Figure 1: Hypotheses Consistent with Observations
There are two main definitions of what a hypothesis is:
Additional meanings, connotations, and synonyms will be discussed in section 5.
Oftentimes, a scientific investigation can be viewed as a process of winnowing. That is, first we construct a set of hypotheses, i.e. tentative explanations, then we delete from the set any hypotheses that do not fit the facts. This is illustrated in figure 1. The black rectangle represents all the hypotheses that are consistent with the first observation; the red rectangle represents all the hypotheses that are consistent with the second observation, and the blue rectangle represents all the hypotheses that are consistent with the third observation. The middle region, where all three rectangles overlap, represents the hypotheses that are consistent with all three observations.
As a specific example of this, in the famous “Twelve Coins” puzzle, we start with 24 hypotheses: the first coin could be heavy (with all other coins normal), the first coin could be light (with all other coins normal), and so on for the remaining coins. At the beginning of the investigation, before any weighings have been performed, all 24 of these hypotheses are in the set of contenders. After the first weighing, we can rule out most of these, leaving us with a smaller number of contenders, typically 8 contenders. After the second weighing, there are typically 3 contenders, and after the third weighing there should be only a single contender, which we annoint as “the” answer, i.e. “the” correct hypothesis.
Note that when we first construct the set of hypotheses, there is no expectation that every hypothesis is correct; indeed we expect that eventually N−1 of the hypotheses will be shown to be incorrect. At the other extreme, it is pointless (but usually harmless) to include hypotheses that have no chance of being correct. A good guideline is to include all the hypotheses that have an appreciable (i.e. more-than-infinitesimal) probability of turning out to be correct. These are called the plausible hypotheses.
You may find Bongard Problems (reference 1) to be good examples and exercises in hypothesis testing.
Let us define a hypothetical scenario as follows: it consists of a group of logical statements called the body of the scenario, plus another statement called the hypothesis of the scenario. We then establish the rule, as a notational convention, that every for statement s in the body of the scenario, if we see it written in the form “s”, we interpret it as meaning “if H then s”, where H is the hypothesis of the scenario.
That’s really all there is to it.
Remark: This is just a notational convention. Any calculation performed using a hypothetical scenario could equally well be performed without it, simply by replacing each statement “s” in the body of the scenario by the conditional statement “if H then s”. The two versions would be equally rigorous. The only difference is that one would be more verbose.
This is analogous to the distributive law of ordinary arithmetic, which tells us that the following two expressions are equivalent:
| (1) |
Similarly, the definition of hypothetical scenario tells us that the following two statements are equivalent:
| (2) |
In particular, if you want to take a result that was calculated within
a scenario and use it outside the scenario, you must explicitly make
it conditional on the hypothesis.
Hint: It is super-important to distinguish what’s inside the body of the scenario and what’s outside. Lots of things that are valid inside the body would not be valid outside. This applies most spectacularly to the hypothesis H itself. Inside the scenario, you may freely assert that H is true ... but that does not mean that H is true in any absolute sense. It is only conditionally true. Like everything else inside the scenario, it is conditional on H.
Suppose you want to take H, which is valid inside the scenario, and move it outside the scenario. If you follow the rules, you end up with the statement “if H then H”, which is a tautology. It is always unconditionally true, whether H itself is true or not.
Remark: As defined here, there is absolutely no requirement that the hypothesis be known in advance to be true or not. It might be known true, known false, or (much more likely) simply unknown, or dependent on unspecified details.
Example: A high-school geometry book can be considered one long hypothetical scenario. The hypothesis is a statement of the axioms of Euclidean geometry. Therefore, when the book asserts the Pythagorean formula a2 + b2 = c2, you must not take that to mean that a2 + b2 = c2 always, but rather it means that a2 + b2 = c2 in a Euclidean space!
As it turns out, there are lots of non-Euclidean spaces where the Pythagorean formula is not valid.
A famous and important application of hypotheses goes by the name proof by contradiction, also known as reductio ad absurdum. This is simply a hypothetical scenario, the point of which is to prove that the hypothesis is false.
As an example, let us build a scenario. We choose as our hypothesis H the assertion that √2 is a rational number. Now you may already know that H is false, but the point of the exercise is to prove this to someone who does not already know it. There is a difference between truth and knowledge, as discussed in reference 2.
On the other side of the same coin, it would be nonsense to assume H is true in the absolute sense, because it turns out that H is not true, i.e. √2 is irrational. If you make false assumptions on line 1 of your calculation, all the rest of the calculation is nonsense.
So we will not assume H. Neither will we assume not-H. We will merely hypothesize H. That is, we will begin a hypothetical scenario, such that every statement in the body of the scenario is conditional on H.
After a few lines of work (which will not be shown here), we come to a contradiction. That is, we derive a statement of the form “q is divisible by 2 and q is not divisible by 2”. This is of the form “X and not-X” and can be further simplified to simply “false”, according to the usual rules of Boolean logic.
Now if we want to use this result outside of the scenario, we must make it explicitly conditional on the hypothesis, so we write “if √2 is rational, then false”.
Next, we need to use some basic facts of logic, which tell us that all the following statements are equivalent:
| (3) |
In particular, the statement “not-B ⇒ not-A” is called the contrapositive of the statement “A ⇒ B”.
In our case, the contrapositive of the statement “if √2 is rational, then false” is the statement “if true, then √2 is not rational” ... or more simply, “√2 is irrational”, which is what we wanted to prove.
This is quite a famous result. It is attributed to Hippasus of Metapontum, about 2500 years ago.
Let us compare and contrast the definitions given in section 2 and section 3.
| They are certainly not equivalent. For example, for hypothesis testing, it is pointless to include hypotheses that cannot possibly be true, while for hypothetical scenarios, proof by contradiction makes very good use of a non-true hypothesis. | They are not wildly inconsistent. For example, for each hypothesis in the tentative solution-set, we could use that hypothesis as the basis of a hypothetical scenario. |
There are various names that can be used for the statements governed by a conditional statement. In particular, for the statement “if H then C” we have:
| H could be called | C could be called |
| antecedent | consequent |
| premise | conclusion |
| proviso | |
| condition | |
| hypothesis |
The meaning of “hypothesis” on the last line of the table is entirely consistent with the meaning in section 3, and simply represents the minimal case where there is only a single statement in the body of the scenario. On the other edge of the same sword, the hypothesis of a scenario may equivalently be called the premise of the scenario or the condition of the scenario.
A conjecture is like a hypothesis, perhaps with the additional connotation that you expect it to be true, or you hope it will be true.
A assumption is assumed to be true. It is fairly common, but sloppy and not recommended, to use the word “assumption” to refer to the hypothesis of a scenario. This is particularly not recommended in the case of proof by contradiction, where the hypothesis is absolutely not an assumption, because it is not true.
Among non-scientists, there is a widespread misconception that hypothesis testing is “the” scientific method. This is quite untrue, and paints a very unhelpful of what science is and what scientists do. In fact scientists use a wide variety of methods, as explained in reference 3.