Abstract
What is a minimal proof-theoretical foundation of logic? Two different ways to answer this question may appear to offer themselves: reduce the whole of logic either to the relation of inference, or else to the property of incompatibility. The first way would involve defining logical operators in terms of the algebraic properties of the relation of inference—with conjunction \(\hbox {A}\wedge \hbox {B}\) as the infimum of A and B, negation \(\lnot \hbox {A}\) as the minimal incompatible of A, etc. The second way involves introducing logical operators in terms of the relation of incompatibility, such that X is incompatible with \(\{\lnot \hbox {A}\}\) iff every Y incompatible with X is incompatible with {A}; and X is incompatible with \(\{\hbox {A}\!\wedge \!\hbox {B}\}\) iff X is incompatible with {A,B}; etc. Whereas the first route leads us naturally to intuitionistic logic, the second leads us to classical logic. The aim of this paper is threefold: to investigate the relationship of the two approaches within a very general framework, to discuss the viability of erecting logic on such austere foundations, and to find out whether choosing one of the ways we are inevitably led to a specific logical system.
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Notes
It might seem more natural to interpret inference taking the reverse perspective: i.e. \(\hbox {A}\vdash \hbox {B}\) as \(\hbox {A}\le \hbox {B}\) (especially if we tend to see propositions as classes of possible worlds), but we will stick to the perspective adopted by most logicians (including both Brandom and Koslow, whose texts are starting points of our considerations).
Koslow’s (1992) approach is closely connected. However, he considers all logical compounds as minima of certain functions w.r.t. this ordering.
As Carnap (1934, p. 2) puts it: “We shall see that the logical characteristics of sentences (for instance, whether a sentence is analytic, synthetic, or contradictory; whether it is an existential sentence or not; and so on) and the logical relations between them (for instance, whether two sentences contradict one another or are compatible with one another; whether one is logically deducible from the other or not; and so on) are solely dependent upon the syntactical structure of the sentences.”
Gentzen (1934; 1936) introduced structural rules by means of which he characterized those relations of inference that he took to be “standard”. In a slightly more contemporary manner, they can be summarized as restrictions on the relation \(\vdash \) between finite sequences of sentences and sentences as follows:
$$\begin{aligned} \begin{array}{ll} \hbox {A}\vdash \hbox {A} &{}\quad (\textit{reflectivity})\\ \hbox {if}\; \hbox {X,Y}\vdash \hbox {A}, \hbox {then X,B,Y}\vdash \hbox {A} &{}\quad (\textit{weakening}\; \hbox {or}\; \textit{extension})\\ \hbox {if}\; \hbox {X,A,A,Y}\vdash \hbox {B}, \hbox {then X,A,Y}\vdash \hbox {B} &{}\quad (\textit{contraction})\\ \hbox {if}\; \hbox {X,A,B,Y}\vdash \hbox {C}, \hbox {then X,B,A,Y}\vdash \hbox {C} &{} \quad (\textit{permutation}\hbox { or } \textit{exchange})\\ \hbox {if}\; \hbox {X,A,Y}\vdash \hbox {B} \hbox { and } \hbox {Z}\vdash \hbox {A}, \hbox {then X},Z,\hbox {Y}\vdash \hbox {B} &{}\quad (\textit{cut})\\ \end{array} \end{aligned}$$Within our framework, two of the conditions, namely contraction and permutation, are implicit to our assumption that inference is a relation between sets (rather than sequences) of sentences and sentences. Reflectivity is obviously a special case of \(({\vdash }1)\), cut is embodied in \(({\vdash }2)\) and weakening follows from \(({\vdash }1)\) and \(({\vdash }2)\): if X \(\vdash \) A, then as A,B \(\vdash \) A by \(({\vdash }1), \hbox {X,B} \vdash \hbox {A}\) follows by \(({\vdash }2)\).
This qualification, which will be omitted in this paper as we will deal only with standard gis’s, is included because it would be possible to study structures in which some of the conditions \(({\vdash }1), ({\vdash }2)\) and \((\bot )\)—and hence some of Gentzen’s structural rules—would be relaxed, thus entering the realm corresponding to that of substructural logics.
Brandom and Aker call this condition—more precisely an equivalent one—defeasibility.
They can be found in Peregrin (2011).
Koslow uses, in effect, “if \(\triangle \hbox {A,B}\), then \(\hbox {B}\vdash \lnot \hbox {A}\)” instead of our \((\lnot \hbox {K} 2)\). Our definition is more general, but in the presence of conjunction or implication, the difference is significant only in cases where the incompatibility or inference dealt with is not compact, which is not a case we will be interested in here.
See Peregrin (2008) for a discussion of this.
See Peregrin (2008). Koslow makes this containment more explicit by replacing the elimination rules by his “extremality conditions”.
Again, the proofs can be found in Peregrin (2011).
The most extensive discussion I know of can be found in a recent book by Garson (2013).
See Peregrin (2010a).
One way to back up this perspective is to see logic as something that is constituted in terms of our discursive practices and especially of what Brandom (1994) calls the game of giving and asking for reasons. Proof theory would then seem to offer us the closest approach to logic’s natural foundation. The point is that, when viewed like this, logic is a matter of the most general and most fundamental rules of our discursive practices. (Hence, it is not so much proof theory in the original Hilbertian sense that would be pertinent, but rather approaches to logic based upon its dialogical nature from the beginning—see, e.g., Lorenzen 1955).
See Peregrin (2008).
See Punčochář (2012) for a discussion of this modal system and its modifications.
Thomason (1973) offers an axiomatization of C which may be seen as capturing it in inferential terms. However, the axiomatization is based on an infinite number of axioms that apparently cannot be captured by means of a finite number of schemas.
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Acknowledgments
Work on this paper has been supported by Research Grant No. 13-21076S of the Czech Science Foundation. I am grateful to V. Punčochář and two anonymous reviewers of this journal for helpful critical comments.
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Peregrin, J. Logic Reduced To Bare (Proof-Theoretical) Bones. J of Log Lang and Inf 24, 193–209 (2015). https://doi.org/10.1007/s10849-015-9214-7
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DOI: https://doi.org/10.1007/s10849-015-9214-7