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Equational reasoning, as distinguished from implicational reasoning, is well-evolved in mathematics today but grievously short-schrifted in contemporary logic textbooks.  Consequently, it may be advisable for me to draw out and place in relief some of the more distinctive characters of equational inference that may have passed beneath the notice of a casual reading of these notes.

Equational reasoning, as distinguished from implicational reasoning, is well-evolved in mathematics today but grievously short-schrifted in contemporary logic textbooks.  Consequently, it may be advisable for me to draw out and place in relief some of the more distinctive characters of equational inference that may have passed beneath the notice of a casual reading of these notes.

By way of a very preliminary orientation, let us consider the distinction between an ''information maintaining process'' (IMP) and an ''information reducing process'' (IRP).  To conform with prudent practice, let's make our first acquaintance with this difference 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.

* Example 1.  Modus Ponens

 

| '''Example 1.  Modus Ponens'''

 

:   ''p'' ⇒ ''q''

|  

:   ''p''

| ''Information Reducing Inference''

:    ''q''

|  

 

 

p \Rightarrow q

:   ''p'' ⇒ ''q''

:   ''p''

: ========

:   ''p q''

|  

| ''Information Preserving Inference''

|  

p \Rightarrow q

p

=\!=\!=\!=\!=\!=\!=

p~q

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: