Inside Logic #24

Some

An idea related to "all" is "some". It will be useful to have a unique way to represent "some". So let us use

(Ex)

to express the idea of "at least one" so that, if Cx stands for x is cold, (Ex)Cx says at least one thing is cold, that is, something is cold. (The "E" was used traditionally for "exists".)

Now if we know that Bob is cold, Cb, obviously it follows that someone is cold, (Ex)Cx, there is at least one thing that is cold. This is an instance of reasoning that clearly is valid, and it gives us a new Rule for "some".

Existential Generalization (EG)

Given Fc as a premise, for any predicate F and name c, we can write (Ex)Fx with all the same assumptions as the premise.

More generally, if a sentence P contains a name c, then we can write (Ex)P* where P* is like P except x replaces at least one occurence of c in P, where (Ex)P* has all the assumptions as P. So, for example, given Fc&Gc as a premise, we can write (Ex)(Fx&Gx).

This rule is called "Existential Generalization" because (Ex) is called the "existential quantifier." As noted (Ex)Fx will be used to mean that at least one thing x is such that Fx. We can use the existential quantifier to represent English statements involving "some" or "something" or "someone" and so forth. (Sometimes these terms are used to mean more than one-- as, for example, in There are some people in this room, which normally would be be understood to mean at least two. It would be odd to say that if you knew there was only one. But for present purposes, we let (Ex) mean "at least one"; and later we can develop a way to say "at least two".)

The rule EG is simple and it makes sense. It enables us, of course, to prove the clearly valid sequent:

151 Cb } (Ex)Cx

  1 1. Cb A        
  1 2. (Ex)Cx 1, EG

Now notice that a sentence like Some lottery winners are happy can be intepreted as

(Ex)(Lx & Hx).

There is at least one thing x such that x is both a lottery winner and happy. That is what it means to say that some lottery winners are happy.

If we know that Edna is a happy lottery winner, we can use the new rule EG to prove (Ex)(Lx & Hx). The proof is just as simple as the earlier one (using the following symbols. Lx: x is a lottery winner; Hx: x is happy; e: Edna).

152 Le & He } (Ex)(Lx & Hx)

  1 1. Le & He A        
  1 2. (Ex)(Lx & Hx) 1, EG

Notice that

(Ex)Lx & (Ex)Hx

says Somebody is a lottery winner and somebody is happy; and this is different from the statement

(Ex)(Lx & Hx).

Why? Because the truth of (Ex)Lx & (Ex)Hx does not guarantee that there is one single person who is both a lottery winner and happy whereas the truth of (Ex)(Lx & Hx) does guarantee that there is at least one person who is both a lottery winner and happy.

We use parenthesis to indicate the difference in the two meanings.

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Practice

24.1 Symbolize the following sentences. Use these symbols. Bx: x is a bicycle. Dx: x is dangerous. Vx: x is a vampire. Tx: x is a tiger. Fx: x is friendly. Mx: x is a marble. Yx: x is yellow. Gx: x is green. Rx: x is red. Px: x is a professor. Cx: x is a criminal.

(a) Some bicycles are dangerous.

(b) Some bicycles are not dangerous.

(c) Some vampires are dangerous.

(d) Some vampires are not dangerous.

(e) Some tigers are friendly.

(f) Some tigers are not friendly.

(g) Some marbles are yellow.

(h) All marbles are green.

(i) No marbles are green.

(j) Some marbles are red and some are green.

(k) Some marbles are both red and yellow.

(l) Some professors are criminals and some are vampires.

24.2 Construct derivations for each of the following valid sequents.

153 Fa, ~Ka } (Ex)(Fx & ~Kx)

154 Fb, ~Kc } (Ex)Fx & (Ex)~Kx

155 (x)(Fx > Kx), Fa, (x)(Gx > ~Kx) } (Ex)~Gx

156 (Ex)Kx > Fc, (x)(Fx > Hx), Gc > Kc, Gc } Hc

24.3 Symbolize the following arguments and construct derivations in each case to show that they are valid.

(a) Alice is a vampire and Alice is a resident of Boston. So, some resident of Boston is a vampire. (Vx: x is a vampire. Bx: x is a resident of Boston. a: Alice)

(b) Every professor is boring. Professor X is a professor. No belly dancer is boring. Therefore, somebody is not a belly dancer. (Lx: x is a professor. Bx: x is boring. Dx: x is a belly dancer. c: Professor X)

(c) Girls just want to have fun. Everyone is either a boy or a girl. Chloe is not a boy. Therefore, somebody just wants to have fun. (c: Chloe. Gx: x is a girl. Bx: x is a boy. Wx: x just wants to have fun.)

24.4 Extra Credit

(a) All green marbles are cat's-eyes. This thing is a marble and it is green. Also this thing is from Indiana. So there are cat's-eyes from Indiana. (Gx: x is green. Mx: x is a marble. Cx: x is a cat's-eye. Ix: x is from Indiana. t: this thing)

(b) If somebody stole a cookie, then it was Mary. Anybody who steals cookies should be put in jail. If the cookie jar is empty, somebody stole a cookie. Indeed the cookie jar is empty. Therefore Mary should be put in jail. (Sx: x steals a cookie. Jx: x should be put in jail. Ex: x is empty. m: Mary c: the cookie jar.)