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Rules for Deductions

Nothing has been said up to now about deductions being in accordance with rules. When we deal with formal deductions, i.e. the deductions of logic, and they are conceived as arrows in categories with additional structure, which is brought by something like the functors corresponding to connectives that we mentioned in the third section, then our deductions are members of families of arrows indexed by

the objects of the category, which are usually natural transformations involving the functors we have just mentioned. Being in accordance with rules here amounts to being members of such families, and the schematic character of the rules is given by the indexing by objects of the members of our families of arrows. With this indexing, the objects that are indices serve to make the sources or targets of the arrows, i.e. the premises or conclusions of the deductions. For example, the natural transformation p1 of the first projection for conjunction elimination with the indices being the objects A and B gives the deduction p1 : ABA.

Is this indexing necessary for the notion of rule for deduction? Can a rule correspond to a family of arrows that is a singleton, without indexing? Can one call rule something which covers a single deduction, with which a single deduction is in accordance? In or outside logic, is generality necessary for rules? Should a rule always cover many cases? Can a rule cover a single case?

Deductions that are not in logic may still resemble the formal deductions of logic by being in accordance with schematically given rules. Such would be the deduction from the premise “The day before yesterday was Thursday” to the conclusion “Tomorrow will be Sunday” (though it is not immediately clear how to formulate the rules in question). If they cannot be found in logic, could one find outside logic deductions that are not instances of something schematic? Shouldn't they be in accordance with rules? What would be the appropriate notion of rule there? When bereft of its psychological or sociological aspects, like compulsoriness, would this notion of rule leave something to be investigated by precise, perhaps even mathematical, means? Grammar and linguistics may give an inspiration for considering such matters, which are close to the concerns of the later Wittgenstein.

In [27] (end of Lecture XIII, Lent Term 1935) Wittgenstein taught that “a rule is something applied in many cases”, but then disparaged this remark off-handedly. He considered it useless for learning how to use a rule. Why must this remark serve that purpose? In another context, where the purpose is to explain what rules are and not to teach how to use them, it may prove important to determine whether generality is necessary for rules. Wittgenstein returned to this question in [26] (Sect. 199) and in other places (for references see [15], Sect. 199, pp. 120–124), with consideration towards the generality of rules, which he put within a wider scheme.

Wittgenstein ended the lecture from which we have quoted above by a nice and enigmatic picture: “A rule is best described as being like a garden path in which you are trained to walk, and which is convenient.” A path is usually something taken many times, by many people. If a rule is like a path, the deductions in accordance with the rule could perhaps be like many particular walks on this path. We will however try to consider more closely this and other matters mentioned in this section on another occasion.

Acknowledgments Work on this paper was supported by the Ministry of Education, Science and Technological Development of Serbia, while the Alexander von Humboldt Foundation has supported the presentation of a part of it at the Second Conference on Proof-Theoretic Semantics, in Tübingen, in March 2013. I am grateful to the organizers of that conference, and in particular Peter SchroederHeister, for their exceptional hospitality. I am also grateful to him and to Thomas Piecha for making some useful comments on this paper.

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