Journal of Philosophical Logic (2005) 34: 217–226 ' Springer 2005
DOI: 10.1007/s10992-004-7805-x
ROY T. COOK
WHAT’S WRONG WITH TONK(?)
1. TONK AND LOGIC
In “The Runabout Inference Ticket” A. N. Prior (1960) examines the idea
that logical connectives can be given a meaning solely in virtue of the stip-
ulation of a set of rules governing them, and thus that logical truth/conse-
quence can be explicated in terms of the meanings (so understood) of
the logical connectives involved. He proposes a counterexample to such a
view, his notorious binary connective tonk (which I will symbolize as ⊗),
whose meaning is given by the following introduction and elimination
rules:



⊗
⊗ I

⊗

⊗ E
Prior noted that acceptance of such a connective as logical, and its rules as
thereby (at least) truth preserving, renders one’s logic trivial
1
since for any
two sentences
and , we can prove that
implies :



⊗

The subsequent literature contains various strategies for holding onto a
view that is at least in the spirit of the inferentialist account being attacked
by Prior, yet which does not allow tonk legitimacy. I do not intend to add
anything further to the project of demarcating the bad operators (or their
rules) from the good ones here, however. Instead, I wish to examine, in a
bit more detail, what underlies the thought that the rules for tonk fail, in
fact, to define a legitimate logical operator.
The particular aspect of Prior’s quick argument that I will focus is
something that Nuel Belnap (1962) was first to notice, writing:
It seems to me that the key to a solution lies in observing that ...we are not defining our
connectives ab initio, but rather in terms of an antecedently given context of deducibility,
concerning which we have some definite notions. By that I mean that before arriving at the

218 ROY T. COOK
problem of characterizing connectives, we have already made some assumptions about the
nature of deducibility. That this is so can be seen immediately by observing Prior’s use of
the transitivity of deducibility in order to secure his ingenious result. But if we note that
we already have some assumptions about the context of deducibility within which we are
operating, it becomes apparent that by a too careless use of definitions, it is possible to
create a situation in which we are forced to say things inconsistent with those assumptions.
(p. 131)
And:
It is good to keep in mind that the question of the existence of a connective having such
and such properties is relative to our characterization of deducibility. If we had initially
allowed A  B(!), there would have been no objection to tonk. (p. 133)
The situation seems to be this: Prior provided a connective, defined in
terms of inference rules, which was prima facie unacceptable (since it
reduces the logic in which it is embedded to triviality). Belnap (and others)
pointed out that Prior’s argument depends on assuming that the conse-
quence relation involved is transitive. Much work was subsequently under-
taken in order to demarcate the acceptable and unacceptable connectives
within such a transitive “antecedently given context of deducibility.”
If Prior’s argument depends on the transitivity of consequence, how-
ever, then the question remains – might there be a (non-transitive) concep-
tion of logic within which the rules for tonk, as given by Prior, define a
legitimate connective? Belnap, immediately after the passage just quoted,
states that:
Also, there would have been no inconsistency had we omitted from our characterization of
deducibility the rule of transitivity. (p. 133)
He does not, however, provide any argument for this claim.
To put things a bit more carefully, Prior and Belnap have provided a
convincing argument for:
(1) For any consequence relation ⇒,if⇒ is transitive (and the resulting
logic contains a theorem) then the addition of tonk produces triviality.
Subsequent discussion of the issue has remained silent on the truth or
falsity of:
(2) There is a consequence relation ⇒ such that (⇒ is intransitive and
contains a theorem and) the addition of tonk fails to produce triviality.
Although not exactly popular, logics where transitivity fails do exist.
2
Thus,
whether or not tonk might be an acceptable connective in such a logic
remains an interesting (and, until now, open) question. The purpose of the
rest of this brief paper is to demonstrate that there is indeed such a notion