Why commutative law, associative law, distributive law ... are considered to be axioms in propositional logic? Though we can prove them using truth tables.

  • $\begingroup$ Not necessarily; we can axiomatize propositional calculus with other axioms and prove the said laws. $\endgroup$ – Mauro ALLEGRANZA Jan 21 '17 at 20:06
  • $\begingroup$ @Allegranza Are there many different kinds of propositional logic? what are the most known ones? the one I'm talking about does it have a particular name? $\endgroup$ – Maykel Jakson Jan 21 '17 at 20:08
  • $\begingroup$ There are many proof systems : Hilbert-style (axioms and rules), Natural Deduction, Tableau (or Truth tree), Sequent Calculus. But they msut be used for "systematize" different sets of "logical laws" : classical logic, intuitionistic, modal logic. $\endgroup$ – Mauro ALLEGRANZA Jan 21 '17 at 20:15
  • $\begingroup$ @MauroALLEGRANZA Thank's, so the one who uses those laws as axioms is Hilbert-style? $\endgroup$ – Maykel Jakson Jan 21 '17 at 20:18
  • $\begingroup$ @MaykelJakson No, the Hilbert system does not use these as axioms. But some other axiomatic system might. Please see my answer below. $\endgroup$ – Bram28 Jan 22 '17 at 18:00

The answer to your question is a bit complicated ... part of it is because we can think about what would make something an 'axiom' in different ways:

First of all, yes, we can prove these laws using the truth-tables ... which really means: we can show that these laws hold on the basis of more fundamental definitions. Typically (but as Mauro says, not always), these more fundamental definitions state that:

  1. Every atomic claim is either true or false (but not both) (or: if you want to go into more abstract binary algebra: every variable takes on exactly one of two values)

  2. $\neg \varphi$ is true iff $\varphi$ is false

  3. $\varphi \land \psi$ is true iff $\varphi$ and $\psi$ are true.

etc. etc. (in other words, these are simply the more formal definitions of what you do in a truth-table)

So yes, from these (i.e. using truth-tables) we can prove all the laws you mention. So, in that sense, laws like commutation, association, etc. typically aren't really axioms, as we can infer them from more basic principles.

On the other hand, sometimes we start out not with the kind of 'formal semantical' definitions as laid out above, but we simply start with a bunch of syntactically defined sentences, and say "these are my axioms, and here are some inference rules that allow you to infer further sentences from that". The Hilbert system is one example of that: in this system we have as one of the axioms $P \rightarrow (Q \rightarrow P)$. And yes, I could of course infer that that statement follows from the semantical definitions from above (e.g. I could show in a truth-table that that statement is always true), but in the context of such 'axiomatic proof systems', this statement is really seen as an axiom ... no deeper semantics is provided.

Now, to make things even more confusing: There are various kinds of axiomatic systems. The Hilbert system actually does not use Commutation, Association, etc. as its axioms. But: you could define an axiomatic system (and there probably are some) where these laws really are its axioms!

  • $\begingroup$ @MaykelJakson you're welcome! $\endgroup$ – Bram28 Jan 22 '17 at 21:19

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