Propositional Inference Rules -Artificial intelligence :

Equivalence rules are specifically useful because of the vice-versa aspect,that means we can discover forwards andbackwards in a search space using them. So, we may perform bi-directional search, which is bonus. However, what if we know that 1 sentence (or set of sentences) being true implies that another set of sentences is true. For example, the following sentence is used ad nauseum in logic text books:

All men are mortal

Socrates was man so,

Socrates is mortal

This is an instance of the application of a rule of deduction call as Modus Ponens. We see that we have deduced the fact that Socrates is mortal from the 2 true facts that all men are mortal and Socrates was a man. Hence , because we know that the rule regarding men being mortal and classification of Socrates as a man are true, we may infer with surely (because we know that modus ponens is sound), that Socrates will be die - which, in fact, he did. Of course, it does not make any sense to go backwards as with equivalences: we would deduce that, Socrates as being mortal implies that he was a man and that all men are mortal!

The common format for the modus ponens rule is following: if we have a true sentence which states that proposition A denotes proposition B and we know that proposition A is true, then we cansuppose that proposition B is true. For this the notation we use is following:

A -> B, A

B

It is an instance of an inference rule. The comma above the line showsin our knowledge base, we know both these things, and the line stands for the deductive step. That is, if we know that the both propositions above the line are true, then we may deduce that the proposition below the line is also true. An inference rule,in general

A/B

is sound if we may be certain that A entails B, for example. B is true when A is true.Tobe More formally, A entails B means that if M is a model of A then M is also a model of B. We write this as A ≡  B.

This gives us a way to examine the soundness of propositional inference rules: (i) draw a logic table for B and A both evaluating them for all models and (ii) check that whenever A is true, then B is also true. We do not care here about the models for which A is false.

This is a small example, but it highlights how we use truth tables: the first line is the just  one where both above-line propositions ( A->B and A) are true. We see that on this line, the proposition B is also true. This indicates us that we have an entailment: the above-line propositions entail the below-line one.

To see why such kind of inference rules is useful, remember what the basic application of automated deduction is: to prove theorems. Theorems are usually part of a big theory, and that theory has axioms. Axioms are special theorems which are taken to be true without question. Therefore whenever we have a theorem statement we want to prove, we should be enabling to start from the axioms and deduce the theorem statement using sound inference rules such as modus ponens.