What is the 'tonk' problem?

Arthur Prior proposed a connective 'tonk' whose introduction rule lets you infer 'A tonk B' from A and whose elimination rule lets you infer B from 'A tonk B', so that anything would follow from anything. Belnap argued this shows inference rules can define a genuine connective only if they meet further constraints such as conservativeness, blocking pathological 'definitions'.

Logical Constants and Logicality

Logic is usually said to be formal and topic-neutral, but that presupposes a principled line between logical vocabulary like 'and', 'all', and 'not' and the rest of language.

Definition

Logical constants are the expressions (such as the connectives and quantifiers) whose meaning is held fixed across all interpretations and in virtue of which an argument's validity depends on its form rather than its subject matter.

Scope

This topic concerns the demarcation problem for logic: which expressions are logical constants and what property makes them so. It covers the leading criteria — invariance under permutations of the domain (the Tarski-Sher thesis), proof-theoretic criteria based on harmonious introduction and elimination rules, and grammatical or inferential accounts — and the bearing of this question on what counts as logical form and hence on logical consequence.

Core questions

Which expressions count as logical constants, and is the list open or closed?
Is logicality marked by invariance under permutations of the domain?
Can logical constants be characterized purely by their inference rules, and what constrains admissible rules?
Does the choice of logical constants determine, or merely reflect, the consequence relation?

Key concepts

topic-neutrality
permutation invariance
introduction and elimination rules
proof-theoretic harmony
conservativeness
logical form

Key theories

Permutation-invariance (Tarski-Sher) criterion: A notion is logical iff it is invariant under arbitrary permutations of the domain of individuals; this captures topic-neutrality by requiring logical notions to be insensitive to which particular objects exist.
Proof-theoretic harmony: A connective is genuinely logical only if its introduction and elimination rules are in harmony, so that no new theorems about the rest of the language are generated; Belnap's response to Prior's 'tonk' shows that arbitrary inference rules cannot define a constant.

History

Gentzen's 1930s natural-deduction rules suggested that connectives might be defined by their inferential role, an idea sharpened by Prior's 1960 'tonk' connective and Belnap's 1962 reply requiring conservativeness and harmony. Tarski's posthumously published 1966 lecture introduced the permutation-invariance criterion, later developed into the Tarski-Sher thesis as the dominant model-theoretic answer.

Debates

Semantic vs. proof-theoretic demarcation: Whether logicality is best fixed by a model-theoretic invariance condition or by constraints on inference rules such as harmony and conservativeness, and whether the two approaches agree on which expressions are logical.

Key figures

Alfred Tarski
Gila Sher
Nuel Belnap
Arthur Prior
Gerhard Gentzen

Seminal works

tarski1986what
belnap1962

Frequently asked questions

What is the 'tonk' problem?: Arthur Prior proposed a connective 'tonk' whose introduction rule lets you infer 'A tonk B' from A and whose elimination rule lets you infer B from 'A tonk B', so that anything would follow from anything. Belnap argued this shows inference rules can define a genuine connective only if they meet further constraints such as conservativeness, blocking pathological 'definitions'.