13 Analytic vs. Synthetic

Let us now appreciate that the three propositions we’ve discussed belong to two different categories. We say that the proposition “one plus two equals three” is an analytic proposition, while the law of gravity and “all swans are white” are synthetic propositions. Analytic and synthetic propositions are two distinct types of propositions.

Now, what makes a proposition analytic? Analytic propositions are either definitions or deducible from definitions. Since analytic propositions are true by definition, they can never contradict the results of experiments and observations. For instance, the proposition “1+2=3” will hold at all times and places regardless of what we happen to observe.

Consider another typical analytic proposition: All bachelors are unmarried. This proposition is true by definition, since “bachelor” is defined as someone unmarried. If someone were to claim to have observed a married bachelor, we would probably conclude that she uses the word “bachelor” differently, for it is impossible to observe a married bachelor, insofar as we stick to our common definition of “bachelor”. To express the same idea differently: an analytic proposition necessarily holds in all possible worlds, i.e. its opposite is inconceivable. Thus, the proposition expressed as “1+2=3” is true in all possible worlds, in the sense that it’s impossible to conceive of a world where this is not the case. Of course, we can always decide to change the definitions of our concepts and, for example, agree that “2” stands for “one, another one, and another one”. But that would effectively change the meaning of “1+2=3”; it would no longer be the same proposition. So as long as “2”, “3”, “+”, and “=” are defined the way they are commonly defined, the proposition “1+2=3” holds, come what may.

This is clearly not the case for synthetic propositions. Synthetic propositions are ones that are not deducible merely from the definitions of their concepts. Because they are not deducible from the definitions of their concepts alone, they can potentially contradict the results of observations and experiments. This is the same as to say that synthetic propositions do not hold in all possible worlds, for their opposites are conceivable/imaginable. Thus, we can easily imagine a world where at least some swans are not white, or a world where the law of gravity is different.

Here are a few additional examples of analytic propositions:

• a (b + c) = ab + ac
• A centaur is a creature that is part human and part horse.
• The sum of the squares of the lengths of the sides of a triangle is equal to the square of the length of its hypotenuse: a2+ b2 = c2.

All of the above propositions are such that their opposites are simply inconceivable, since they are either definitions or logically follow from definitions. And here are some examples of synthetic propositions:

• In combustion, methane combines with oxygen to form carbon dioxide and water: CH4+ 2O2 → CO2 + 2H2O.
• The game of football is played in four 15-minute-long quarters.
• Paris is the capital of France.
• There are no centaurs on planet Earth.

All these propositions are attempting to say something about the world we happen to inhabit and, consequently, they may or may not hold in other possible worlds. We can easily conceive of worlds where these propositions are false. Thus, we can imagine a strange world full of centaurs, or a bizarre world where Paris is the capital of England. We can also conceive of an absolutely ridiculous world where the game of American Football is played in two 45-minute halves.

This brings us to a useful rule of thumb for distinguishing analytic from synthetic propositions: if the opposite of a proposition is conceivable, i.e. if it doesn’t lead to logical contradictions, then the proposition is synthetic. Conversely, if the opposite of a proposition is inconceivable, i.e. if it contains logical contradictions such as the idea of a married bachelor, then the proposition is analytic.