[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”[1]

 

	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, so they must have opposite truth values.

·         A and O sentences contradict each other; a pair of such sentences is contradictory;

·         E and I sentences contradict each other; a pair of such sentences is contradictory.

 

“Most of the sentences you will be asked to symbolize in this chapter are simply elaborations of these four basic sentence patterns.” (p.179)

 

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

 

(x)[(Dx Ú Cx) É (Sx Ú Nx)]

 

This is just a complicated A sentence: 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.

 

We saw one example of this earlier: the phrase “elected official” stands for a single property, being an elected official, rather than two separable properties, being elected and being an official. Someone can be elected (i.e., chosen by way of a vote) without being an official.  (see exercise 7-5, #4 and 5, in which a single property constant is chosen to stand for being an elected official).

 

Consider

 

“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 medical doctor) 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 property 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.

 

Further clarification on when to use two (or more separate property constants)…

 

“All female marsupials have pouches.”                     (x)[(Fx · Mx) É Px]

 

It is impossible to be both female and a marsupial without being a female marsupial. Hence, it is safe to use two property constants, one for being female and another for being a marsupial. Since it is safe to do so, we should—this reveals more logical structure than would be revealed were we to use a single property constant to represent being a female marsupial. We would not change the meaning of the English sentence by representing it as

 

(x)(Fx É Px)   [where “Fx” means being a female marsupial]

 

But we would fail to show as much of the internal logical structure of that sentence as possible.

 

 

“All professional athletes are healthy.”                     (x)(Px É Hx)

 

It is possible to be both a professional and an athlete without being a professional athlete (e.g., one can be a medical professional and an amateur athlete). Hence, to use two property constants, one for being a professional and one for being an athlete, would result in a sentence that attributes health to too many individuals—it includes professional doctors, lawyers, etc. who are amateur (non-professional) athletes:

 

(x)[(Px · Ax) É Hx]

 

This attributes health to, e.g., the doctor who is an amateur marathon runner. So using two property constants changes the meaning of the original sentence. So we must use only one property constant.

 

Other examples:

·         “All short circus clowns are funny.” [requires separate constants for Short and Circus Clown]

·         “All rabid possums should be shot.” [requires separate constants for Rabid and Possum]

·         “All French instructors work hard.” [requires a single constant for French Instructor]

 

 

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 in order 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.182-83

·         [we will do a few even problems in class]

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

 

Exercise 7-7, p.183

·         [we will do a couple of evens in class]

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

 

 

[6.8.] Difficult Symbolizations [Ch.7:9]

 

[6.8.1.] “Only”, “None But”, “None Except.”

 

Remember how “p only if q” gets symbolized in sentential logic: p É q.

 

So in sentential logic, “John passes the course only if he studies” would get symbolized as: P É S. In other words, John’s studying is a necessary condition of his passing. If he passes, then that’s a guarantee that that necessary condition has been met. In other words, his passing is a guarantee that he studied.

 

Keep that in mind as we consider how the following sentence is to be symbolized in predicate logic:

 

“Only those who study will pass.”

 

This conveys the same idea, that studying is a necessary condition of passing the course. So for anyone who passes, his or her passing is a guarantee that he or she studied: if x passed, then x studied.

 

To symbolize this sentence correctly, it is helpful to recognize it as a universal affirmative statement (an “A” form statement). It attributes a property to all members of a certain group. The tricky part is recognizing which trait is being attributed to the members of which group. But what we said above about the logic of “only” should help you figure out which group is having a property attributed to all of its members. The sentence is saying something about everyone who passes the course, namely, that he or she studied.

 

So the property being attributed is: studying (represent this with “Sx”).

 

And the group to which it is being attributed is: those who pass the course (represent this with “Px”).

 

So a correct symbolization is: (x)(Px É Sx)

 

Another correct symbolization is: (x)(~Sx É ~Px) . This is the contrapositive of the first symbolization and is thus equivalent to that symbolization by way of the rule of contraposition (Contra).

 

The phrases “none but” and “none except” are treated in the same way as “only”:

 

“None but those who study will pass.”

and

“None except those who study will pass.”

(i.e., “None will pass except those who study.”)

 

are equivalent to “Only those who study will pass” and so is symbolized in the same way:

(x)(Px É Sx) .

 

Other examples:

·         “No one understands reality except for philosophers.”                  (x)(Ux É Px)

·          “Braves fans are the only fans who are loyal.”                             (x)[(Fx · Lx) É Bx]

·         “Only smokers are angry about the new tax cigarette tax.”                        (x)(Ax É Sx)

·         “People carry umbrellas only when they suspect rain.”                 (x)[(Px · Cx) É Sx]

 

 

[6.8.2.] “Unless.”

 

Recall from our treatment of sentence logic that “unless” is equivalent to “if not”:

 

p unless q                 =

 

p if not-q                  =

 

if not-q, then p          =

 

~q É p                      =

 

q Ú p                         =

 

p Ú q

 

An aside: When we first studied symbolization, we saw that sentences of the forms “~q É p” and “p Ú q” were equivalent, and equally good translations of “p unless q”. This equivalence should be more obvious to you now that you have learned the rules of implication (Impl), double negation (DN), and commutation (Comm):

	1. p Ú q                                    p 2. q v p                                    1 Comm 2. ~~q Ú p                                2 DN 3. ~q É p                                 3 Impl

 

1. ~q É p                              p

2. ~~q v p                             1 Impl

3. q Ú p                                 2 DN

4. p v q                                 4 Comm

 

 

We can use the word “unless” to express the same claims as those we have just seen:

 

Only those who study will pass.          =

 

No one will pass unless he or she studies. =

(not-pass unless study)

(not-pass if not-study)

(if not-study, then not-pass)

 

(x)(~Sx É ~Px)  =

 

(x)(Px É Sx) =

 

 

But remember that “unless” can also serve as a truth-functional sentence connective, as in:

 

All the questions will be difficult unless Professor Smith is merciful.  =

 

[letting the individual constant “s” stand for Smith]

 

(x)(Qx É Dx) unless Ms   =

 

(x)(Qx É Dx) if not Ms   =

 

~Ms É (x)(Qx É Dx)   =

 

Ms Ú (x)(Qx É Dx)

 

 

 

 

Exercise 7-9 (pp.188)

·         do all problems for next time; we’ll check some of the odds in class

Exercise 7-10 (p.188-189)

·         do all problems for next time; we’ll check some of the odds in class

 

 

Stopping point for Tuesday March 6. For next time:

·         complete exercises 7-6, 7-7-, 7-9 and 7-10;

·         read Ch.9:1-2 (200-208).