Part I: Lesson

“We must determine what a premise is, what a term is, and what sort of deduction is complete and what sort incomplete.’”

(Aristotle, Prior Analytics, transl. Robin Smith, 2012)

A few lessons ago, we distinguished between categorematic and syncategorematic terms. In this course, we will be primarily concerned with the former. To review:

Categorematic Term: any term that designates a category of things in the world.

Remember that categories are roughly collections of things with a common, exclusive property. The properties are exclusive in the sense that if anything does not have a certain property, then those things are not in the category corresponding to that property. So, when we say that something has a certain property, we are saying that it is part of a certain category. Thus, when we say “Socrates is mortal”, we are saying that Socrates has the property of being mortal, which means that Socrates belongs to the category “mortal things.”

Note that if this diagram was perfect, there would be many more things inside of the category “mortal things”.

Remember, terms themselves are never true or false. However, terms can be vague or ambiguous.

A) Vague Terms

As is often the case with language, terms can be imprecise, unclear, or confusing. We need to be especially wary of this when we do logic, since we aim for our study to be exactly the opposite. One feature that demands close attention is vagueness. A term, usually a predicate, is called vague if it has particular borderline cases where it is not clear whether it can be correctly predicated of the subject or not. In other words, a term is vague if it is unclear which things are members of the category it refers to.

Some classic examples are words like ‘bald’, ‘short’, ‘tall’, ‘beautiful’, etc. We do not have a precise height after which we consider a person a member of the category ‘tall’ and before which we consider them a member of the category ‘short’. We do not have a precise number of hairs, say, after which we consider a person not-bald and before which we consider them bald. Suppose that Socrates is balding. When would it be true to say “Socrates is balding”? In other words, when is Socrates a member of the category “bald things”?

In logic, we never want to have diagrams like the one above. We need to make sure we are clear enough to understand what exactly the members of each category of the terms we use are.

Consider the famous example of a heap of sand. How many grains of sand does it take to form a heap? If we had a thousand grains of sand in a pile, we would probably call it a “heap” of sand. But let’s say we start taking away grains of sand one by one. When is it no longer a heap? When there are only 5? 3? 2? This is an example of the vagueness of the word “heap.” 

[Activity] Think of 3 more vague terms. Find a partner and explain to them what borderline cases each of them has.

B) Ambiguous Terms

Another problematic feature of certain words is ambiguity. A term is called ambiguous if it can have multiple meanings. In other words, a term is ambiguous if it can refer to multiple different categories. Consider the following words: ‘bank’, ‘star’, ‘cold’, ‘ruler’. Each of them has more than one meaning. ‘Bank’ could designate the category of all river banks or it could designate the category of all of the financial institutions we call “banks”:

Notice that both of the categories above are called “Banks”, but they have different members! So, which category are we referring to when we say “bank” or “that is a bank”?

For another example, ‘Star’ could refer to a category of the celestial bodies we call ‘stars’ or to a category of actors famous enough that we call them ‘stars’. 

If these words need to be ‘disambiguated’ — we need to specify which meaning we want to use — in a logical context. Only when we know what the word refers to can we evaluate the truth of various sentences. The sentence “That bank is beautiful”, when expressed by a speaker in a city with a beautiful river and terribly unappealing buildings, could be true when said of the beautiful river bank but false when said of the financial institution. 

[Activity] Write down a few more examples of ambiguous terms. Use each of the term’s meanings in a sentence and share with a partner! Draw some diagrams if it helps!

[Activity] With a partner, take turns coming up with one term and decide whether it is vague or ambiguous. Then, share it with your partner and ask them to decide whether it is vague or ambiguous. Your partner should then explain their reasoning; if you disagree, debate!