Language as a Way of Knowing: Three Logic Puzzles

By Louis Cheng

(Test Preparation (ACT/SAT/SSAT/ISEE/IELTS/TOEFL) , IB ToK (Theory of Knowledge), Economics tutor at The Edge Learning Center)

Every IB student needs to take Theory of Knowledge, a course that aims to provoke critical examination into the ways through which we know what we claim to know. It is a simplified version of what is traditionally called “epistemology in philosophy. Each year, many students come to me for assistance with the two assessments in their ToK courses – namely, the essay and the presentation – and most of them complain that they do not really understand what is at stake in ToK as their ToK teachers at school are sometimes similarly confounded. Therefore, in this post, I wish to take a step back from the assessments and briefly look at the subject matter ToK is concerned with. It may turn out that, contrary to popular opinion, ToK can be quite fun and thought-provoking to learn.

Let’s look at language. Language is one of the primary “Ways of Knowing” (the ToK course identifies eight of them). Naturally, to examine the validity of what we claim to know, the first place to look is the way through which we acquire the information. Every day we use language to communicate with friends, read news, and even formulate thoughts. However, ordinary speech is imbued with ambiguities and nuances that are sometimes not immediately comprehensible by logical reasoning, which gives rise to interesting problems for logicians and philosophers of language. Let’s look at three such puzzles:


Otherwise known as the sorites paradox, this puzzle is concerned with everyday words which lack a precise meaning, such as “tall”, “short”, “hot”, and “cold.” The puzzle is formulated in terms of an induction:

  1. A person who is one-day-old is young.
  2. If a person who is n-day-old is young, then a person who is (n + 1)-day-old is also young.
  3. Recursively applying (2) onto (1) yields the conclusion that a 36,500-day-old (~100-year-old) person is young. (i.e., one-day old person is young, two-day old person is young, three-day-old person is young, …, 36,500-day-old person is young.)

It would seem that conclusion (3) is unacceptable as it uses the word “young” in a decidedly wrong manner. However, it also seems that premises (1) and (2) are true statements. What gives? What implications does this discovery have for the status of vague words? Can we still rely on them in making knowledge claims about the world when their meanings are permeated with uncertainties?

One is never too old (young?) to start saving for retirement. (Source: Glasbergen)

Number Words

We use number words (e.g., “one” and “two”) all the time, and their meanings seem intuitively obvious to us. Nevertheless, logicians have discovered that the semantics of individual number words differ in significant ways in different utterances. Consider the following three statements:

  1. You have to correctly answer three questions to pass the test.
  2. You can still pass the test with three mistakes.
  3. You made three mistakes in the test.

Even though these sentences look unproblematic at first glance, an examination into the meaning of “three” in each reveals a problem. In (1), “three” means “at least three”; in (2), “three” means “at most three”; in (3), “three” means “exactly three.” For example, a competent language user wouldn’t falsely understand (1) to mean that you must correctly answer exactly three questions to pass test, not more, not fewer (i.e., you will fail the test by answering four questions correctly). Similarly, we would not take (2) to mean that you cannot pass the test if you make only two mistakes.

What explains the incongruent semantics of number words? How do competent language users distinguish between the three possible meanings? Having discovered this instability in the semantics of number words, one may question how our use of these words may depend on cultural assumptions and how they may distort or reinforce knowledge claims in various ways.

It turns out that number words definitely take on different meanings when describing one’s age. (Source: Dumpaday)


Generics are sentences that express general claims about categories of objects. These are sentences such as “Tigers are striped.” These sentences are used very frequently in everyday speech. Consider the following generic statements:

  1. Ducks lay eggs.
  2. Ducks are female.

Most competent language users will say that (1) is a true statement while (2) is false. However, upon scrutiny we discover strange truth conditions for the two statements. Namely, only female ducks lay eggs, and in fact only a portion of female ducks lay eggs. In other words, there are more ducks satisfying the condition stated in (2) than that in (1), but (1) is true while (2) is false. Similar perplexities can be found in sentences such as “sharks attack swimmers” (true, even though most sharks do not attack swimmers) and “books are paperbacks” (false, even though the majority of books are paperbacks).

Other interesting cases concern the description of persons in a similar sentence type called “habituals.” For instance, the statement “Louis murders children” is taken as true even if Louis has only murdered one child once in his life, while “Louis reads the New York Times” is false if Louis has only read the Times once in his life. What do these peculiarities say about our claim to knowledge about objects and persons? The fact that knowledge claims are oftentimes expressed in a generic form should raise eyebrows upon this discovery of the inconsistent truth conditions of these statements.

“Donald Duck lays eggs” is a false statement. (Source: ClipartFest)

These three puzzles are not comprehensive by any means, but they illustrate logical difficulties in our everyday use of language. Importantly for the purpose of ToK, perhaps the awareness of these unresolved issues in the philosophy of language forces us to more critically examine the ways we use words to make claims about the world. What can we say about the status of personal and shared knowledge when most of what we know come from and are indeed expressed in language full of unresolved ambiguities? One could also question whether the application of formal logic onto everyday speech is the right move to make – perhaps the ways we use language on a daily basis are governed by a logic that is not fully captured by the strict, quantitative rationality favored by logicians. In any case, these questions provide food for thought for any inquiring ToK student.