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Most people on this site are familiar with the idea of the "strength" of a statement. Ignoring the validity of the statements themselves, "All primes are even." is a logically weaker statement than "All integers are even." Even the statement "All integers except 7 are even." still seems qualitatively stronger than "All primes are even." even though neither implies the other.

What name describes the theory of the relative strength of logical statements? I've tried searching for a variety of keywords but have turned up surprisingly little besides references to the "intuitive" definitions like that above.

MooseBoys
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  • I have mainly seen the use of the word weaker in the context, that a statement $A$ is weaker than a statement $B$, if $B\Rightarrow A$. This fits your example since $\text{"All integers are even" $ \Rightarrow $ "All primes are even"}$. – Leander Tilsted Kristensen Nov 07 '21 at 17:30
  • "A is at least as strong as B" is equivalent to "A implies B". And arguably all of math is about investigating which statements imply which other statements. – Karl Nov 07 '21 at 17:42
  • @Karl OK but consider a modified second statement, "All integers except 7 are even." The latter no longer implies the former, but it still intuitively seems to be a "stronger" statement, which is why I would expect there to be some study of the non-obvious ordering of "strength". – MooseBoys Nov 07 '21 at 18:37
  • In addition, implication from a false premise like "all primes are even" can always be proved. But it still seems like there could be interesting analysis of the statement itself and their relative weakness. – MooseBoys Nov 07 '21 at 18:45

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I agree with the above comments; $B{\implies}A$ in conjunction with $A\not\equiv B$ is the only context I can think of offhand where it is unambiguous to say that statement $B$ is stronger than $A,$ so I think that's a fair characterisation of statement strength. If there is a specific descriptor for this, I have never heard of it.

Addendum

I think the more interesting cases are where one statement is not a clear superset of the other. I've updated my question to clarify this.

In this case (including in your specified example), any assertion about the two statements' relative strength would be ambiguous in sense; on the other hand, there is no controversy in claiming that one statement is satisfied by more entities than the other statement.

ryang
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  • I think the more interesting cases are where one statement is not a clear superset of the other. I've updated my question to clarify this. I've actually seen a couple other questions here that ask this for specific cases, but none point to a general structure in which to analyze strength of a statement. – MooseBoys Nov 07 '21 at 18:54
  • I don't deny that there's ambiguity in the relative strength of the statements, but I kind of assumed there would be some named branch of logic theory that attempts to compare them in a meaningful way. In the same way that the sum of all positive integers can be classified as "-1/12" instead of just "infinite", I was hoping that the examples presented could have some useful classification instead of just being "incomparable". – MooseBoys Nov 08 '21 at 03:47
  • @MooseBoys In Mathematics, we speak of a function being a restriction of another, but we don't say that $|x|$ is stronger than $x,$ or $\log x.$ Apologies for rambling. – ryang Nov 08 '21 at 04:14
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    @MooseBoys Perhaps reverse mathematics addresses your interests. – BrianO Nov 08 '21 at 05:58
  • @ryang Maybe it would be better to start from a different perspective. Are there any fields of logic or mathematics that would distinguish between the statements $\lnot(a \land b)$ and $\lnot a \lor \lnot b$? Obviously these are identical in the domain of formal logic, but surely there are fields of study that analyze the formulation of these kinds of statements themselves. – MooseBoys Nov 08 '21 at 05:59
  • @BrianO Not exactly, but that link fairly quickly led me to the field of metamathematics which seems much closer to what I'm looking for. Unfortunately I'm familiar with most of the topics there insofar as I remember having encountered them before and been completely baffled by them. Still, it's a useful term and may still help me find what I'm looking for in this question. – MooseBoys Nov 08 '21 at 06:07
  • @MooseBoys Kleene did a famous book "Introduction to Metamathematics", a a detailed presentation of first-order logic and its metatheory — that is, what has been proved about it. – BrianO Nov 08 '21 at 06:25