Changes

The next Example is extremely important, and for reasons that reach well beyond the level of propositional calculus as it is ordinarily conceived.  But it's slightly tricky to get all of the details right, so it will be worth taking the trouble to look at it from several different angles and as it appears in diverse frames, genres, or styles of representation.

The next Example is extremely important, and for reasons that reach well beyond the level of propositional calculus as it is ordinarily conceived.  But it's slightly tricky to get all of the details right, so it will be worth taking the trouble to look at it from several different angles and as it appears in diverse frames, genres, or styles of representation.

In discussing this Example, it is convenient to observe that the implication relation ordinarily indicated by the propositional form <math>x \Rightarrow y</math> is equivalent to an order relation <math>x \le y</math> on the boolean values <math>0, 1 \in \mathbb{B},</math> where <math>0\!</math> is taken to be less than <math>1.\!</math>

In discussing this Example, it is useful to observe that the implication relation indicated by the propositional form <math>x \Rightarrow y</math> is equivalent to an order relation <math>x \le y</math> on the boolean values <math>0, 1 \in \mathbb{B},</math> where <math>0\!</math> is taken to be less than <math>1.\!</math>

* Example 2.  Transitivity

 

| colspan="3" | '''Example 2.  Transitivity'''

 

: &nbsp;&nbsp;''p'' &le; ''q''

| &nbsp;

: &nbsp;&nbsp;''q'' &le; ''r''

| width="1%" | &nbsp;

| colspan="2" | ''Information Reducing Inference''

: &nbsp;&nbsp;''p'' &le; ''r''

 

| &nbsp;

| &nbsp;

 

| width="1%" | &nbsp;

: &nbsp;&nbsp;''p'' &le; ''q''

: &nbsp;&nbsp;''q'' &le; ''r''

: ============

: &nbsp;&nbsp;''p'' &le; ''q'' &le; ''r''

q \le r

p \le r

| &nbsp;

| &nbsp;

| colspan="2" | ''Information Preserving Inference''

| &nbsp;

| &nbsp;

| &nbsp;

q \le r

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

p \le q \le r

In stating the IMP analogue of transitivity, I have taken advantage of a common idiom in the use of order relation symbols, one that represents their logical conjunction by way of a catenated syntax.  Thus, ''p''&nbsp;&le;&nbsp;''q''&nbsp;&le;&nbsp;''r'' means that ''p''&nbsp;&le;&nbsp;''q'' and that ''q''&nbsp;&le;&nbsp;''r''.  The claim that this 3-adic relation holds among the 3 propositions ''p'', ''q'', ''r'' is a stronger claim — contains more information — than the claim that the 2-adic relation ''p''&nbsp;&le;&nbsp;''r'' holds between the 2 propositions ''p'' and ''r''.

In stating the IMP analogue of transitivity, I have taken advantage of a common idiom in the use of order relation symbols, one that represents their logical conjunction by way of a catenated syntax.  Thus, ''p''&nbsp;&le;&nbsp;''q''&nbsp;&le;&nbsp;''r'' means that ''p''&nbsp;&le;&nbsp;''q'' and that ''q''&nbsp;&le;&nbsp;''r''.  The claim that this 3-adic relation holds among the 3 propositions ''p'', ''q'', ''r'' is a stronger claim — contains more information — than the claim that the 2-adic relation ''p''&nbsp;&le;&nbsp;''r'' holds between the 2 propositions ''p'' and ''r''.