The Stanford Encyclopedia of Philosophy, s.v. "Modal Logic," states,

The most familiar logics in the modal family are constructed from a weak logic called K (after Saul Kripke)...

References to 'weak' and 'weakness' are relatively frequent in questions posed on this website, e.g. Weak second order logic. But neither the answers given under that particular heading nor explanations of weakness such as that given under What is meant by one axiom being "weaker" than another? explain to me what might be the weakness of "a logic," or of some other entire system.

What is meant by "a weak logic"?

  • 2
    $\begingroup$ Loosely and informally speaking, a logic is weak if it can't prove that many things. I've also heard the term refer to the consistency or (or lack thereof) of a system. In the cases I've seen, it's a relative term, and not an absolute term. $\endgroup$
    – Graviton
    Aug 18, 2020 at 9:32
  • $\begingroup$ In other articles in the website, such as Weakness of Will, they seem to use the word there to describe something that is logically unjustified. $\endgroup$
    – Graviton
    Aug 18, 2020 at 9:38

1 Answer 1


A logic is stronger the more theorems it proves, and as a corollary, the fewer models it has.
The more axioms there are, and the more specific an axiom is (in the sense that A is more specific than B if A entails B but B does not entail A), the more formulas will be deducible from these axioms: A logic is strong in the sense that it manages to prove many sentences.
On the other hand, the more a theory requires to be true, the more difficult it gets for a structure to satisfy all of the axioms, so the fewer models there will be: A logic is strong in the sense that it manages to kick out many structures and leaves only few possibilities of what the universe could look like.

Modal logic K has only one rule and one axiom, or in terms of the accessibility relation, no constraints at all. So any modal structure can satisfy this theory, and there are not that many theorems that can be derived from just this one axiom, and manage to be universally true in all of these many structures, in this more general setting.
By adding more axioms, or constraints on the accessibility relation, more structures are ruled out. Thus more sentences can be proved, and manage to be true in all of those fewer models, in this more specific theory. Theories such as T, S4, S5 are hence stronger than K.

Note that this definition breaks down if the logic is inconsistent and incorporates the classical law of explosion: Then the logic proves every statement, and it has no models -- which by the above criteria would make it indefinitely strong; but this not what we'd want intuitively, because such a logic is trivial. (Though note that this classical treatement is not a necessity: there are logics which don't automatically make inconsistent theories explode; cf. paraconsistent logic).

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    $\begingroup$ +1. That's absolutely the right answer in this question, but let me note to the OP that weak/strong is also used to refer to expressive strength - e.g. weak second-order logic is a proper sublogic of full second-order logic, first-order logic is weaker than second-order logic, etc. $\endgroup$ Aug 19, 2020 at 0:36

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