Categorical Propositions with Complex Subjects and Predicates

We have restricted ourselves to categorical propositions with simple subjects and predicates, such as:

All mustard is yellow translated as (x) (Mx ⊃ Yx)

where "Mx" abbreviates "x is mustard" and "Yx" abbreviates "x is yellow."  Now we can lift the restriction and consider more complex propositions which are still categorical.

All spicy mustard is yellow translated as (x) ((Mx & Sx) ⊃ Yx)

Here the subject is not simply, "Mx," for  "x is mustard," but "Mx & Sx" for "x is mustard and x is spicy." It should be obvious why we translate the complex subject as a conjunction.

Both the subject and the predicate may be complex:

Some dark chocolate is neither sweet nor high in carbohydrates.

Let's do an intermediate translation which makes the propositional functions explicit:

There is an x which is dark and chocolate, and  is neither sweet nor high in carbohydrates.

Notice that we use a comma to separate the subject from the predicate. Using our knowledge of propositional logic, it should be clear that the predicate can be translated as "x is not sweet and x is not high in carbohydrates" or as "it is not the case that either x is sweet or x is high in carbohydrates." Our two correct translations are:

(∃x) ((Dx & Cx) & (~Sx & ~Hx))
(∃x) ((Dx & Cx) & ~(Sx v Hx))

We take care to make sure that our parentheses encase the entire categorical proposition within the scope of the quantifier.

Another kind of sentence with a complex subject appears to be translatable as a conjuction, but this appearance is illusory. Consider the sentence:

All Democrats and Republicans are politically sensitive.

Can we translate this as:

(x)((Dx & Rx) ⊃ Px)

Clearly not: This says: All things which are both Democrats and Republicans are politically sensitive. What the sentence really expresses is a disjunctive subject, that for all x, if x is either a Democrat or a Republican, x is politically sensitive:

(x)((Dx v Rx) ⊃ Px)

We could also translate this sentence as the conjunction, "All Democrats are politically sensitive and all Republicans are politically sensitive."

(x)(Dx ⊃ Px) & (x)(Rx ⊃ Px)

Note that this means that the following equivalence holds:

(x)((Dx v Rx) ⊃ Px) ≡  ((x)(Dx ⊃ Px) & (x)(Rx ⊃ Px))

When you've acquired the skill of judging validity and other semantic properties for QL, you can confirm, using the truth-tree method, that this wff is logically true.

In the example above, we expressed "all Democrats and Republicans" as a disjunctive subject. While we can do this for universal categorical propositions, we cannot do so for particular, or existentially quantified propositions. Let's look at the following English sentence:

Some Democrats and Republicans are politically sensitive.

Notice that this commits us to the existence of at least one Democrat who is politically sensitive, and one Republican who is politically sensitive. Translating this as:

(∃x)((Dx v Rx) & Px)

says that there is something that is either a Democrat or a Republican who is politically sensitive. And that would be true if, say, there were no Republicans who are politically sensitive. So we have to translate the sentence as a conjunction of I type categorical propositions:

 
(∃x)(Dx & Px) & (∃x)(Rx & Px)

Another situation to watch out for are sentences such as this one:

Philosophers are lonely only if they are mean.

The difficulty here is figuring out whether the subject is "philosophers" and the predicate "those who are lonely only if they are mean" or the subject is "philosophers who are lonely" and the predicate is "those who are mean".  Our two candidate translations are:

(x)(Px ⊃ (Lx ⊃  Mx))

or

(x)((Px & Lx) ⊃  Mx)

Which translation is correct? To answer this, let's take a look at an I categorical sentence using the same propositional functions:

Some philosophers are lonely only if they are mean.

Is the subject some "philosophers" or some "philosophers who are lonely"? That is, does the sentence commit us to the existence of at least one lonely philosopher? So is it:

(∃x)(Px & (Lx ⊃  Mx))

or

(∃x)((Px & Lx) ⊃  Mx)

To find the answer, consider this: Could  "some philosophers are lonely only if they are mean" be true if there were no lonely philosophers? Sure - there could be a philosopher, but no mean philosophers. It would still be true that if there were a lonely philosopher, that individual would be mean. So the correct translation of the I categorical version is:

(∃x)(Px & (Lx ⊃  Mx))

and the correct translation of the A categorical version is:

(x)(Px ⊃ (Lx ⊃  Mx))

In both cases, the predicate of the categorical form is of the form "A only if B".

 

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