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

 

 

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

 

 

“Only those who study will pass the course.”

 

The key to symbolizing this correctly is 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.

 

This sentence is not saying something about everyone who studies. In particular, it is not saying that everyone who studies will pass the course.

 

This 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 “S”).

 

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

 

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 the course”

 

and

 

“None except those who study will pass the course”

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

 

are equivalent to “Only those who study will pass the course” and so is symbolized in the same way: (x)(Px É Sx) .

 

 

[6.8.2.] “Unless.”

 

We can use “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. =

 

Only those who study will pass. =

 

If someone passes, then he or she has studied. =

 

(x)(Px É Sx) =

 

(x)(~Sx É ~Px)

 

 

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

 

All the questions will be difficult unless the professor is merciful.  =

 

[letting the individual constant “p” stand for the professor]

 

(x)(Qx É Dx) unless Mp   =

 

(x)(Qx É Dx) if not Mp   =

 

~Mp É (x)(Qx É Dx)   =

 

Mp Ú (x)(Qx É Dx)

 

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 we’ve learn the rules of implication (Imp), 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

 

 

 

 

Exercise 7-9 (pp.185-86)

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

Exercise 7-10 (p.186)

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

 

 

Stopping point for Monday March 10. For next time, complete exercises 7-9 and 7-10, and read Ch.8:1-2, 4-5 (pp.189-93, 195-96).