[10.5.] “Properties of Relations.” [Ch. 13:3]

 

In this section, we examine three properties of that dyadic (two-place) relations can possess:

·         reflexivity

·         symmetry

·         transitivity

 

 

[10.5.1.] Reflexivity.

 

A relation Fxy is totally reflexive if and only if the following is true: for all x, x is F to itself. I.e., 

 

(x)Fxx

 

Examples:

·         x is numerically identical to x

·         x is qualitatively identical to x

 

 

A relation Fxy is (not totally reflexive, but nonetheless) reflexive if and only if the following is true: if x is F to something, then x is F to itself., i.e.,

 

(x)[($y)Fxy É Fxx]

 

Examples:

·         x is in the same political party as y

·         x uses the same barber as y

·         x likes the same thing as y

·         x is the same size as y

 

 

A relation Fxy is irreflexive if and only if the following is true: for all x, x is not F to itself. I.e.:

 

(x)~Fxx

 

Examples:

·         x is the father of y

·         x is the child of y

·         x is taller than y

 

 

A relation Fxy is non-reflexive if and only if it is neither reflexive nor irreflexive.

 

Examples:

·         x loves y [the fact that x loves y implies neither that x loves x nor that x does not love x]

·         x hates y

·         x admires y

 

[10.5.2.] Symmetry.

 

A relation Fxy is symmetrical if and only if the following is true: if x is F to y, then y is F to x. I.e., a relation is symmetrical if the following is true:

 

(x)(y)(Fxy É Fyx)

 

Examples:

·         x is married to y

·         x is a sibling of y

·         x belongs to the same political party as y

 

A relation Fxy is asymmetrical if and only if the following is true: if x is F to y, then y is not F to x. I.e., a relation is asymmetrical if the following is true:

 

(x)(y)(Fxy É ~Fyx)

 

Examples:

·         x is a child of y

·         x is the father of y

·         x is underneath y

·         x is taller than y

 

A relation Fxy is non-symmetrical if and only if it is neither symmetrical nor asymmetrical.

 

Examples:

·         x loves y [the fact that x loves y implies neither that y loves x nor that y does not love x]

·         x hates y

·         x admires y

 

 

[10.5.3.] Transitivity.

 

A relation Fxy is transitive if and only if the following is true: if x is F to y, and y is F to z, then x is F to z. I.e., a relation is transitive if the following is true:

 

(x)(y)(z)[(Fxy · Fyz) É Fxz]

 

Examples:

·         x is taller than y

·         x is wealthier than y

·         x is underneath y

 

See the proof on p.293: in order to show that “Art is taller than Betsy” and “Betsy is taller than Charles” deductively imply that “Art is taller than Charles” in our system of logic, you have to add a premise that states that being taller than is a transitive relation:

 

(x)(y)(z)[(Txy · Tyz) É Fxz]

 

 

A relation Fxy is intransitive if and only if the following is true: if x is F to y, and y is F to z, then x is not F to z. I.e., a relation is intransitive if the following is true:

 

(x)(y)(z)[(Fxy · Fyz) É ~Fxz]

 

Examples:

·         x is a child of y

·         x is the father of y

 

 

A relation is non-transitive if and only if it is neither transitive nor intransitive.

 

Examples:

·         x loves y [the fact that x loves y and that y loves z implies neither that x loves z nor that x does not love z]

·         x hates y

·         x admires y

 

 

Exercise 13-4 (A): we will work all of these in class next time.

 

 

 

Stopping point for Friday April 25. Next time we will work through 13-4 A in class. Over the next two days we’ll review exercises, have a Q&A for last test, and have student evaluations.