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Let us examine these two types of inference in a little more detail.  A display of the form:

Let us examine these two types of inference in a little more detail.  A ''rule of inference'' is stated in the followed form:

{| align="center" cellpadding="8" width="90%"

{| align="center" cellpadding="8" width="90%"

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is used to state a ''rule of inference'' (ROI).  The expressions above the line are called ''premisses'' and the expression below the line is called a ''conclusion''.

The expressions above the line are called ''premisses'' and the expression below the line is called a ''conclusion''. If the rule of inference is simple enough, the ''proof-theoretic turnstile symbol'' <math>{}^{\backprime\backprime} \vdash {}^{\prime\prime}</math> may be used to write the rule on a single line, as follows:

 

If the rule of inferenced is succinct, the ''proof-theoretic turnstile symbol'' <math>{}^{\backprime\backprime} \vdash {}^{\prime\prime}</math> may be used to write it on a single line, as follows:

{| align="center" cellpadding="8" width="90%"

{| align="center" cellpadding="8" width="90%"

Either way, one reads such a rule in the following manner:  "From <math>\textit{Expression~1}</math> and <math>\textit{Expression~2}</math> infer <math>\textit{Expression~3}.</math>

Either way, one reads such a rule in the following manner:  "From <math>\textit{Expression~1}</math> and <math>\textit{Expression~2}</math> infer <math>\textit{Expression~3}.</math>

Looking to our first Example, the rule that is classically known as ''modus ponens'' says the following:  If one has that ''p'' implies ''q'', and one has that ''p'' is true, then one has a ''way of putting it forward'' that q is true.

Looking to Example&nbsp;1, the rule of inference known as ''modus ponens'' says the following:  From the premiss <math>p \Rightarrow q</math> and the premiss <math>p\!</math> one may logically infer the conclusion <math>q.\!</math>

Modus ponens is an ''illative'' or ''implicational'' rule.  Passage through its turnstile incurs the toll of some information loss, and thus from a fact of ''q'' alone one cannot infer with any degree of certainty that ''p'' &rArr; ''q'' and ''p'' are the reasons why ''q'' happens to be true.

Modus ponens is an ''illative'' or ''implicational'' rule.  Passage through its turnstile incurs the toll of some information loss, and thus from a fact of ''q'' alone one cannot infer with any degree of certainty that ''p'' &rArr; ''q'' and ''p'' are the reasons why ''q'' happens to be true.