[9.4.] “Symbolizing ‘Someone’, ‘Somewhere’, ‘Sometime’, and So On.” (Ch.10:5)

 

Some uses of the terms “someone,” “somewhere,” etc., are misleading. At this point, you are probably tempted to reach for an existential quantifier — “($x)” — anytime you see these words. But some sentences that use these words are better symbolized using a universal quantifier: “(x)”.

 

For example:

 

“If someone is late, then everyone is annoyed with that person.”

 

You may be tempted to symbolize this sentence as follows:

 

dod = persons

 

($x)[Lx É (y)Ayx]

 

But this says: there is at least one person for which the following is true: if he or she is late, then everyone is annoyed with him or her. And this sentence is true if there is only one person in the world who would annoy everyone by being late. E.g., this sentence would be true if Homer would annoy everyone by being late but Marge would not (maybe everyone likes Marge more and so is willing to put up with her tardiness).

 

This is not what our original sentence means. Our original sentence, even though it uses the word “someone,” is attributing a property to every individual in the domain of discourse (viz., that he or she will make everyone annoyed if he or she is late). So we should use a universal quantifier:

 

(x)[Lx É (y)Ayx]

 

Important points to notice:

 

·         The mistake of using “($x)” rather than “(x)” is another example of incorrect combination of existential quantifier and horseshoe. Remember: if your symbolization is an existential quantification of a conditional, you should be suspicious. It is very unusual for a sentence to be accurately symbolized in that way, just as it is unusual for a sentence to be symbolized as a universal quantification of a conjunction, as in “(x)(Fx · Gx)”.

 

·         The symbolizations “($x)Lx É (y)Ayx” and “(x)Lx É (y)Ayx” would also be incorrect, since in each one, the second “x” is a free variable, i.e., a variable not bound by a quantifier. This is because the second “x” falls outside the scope of the “x” quantifier.

 

·         The sentence “If someone is late, then everyone is annoyed,” if interpreted strictly (to mean that if any person anywhere is late, then everybody everywhere will be annoyed) can be symbolized with an existential quantifier:

 

($x)Lx É (y)Ay                  which is equivalent to                 (x)[Lx É (y)Ay]

 

 

 

A more complicated example (from p.232):

 

"If someone is too noisy, then if everyone in the room is annoyed, someone will complain."

 

We can partially translate this correctly with an existential quantifier:

 

If ($x)Nx then, if (y)(Ry É Ay) then ($z)Cz

 

and then complete the translation as:

 

($x)Nx É [(y)(Ry É Ay) É ($z)Cz]

 

But the following cannot be translated that way:

 

"If someone is too noisy, then if everyone in the room is annoyed, someone will complain about that person."

 

We cannot translate this sentence as:

 

($x)Nx É [(y)(Ry É Ay) É ($z)Czx]

 

because the final x is unbound, i.e., it is a free variable, not within the scope of the “x’ quantifier.

 

And we cannot solve the problem by bringing the “x” within the scope of the “($x)” with braces:

 

 ($x){Nx É [(y)(Ry É Ay) É ($z)Czx]}

 

…because what that symbolization says is: there is at least one person about whom the following is true: if he or she is late and everyone in the room is annoyed, someone will complain about him or her. And this is consistent with there being only one person whose tardiness would cause annoyance and a complaint , as in the Homer and Marge example (again, notice the suspicious pairing of the existential quantifier with a conditional).

 

Our original sentence does not mean that there is at least one such person. Rather, it means that anyone whose tardiness causes everyone in the room to be annoyed will be complained about by someone. So as with the first example, we need to use a universal quantifier:

 

(x){Nx É [(y)(Ry É Ay) É ($z)Czx]}

 

Review a few examples from the box on pp.233-34.

 

Exercise 10-6 (p.235)

·         complete all for next time; we will cover at least the odds in class

Exercise 10-7 (p.235-36)

·         [we will do at least the first four in class]

·         complete all for next time; we will cover at least the odds in class

 

Stopping point for Monday April 14. For next time, complete ex.10-6 and 10-7, and read read ch.10:7 (pp.233-8)