[6.6.] “Basic Predicate Logic Symbolization.” [Ch. 7:5-6]

 

These sections of your textbook cover four types of sentence, all of which we’ve seen before. Here they are reintroduced in a more systematic way:

 

“The Square of Opposition”

 

	AFFIRMATIVE	NEGATIVE
U N I V E R S A L	A   ATTRIBUTE a trait to ALL members of a group   “All As are Bs”   (x)(Ax É Bx)   ~($x)(Ax · ~Bx)	E   DENY a trait of ALL members of a group   “No As are Bs”   (x)(Ax É ~Bx)   ~($x)(Ax · Bx)
P A R T I C U I L A R	I   ATTRIBUTE a trait to SOME member(s) of a group   “Some As are Bs”   ($x)(Ax · Bx)   ~(x)(Ax É ~Bx)	O   DENY a trait to ALL members of a group     “Some As are not Bs”   ($x)(Ax · ~Bx)   ~(x)(Ax É Bx)

 

Notice that the members of each pair of same-colored sentences contradict each other: one is simply the negation of the other, so they must have opposite truth values.

 

“Most of the sentences you will be asked to symbolize in this chapter are simply elaborations of these four basic sentence patterns.” (p.177)

 

E.g.: “All dogs and cats should be spayed or neutered.”

 

(x)[(Dx Ú Cx) É (Sx Ú Nx)]

 

This is just a complicated sentence of the first type: it attributes a trait (needing to be spayed or neutered) to every member of a group (the group consisting of all dogs and cats).

 

**Notice that the antecedent is a disjunction, not a conjunction. This symbolization

 

(x)[(Dx · Cx) É (Sx Ú Nx)]

 

would be incorrect; it means: “Anything that is both a dog and a cat should be spayed or neutered” (this is probably true, but it is not at all the same claim as the sentence with which we began).

 

 

[6.7.] “Common Pitfalls in Symbolizing with Quantifiers.” [Ch.7:7]

 

 

1.      Expressions that seem to represent compounds of properties but really do not.

 

“Some professional athletes excel in both football and baseball.”

 

The expression “professional athletes” should NOT be symbolized with separate property constants, as in:

 

($x) [(Px · Ax) · (Fx · Bx)]

 

This is because you can be a professional (e.g., a professional philosopher) AND an athlete (e.g., an amateur racquetball player) without being a professional athlete. The correct symbolization is:

 

($x) [Px · (Fx · Bx)]

 

where “P” stands for the property being a professional athlete. So when you are symbolizing, you must think about each of the properties you are symbolizing and whether it is accurate to break any of them down into separate properties.

 

Another example: Some soldiers fought in Iraq.  “Fought in Iraq” must be symbolized with a single constant, since it is possible to have fought (e.g., to have fought on the playground as a child), and to be in Iraq (e.g., as a civilian contractor), without having fought in Iraq.

 

 

2.      The words “a” and “any.”

 

Sometimes these words mean “all”--

 

“A day without rain is a pleasure.”                     (x)[(Dx · ~Rx) É Px]

 

“Any friend of yours is a friend of mine.”            (x)(Yx É Mx)

 

but sometimes they do not--

 

“A mouse ate that cheese.”                               ($x)(Mx · Ax)

 

“There isn’t any cheese in the trap.”                   ~($x)(Cx · Tx)

 

You must pay close attention to the meaning that the sentence conveys to detect whether a universal quantifier or an existential quantifier is appropriate.

 

 

 

3.      Misleading “and”s.

 

We already saw one of these with the “it’s a dog-cat!” example above. Here are two more:

 

“Juniors and seniors are exempt from finals.”      (x)[(Jx Ú Sx) É Ex]

 

This could also be symbolized as a conjunction of conditionals, thus making the “and” explicit:

 

(x)[(Jx É Ex) · (Sx É Ex)]

 

 

 

“Some juniors and seniors bought beer.”                         ($x)(Jx · Bx) · ($x)(Sx · Bx)

 

This MUST be symbolized as a conjunction of conjunctions. It cannot be symbolized:

($x)[(Jx Ú Sx) · Bx], because that symbolization is true if only juniors bought beer, and it is true if only seniors bought beer.

 

 

 

Exercise 7-6, pp.180-81

·         do all problems for next time (number 9 is especially tricky); we’ll go through the odds next class

 

Exercise 7-7, p.181

·         do all problems for next time; we’ll check the odd ones in class

 

 

 

Stopping point for Friday March 7. For next time, do ex.7-6 & 7-7, and read ch.7:9 (pp.181-82).