very much like what Peirce envisions as his endopereutic, and they embody insights of certain expressions of the pragmatic maxim that Peirce offers around 1905. Taken together, this suggests to me that Peirce would have embraced such a system of logic, and so I find it especially unfortunate that he was evidently unaware of Lewis Carroll’s pioneering efforts to develop tree-like proof systems to solve logical puzzles with multiliteral sorites.
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Notes
- 1.
With that in mind, one can readily discern what the Tableau rules for Belnap’s infamous “Tonk” operator would be, even though it would be impossible to read such rules off of a truth table: namely the left-hand rule of a conjunction, paired with the right-hand rule of a disjunction.
- 2.
Chapters 5–6 of Wilfried Sieg’s Logic and Proofs [13] contains perhaps the best discussion of strategies for going about constructing natural-deduction derivations using standard introduction and elimination rules.
- 3.
Axiomatic systems, of course, fare even worse by comparison. For instance, an axiomatic derivation of Peirce’s law using Mendelson’s three-axioms (which of course do not include Peirce’s Law) takes over 30 steps, and requires a rather ingenious selection of initial forms of the axiom schema.
- 4.
One can actually shorten this proof to seven steps by combining the insertions in steps 5 and 8. I find the proof in Fig. 2 to be more perspicuous in that it shows why the content of the consequent in the most deeply embedded conditional (the Y that is the second insertion in step 8) is immaterial to the overall truth of the schema.
- 5.
Consider, for instance, the respective proofs of conditional exchange and the DeMorgan’s equivalencies.
- 6.
By contrast, compare this formulation with the one Peirce gives in his Baldwin’s Encyclopedia entry on pragmatism (1902): “In order to ascertain the meaning of an intellectual conception one should consider what practical consequences might conceivably result by necessity from the truth of that conception; and the sum of these consequences will constitute the entire meaning of the conception” (CP 5.9).
- 7.
Amirouche Moktefi has pointed out to me (in correspondence) that this raises the question of whether Peirce himself appreciated the originality of Franklin-Ladd’s antilogism. As noted above, Peirce’s system of permissions does not take advantage of potential incompatibilities in the way that tree systems (including Carroll’s) do.
- 8.
While it is rather hard to imagine how Peirce could have been exposed to Carroll’s later work on logical diagrams, there nevertheless was at most only two degrees of separation between them. Not only did Peirce maintain a substantial correspondence with Victoria Welby from 1903–11, Welby (who championed Peirce’s thought in the UK) is also known to have corresponded with Cook Wilson (see [8], p. 83). That then might be one place to look for any hint of a flow of information back from Carroll to Peirce.
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Beisecker, D. (2018). Peirce and Proof: A View from the Trees. In: Chapman, P., Stapleton, G., Moktefi, A., Perez-Kriz, S., Bellucci, F. (eds) Diagrammatic Representation and Inference. Diagrams 2018. Lecture Notes in Computer Science(), vol 10871. Springer, Cham. https://doi.org/10.1007/978-3-319-91376-6_49
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