Abstract
This Chapter is devoted to the description of standard systems of natural deduction (ND). After historical introduction in Section 2.1 we present some preliminary criteria which should be satisfied by any system of natural deduction. Sections 2.3 and 2.4 develop a systematization of existing systems based on two features: the kind of data used by a system and the format of proof setting. As a result we divide traditional ND systems into F- and S-systems (rules defined on formulae or sequents), and on L- and T-systems (linear or tree proofs). The division is not exhaustive as there may be systems operating on other types of items (e.g. labelled formulae – cf. Chapter 8) The main conclusion of this part is that almost all of the known variants of ND may be traced back to the independent works of Jaśkowski and Gentzen who started the investigation on non-axiomatic deductive systems.
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- 1.
This is why we do not treat sequent calculi and tableau systems as examples of natural deduction systems. In ordinary SC we have only cumulative proofs and introduction rules, in tableau systems only indirect proofs and elimination rules.
- 2.
Note that Jaśkowski did not use ⊥, so it is present in the formulation of rules as a metalinguistic sign of inconsistency. Also the names of rules are not original; he just used on the level of realization names like: rule I, rule II, ֵֵ. Note that in general we do not use the original names of rules from described systems since several authors either use different notation for the same things or the same names for different things.
- 3.
In rules of elimination like \((\rightarrow E)\) we will ocassionaly use traditional distinction between major premise which contains eliminated constant, and minor premise.
- 4.
Jaśkowski formulated also ND system for propositional logic with quantifiers.
- 5.
In fact, Gentzen defined antecedents of sequents rather as lists of formulae, so he needs also a rule of permutation and contraction for elements of antecedents.
- 6.
This is, for example, a solution of Dyckhoff in [81] with H denoting hypotheses (assumptions) and F denoting facts, i.e. assumptions and formulae inferred from them.
- 7.
Similar way of description is in Wójcicki [285], where every deductive system consists of axioms, H-rules (i.e. sequents) and G-rules (i.e. sequent rules).
- 8.
Cf. e.g. an algorithm in Chapter 4.
- 9.
Additionally Jaśkowski has used a prefix S (from supposition) in front of any assumption.
- 10.
It must be said that Jaśkowski was not as lucky as Gentzen, whose contribution into development of proof methods is widely known. There are a lot of books and papers using some variant of ND in Jaśkowski format but crediting it to Fitch or to Gentzen.
- 11.
This remark applies to practically oriented ND; in systems constructed for the needs of theoretical investigation other solution may work better, e.g. von Plato/Negri idea of generalized elimination rules in [193] leading to smooth proof of normalization.
- 12.
In Chapter 4 we will introduce some drastic modification of ND leading to “flat” (no additional subderivations) proofs.
- 13.
For simplicity we keep the same names as for suitable rules in the calculus, but note that “rules” from the level of realization correspond only partially to them; cf. Section 2.5.4.
- 14.
The version from [158] differs in even more respects; e.g. there are no boxes, no specified inference rules for connectives, no [RED].
- 15.
- 16.
There is no risk of confusion with the notion concerning formulae in sequents so, in this book, we will call them shortly parameters in accordance with the terminology of Fitting.
- 17.
- 18.
Here and then we follow the convention of Section 2.5, where two instructions for assumption introduction were signed by (a) and (b).
- 19.
But it may be the same since the depth is measured not by the number of boxes in general but by the maximal number of nested boxes, and the eliminated box may not be a member of this maximal sequence.
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Indrzejczak, A. (2010). Standard Natural Deduction. In: Natural Deduction, Hybrid Systems and Modal Logics. Trends in Logic, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-8785-0_2
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