Abstract
In the Tractatus, it is stated that questions about logical formatting (“why just these rules?”) cannot be meaningfully formulated, since it is precisely the application of logical rules which enables the formulation of a question whatsoever; analogously, Wittgenstein’s celebrated infinite regress argument on rule-following seems to undermine any explanation of deduction, as relying on a logical argument. On the other hand, some recent mathematical developments of the Curry-Howard bridge between proof theory and type theory address the issue of describing the “subjective” side of logic, that is, the concrete manipulation of rules and proofs in space and time. It is advocated that such developments can shed some light on the question of logical formatting and its apparently unintelligible paradoxes, thus reconsidering Wittgenstein’s verdict.
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Notes
- 1.
This way of reading the Tractatus presupposes the thesis that there is a theoretical unity tying together the theory of the logical form and what Wittgenstein writes later on the following rules: in both cases, one indeed finds that constitutive conditions of “saying” cannot be expressed linguistically without falling into circularity or regress.
- 2.
This expectation clearly goes well beyond Wittgenstein’s view on rules and logic, since he believed that logical rules are truly arbitrary ones, though enjoying a special position in language.
- 3.
And by that logical correctness, since the eliminability of cuts (which implies the termination of all reductions) has as a corollary the logical coherence of the calculus: since no introduction rule is given for the absurdity \(\perp \), if a proof of \(\perp \) existed, then it would reduce to a proof whose last rule is an introduction for \(\perp \), thus no such proof can exist.
- 4.
In this section, I’ll adopt the convention to name sequent calculus proofs with capital Greek letters \(\Pi,\Lambda,\ldots\) and with small letters π, λ, … the programs, respectively, associated.
- 5.
We neglect for simplicity matters regarding the discipline of contexts, which would require a more sophisticated argument to reach exactly the same conclusion.
- 6.
Along with the rule (Ex) of exchange which allows to permute the order of appearance of occurrences of formulas in a sequent and whose rejection leads to non-commutative logic N L, see Abrusci and Ruet (2000).
- 7.
Which happens to be very well organized, with classical connectives splitting into a multiplicative and an additive version, and with modalities (known as exponentials) reintroducing, in a linear setting, controlled versions of weakening and contraction.
- 8.
Where
denote, respectively, linear negation, multiplicative conjunction (tensor) and multiplicative disjunction (par). - 9.
In this section, I’ll adopt the convention to name sequent calculus proofs with capital greek letters \(\Pi,\Lambda,\ldots\) and with small letters \(\pi,\lambda,\ldots\) the wires, respectively, associated.
- 10.
A simple calculation shows that, for u a linear operator, \((I - u)^{-1} =\sum _{ n}^{\infty }u^{n}\), given the convergence of the series on the right or, equivalently, the invertibility of I − u.
- 11.
To be precise, the wire corresponds to a linear operator u whose support is a projection π of dimension 6, i.e. such that u = π u π, and which can thus be thought of as a 6 × 6 matrix. In the following, I’ll freely use matrices to represent linear operators of finite support.
denote, respectively, linear negation, multiplicative conjunction (tensor) and multiplicative disjunction (par).References
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Pistone, P. (2015). Rule-Following and the Limits of Formalization: Wittgenstein’s Considerations Through the Lens of Logic. In: Lolli, G., Panza, M., Venturi, G. (eds) From Logic to Practice. Boston Studies in the Philosophy and History of Science, vol 308. Springer, Cham. https://doi.org/10.1007/978-3-319-10434-8_6
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