Intro to logic-sec, 4.4 operations on categorical propositions | Introduction to Logix | Bergen Community College

This week, our topic is operations that can be performed on categorical propositions.  Remember that there are four standard form categorical propositions, and that was because there are four ways to relate one category to another.  (Total inclusion, total exclusion, partial inclusion, partial exclusion.)  But this relation is not symmetrical.  There are four ways a subject can relate to a predicate, but how the predicate relates back to the subject is somewhat undetermined.  To be precise about all the possible ways that two categories can relate, the propositions have to be manipulated a little bit.  And that is done with the operations we will talk about.  

There are three operations: Conversion, Obversion, and Contraposition.  They all involve subjecting a proposition to a specific set of rules, which yields a new proposition.  There are only one or two rules each, so it’s not complicated.  But the rules do involve remembering the definitions of quality and quantity that we looked at earlier. 

The new proposition–which again is obtained by applying the rules of whatever operation you’re performing–has a specific logical relation to its original.  As a result, we can understand all conceivable ways one category can relate to another.  And, it is also possible to know whether arguments that contain different permutations of propositions are valid or invalid. 

So, here are the operations.

1. CONVERSION:  It’s simple, one rule.  Reverse the order of the subject and the predicate.  (You might have expected that.)  All A are B becomes All B are A.  The same for the rest.  This second proposition is said to be the ‘converse.’   You will note that in the textbook, pg. 222, Venn diagrams are given to illustrate the logical relations between each proposition and its converse.

2. OBVERSION:  A little less simple, two rules:  1. Change the quality, without changing the quantity.  It’s easier than it sounds, and by the way, it’s good mental practice correctly applying rules to situations.   Rule 2. Replace the predicate term with its ‘term compliment.’ Ugh.  What is a ‘term compliment?’  A term compliment is the term that denotes all things in the world that are not denoted by the term in the original proposition.  If I have a category, houses (H), then its term compliment is all things that are not houses (non-H).  Why is this important?  Off the top of my head, I’m not sure.  If I think of something, I’ll let you know.  Or, maybe you can think of a reason.  In any case, when you perform the obversion operation on a proposition, you form what is called the ‘obverse.’  To see what the obverse of the four categorical propositions look like, see page 224.  Here is the first one, though.  

1. All A are B.  Obverse:  No A are non-B.  Look, I changed the quality from affirmative to negative, but I kept the quantity as universal.  And then I replaced the predicate with its term compliment.  Ok, one more.

2. No A are B.  Obverse:  All A are non-B.   I did the same thing again.  An interesting thing about the obverse of a proposition.  Look at the Venn Diagrams.  The obverse of each proposition is logically equivalent to the original proposition.  That’s good to know.  

3. CONTRAPOSITION:  Again, two rules.  1. Reverse the order of the subject and predicate terms.  (That is, form the converse.)  But then 2.  Replace both subject and predicate terms with their term compliments.  So All A are B becomes All non-B are non-A.  No A are B becomes No non-B are non-A.  And so on.   Look at the Venn Diagrams for the logical relations between these propositions.  They look a little strange.  In our Zoom meeting, I will explain these diagrams.  

We now have all conceivable ways two categories can relate to each other, as well are their logical relations.   In the assignment window, I will put up some study questions, which, if you answer them, you should understand the material.