Tonk- A Full Mathematical Solution

There is a long tradition (See e.g. [9, 10]) starting from [12], according to which the meaning of a connective is determined by the introduction and elimination rules which are associated with it. The supporters of this thesis usually have in mind natural deduction systems of a certain ideal type (explained in Section 3 below). Unfortunately, already the handling of classical negation requires rules which are not of that type. This problem can be solved in the framework of multiple-conclusion Gentzen-type systems (also first introduced in [12]), where instead of introduction and elimination rules there are left introduction rules and right introduction rules. The thesis according to which the meaning of a connective is given by its introduction (and elimination) rules was strongly challenged by Prior in [13]. In that paper he introduced his famous “connective” Tonk (denoted below by T ). This “connective” has two rules of the ideal type. The introduction rule allows to infer φTψ from φ. The elimination rule allows to infer ψ from φTψ. In the presence of Tonk every formula can be derived from any other formula, making trivial the “logic” which is “defined” by any system which includes this “connective”. Prior’s paper has made it clear that not every combination of “ideal” introduction and elimination rules can be used for defining a connective. Some constraints should be imposed on the set of rules. Such a constraint was indeed suggested by Belnap in his famous [6]: the rules for a connective ⋄ should be conservative, in the sense that if T ⊢ φ is derivable using them, and ⋄ does not occur in T ∪ φ, then T ⊢ φ can also be derived without using the rules for ⋄. This solution to the Tonk problem has at least two problematic aspects: