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Laws of logic are propositions, and more precisely, propositions that are always true ( true in all interpretations).

For example : [ (A OR B) & ~B ] --> A

Rules of logic are " commands" , " imperatives" ; as such they are not propositions ( declarative sentenses) for they are neither true nor false; for example :

" from (A OR B) and ~B , infer A".

Is this account correct?

For example, is it helpful to distinguish the "law of modus ponens" and the "modus ponens rule"?

Are there logical languages / systems for which this distinction does not hold?

  • (-1) should I understand that the distinction is obviously pertinent or that it obviously isn't? –  Dec 21 '19 at 20:01
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    Do you have a source for the definitions of law vs. rule you give here? To the best of my knowledge those aren't technical terms (and I suspect that's what the downvote is about). That said, the distinction you're pointing at is indeed an important one (and I've upvoted). – Noah Schweber Dec 21 '19 at 20:13
  • I think I encountered this distinction in Robert Blanché's Introduction to contemporary logic ( Introduction à la logique contemporaine): " The first way [to conceive of logic] is to consider it as a deontology of reasoning. Laws [here] have a lower status than rules , and often are not well distinguished from rules. (...) Such a logic is not properly a science, for rules are, as such, neither true nor false. (...) Logic [ thus understood ] will be classified amongst "normative sciences",... –  Dec 21 '19 at 20:38
  • .... in the same way as ethics or esthetics. This conception could be considered as old-fashioned: [however] it is newly in favour with recent systems of natural deduction". ( Chapter 1, §3) - Blanché establishes a link between the law/rules distinction and the traditional disputed question " is logic a science or an art?" –  Dec 21 '19 at 20:38
  • For FOL, we may equate "laws of logic" with logical truth (i.e. valid formulas) – Mauro ALLEGRANZA Dec 22 '19 at 10:59

1 Answers1

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To the best of my knowledge, neither "law of logic" nor "rule of logic" has a specific technical meaning - at least in the context of mathematical logic (I don't know about the philosophy side of things). That said, the difference between (in a given system) axioms, theorems, and tautologies on the one hand and inference rules on the other is extremely important in proof theory; see e.g. this Mathoverflow question.

Logic is broad enough that there isn't a single "most general" situation, but the following is a pretty good summary of a wide range of cases:

  • We have a set $S$ of things called sentences, and a deduction relation $\vdash$ on $S$. That is, $\vdash$ is a subset of $\mathcal{P}(S)\times S$, with "$(X,y)\in$ $\vdash$" being interpreted as "from $X$, infer $y$," and abbreviated by "$X\vdash y$." (Usually $S$ is more than just a set, and in particular $S$ is often a free algebra over some signature whose elements are our logical connectives.)

  • The tautologies of the deductive system $(S,\vdash)$ are the sentences $y$ such that $\emptyset\vdash y$. For a given set $\Gamma\subseteq S$, the theorems of $\Gamma$ are the sentences $y$ such that $\Gamma\vdash y$, and when focusing on a single such $\Gamma$ we call elements of $\Gamma$ axioms.

  • An inference rule is a pair $(\Gamma,y)$ with $\Gamma\subseteq S$ and $y\in S$ (or a set of such pairs closed under appropriate substitutions). Generally $\vdash$ is "generated by" a collection of inference rules in the obvious way, and we can talk about a given inference rule being admissible (with respect to $\vdash$).

The relevant topic is algebraic logic, and I strongly recommend this paper of Blok and Pigozzi as a good introduction.

Noah Schweber
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