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By way of a very preliminary orientation, let us consider the distinction between ''information reducing inferences'' and ''information preserving inferences''.  It is prudent to make make our first acquaintance with this distinction in the medium of some concrete and simple examples.

By way of a very preliminary orientation, let us consider the distinction between ''information reducing inferences'' and ''information preserving inferences''.  It is prudent to make make our first acquaintance with this distinction in the medium of some concrete and simple examples.

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| '''Example 1.  Modus Ponens'''

| '''Example 1.  Modus Ponens'''

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 display of the form:

: &nbsp;&nbsp;''Expression''₁

: &nbsp;&nbsp;''Expression''₂

<math>\begin{array}{l}

: &nbsp;&nbsp;''Expression''₃

\textit{Expression~1}

\textit{Expression~2}

\textit{Expression~3}

\end{array}</math>

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

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''.

If the ROI in question is succinct enough, one may write it in-line, as ''Premiss''₁, ''Premiss''₂ |&ndash; ''Conclusion'', where the symbol "|&ndash;" is called the ''(proof-theoretic) turnstile''.

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:

 

<math>\textit{Premiss~1}, \textit{Premiss~2} ~\vdash~ \textit{Conclusion}</math>

|}

Either way, one reads such a ROI in the following manner:  "From ''Expression''₁ and ''Expression''₂, infer ''Expression''₃".

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 ROI 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 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.

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.