Superlatives and Exceptives

In addition to expressing number, we can also translate superlatives, such as "Blanche is the best mathematician" and exceptives, such as "Blanche is the only mathematician in the room" and "Every mathematician in the room except Blanche is wearing tennis shoes."

Let's start with our superlative:

1. Blanche is the best mathematician.

Let's begin with our dictionary:

a: Blanche

Mx: x is a mathematician

Bxy: x is better than y.

Here, as throughout our translations in QL, the categorical form of propositions is an enormous aid to translation. So what is the categorical form of (1)?  It asserts that Blanches is better than all (other) mathematicians. So, for all mathematicians who are not Blanche, Blanche is better than them. Or:

1t. (x)((Mx & x ≠ a)) ⊃ Bax) & Ma

This says, "For all x, if x is a mathematician and x is not identical to Blanche, then Blanche is better then x, and Blanche is a mathematician." Don't forget the last conjunct. It may seem to be presupposed in what you've already translated, but it is not.  Superlatives involve comparisons, so expect to see two-place relations when you translate them.

Since this is an "A" type categorical proposition, there is a negated O categorical proposition which is logically equivalent to it. What is that proposition?

Let's turn to exceptives. We'll translate:

2. Blanche is the only mathematician in the room.

Remember how we dealt with "the only" in regular categorical propositions? That knowledge is about to come in handy.  Let's begin by rewriting (2) as:

The only mathematician in the room is Blanche.

which, categorically, reads as:

No mathematicians (other than Blanche) are in the room and Blanche is a mathematician and Blanche is in the room.

Admittedly, we would never rephrase (2) this way in ordinary conversation, but now we can see exactly how to translate into QL:

2t. (x)((Mx & x ≠ b)  ⊃ ~Rx) & (Mb & Rb)

 Similarly, we can translate:

3. Every mathematician in the room except Blanche is wearing tennis shoes.

This is equivalent to "All mathematicians in the room who are not Blanche are wearing tennis shoes, and Blanche is a mathematician and Blanche is not wearing tennis shoes." So this is an A categorical sentence, with the conjoined bit about Blanche. We need to add the following to our dictionary:

Tx: x is wearing tennis shoes

Note that we could make our dictionary more precise, by including "x wears y" as a relation, and the predicate "x is (a pair of) tennis shoes", but we'll stick to the simpler formulation here.

3t. (x)(((Mx & Rx) & x ≠ b) ⊃ Tx) & ((Mb & Rb) & ~Tb)

You're now ready for an exercise on Superlatives and Exceptives.

 

 

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