Other examples: "is the same age as" "is the brother or sister of"

 

Other examples: "is the parent of", "is less than."

A weaker relation involving symmetry is antisymmetry. In an antisymetrical relation, the relata are symmetrical only if they are identical.

 

examples: "less than or equal to"

 

Results:

 

Any asymmetric relation is antisymmetric:

 

(x)(y)(Rxy ⊃ ~Ryx) // (x)(y)((x ≠ y & Rxy) ⊃ ~Ryx)

 

2. Transitivity

If 2 is less than 5 and five is less than 10, then 2 is less than 10. This fact is an instance of the transitivity of the "less than" relation.

 

Examples: "is the same age as", "is less than"

Notice that transitivity is really an A categorical proposition. What about intransitivity?

That's an E categorical proposition!

Examples: "is the parent of", "differs by 1 from"

We know that E categorical propositions are not the negations of A categorical propositions. So the notion of non-transitivity is different from the notion of intransitivity. We expect, and confirm, that "not transititive" is the negation of an A categorical proposition, namely, an O categorical proposition:

     

The final possibility is the negation of the E form, when a relation R is not intransitive:

R is not intransitive    

 

3. Reflexivity

 

R is reflexive            

 

R is irreflexive            (x)~Rxx