Normality, Non-contamination and Logical Depth in Classical Natural Deduction

Abstract

If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections.

Classical logic, argument and dialectic

“…23 We use a non-standard version of classical natural deduction [20,22,23], that we call "C-intelim" (for "classical intelim"). As discussed in [20,24], this version is more faithful to the intuitive classical meaning of the logical operators and naturally suggests a simple measure of the "depth" of an argument. This is used to define a hierarchy of tractable, albeit increasingly complex, approximations to classical propositional logic that converge to it in the limit (for ideal "unbounded" agents).…”

“…The intelim and falsum rules contain no discharge rules, namely rules that involve the temporary introduction of assumptions that are subsequently "discharged", to the effect that their conclusion no longer depends on them. 24 To obtain a complete set of rules for classical propositional logic we only need add a single discharge rule: if we have a deduction D 1 of ψ depending on assumptions Γ ∪ {ϕ} and a deduction D 2 of ψ depending on assumptions ∆ ∪ {¬ϕ}, we thereby have a deduction of ψ depending on Γ ∪ ∆. This typical pattern of classical case reasoning relies on the Aristotelian principle of bivalence; a cornerstone of classical semantics whereby any sentence is either true or false, and there are no other possibilities.…”

“…It can be shown (see [24]) that restricting to normal C-intelim proofs involves no loss of deductive power in that for each C-intelim proof there exists a normal one with the same assumptions and the same conclusion. The simple restrictions on the construction of normal C-intelim proofs ensure that such proofs are not trivially redundant (see [24] for a discussion). This allows us to bar convoluted proofs such as those shown on the left in each of Figure 5 (a), (b), (c) and (d) and to generate the normal ones on the right instead.…”

“…Moreover [24] shows that if an argument associated with T is contaminated, and so given Proposition 48, can only by E-contaminated (e.g. a proof of q from p, ¬p), then: Proposition 49.…”

“…Remark 50. [24] shows that it is easy to turn an improper proof into a proof of (i.e., a refutation) of the same depth.…”