Rules of Inference

• The main issue for proof methods is the use of correct rules of inference that can be applied directly to sentences to derive conclusions that are guaranteed to be correct under all interpretations.
• Since the interpretations are not enumerated, time and space can often be saved.
• A rule of inference is a pattern of reasoning consisting of
  1. one set of sentence schemata, called premises, and
  2. a second set of sentence schemata, called conclusions.
• A rule of inference is sound if and only if, for every instance, the premises logically entail the conclusions.
• The following is a rule of inference called Modus Ponens.
  

  $\phi$	$\rightarrow$	$\psi$
  $\phi$
  $\psi$

  
  
  The premises are written above the horizontal line and the conclusions under the horizontal line.
• An instance of a rule of inference is the rule obtained by consistently substituting expressions for the metavariables in the rule, so that all premises and conclusions are legal sentences.
• The following are all legal instances of MP:
  
  

  $p$	$\hspace{-.2cm}\rightarrow$	$(q \rightarrow r)$
  $p$
  $q \rightarrow r$

          

  $(p \rightarrow q)$	$\rightarrow$	$r$
  $p \rightarrow q$
  $r$

          

  raining	$\rightarrow$	wet
  raining
  wet

• The following is a rule of inference called Modus Tollens:

  $\phi$	$\rightarrow$	$\psi$
  $\neg \psi$
  $\neg \phi$

  
  

The Stoics are accredited by historians of logic who did the early work on the nature and the theory of conditionals (in which Chrysippus, Diodorus Cronus, and Philo of Megara can further be distinguished). In Diogenes Laertius or Sextus Empiricus, one can find and read the first inscriptions related to this matter.
According to the Stoic logicians, the first kind of indemonstrable statements is as follows:

If the first, then the second; but the first; therefore the second.

We call this basic argument form as modus ponendo ponens, in abbreviation modus ponens, the mood that by affirming affirms.
The second kind of indemonstrable statements of the Stoics is:

If the first, then the second; but the second is not; therefore the first is not.

This basic argument form is called as modus tollendo tollens, in abbreviation modus tollens, the mood that by denying denies.

• The following is a rule of inference called Equivalence Elimination EE:

  $\phi$	$\leftrightarrow$	$\psi$
  $\phi$	$\rightarrow$	$\psi$
  $\psi$	$\rightarrow$	$\phi$

  
  
• The following is a rule of inference called Double Negation DN:

  $\neg \neg \phi$

  $\phi$

A contextless inference which "looks like" a counter-example to modus tollens:

(1) If it rained, it didn't rain hard.
(2) It rained hard.
(3) So, it didn't rain.

If such a conversation occurs in our everyday lives, the person who uttered (1) will not say (3), after learning (2) from his/her friend related to the current situation outside. The inferences which are similar to the above are criticized in a lot of books about logic.