[M07] Types of possibility

The concepts of necessary and sufficient conditions relate to the concept of possibility. To say that X is necessary for Y is to say that it is not possible for Y to occur without X. To say that X is sufficient for Y is to say that it is not possible for X to occur without Y. There are, however, different senses of "possibility", and corresponding to these different meanings there are different kinds of necessary and sufficient conditions.

Consider these statements :

• It is impossible to be a tall man without being tall.
• It is impossible to dissolve gold in pure water.
• It is impossible to travel from Hong Kong to New York in less than ten minutes.
• It is impossible to visit the army barracks without a permit.

The word "impossible" has different meanings in each of these statements. In the first statement, what is being referred to is logical impossibility. Something is logically impossible if it is contradictory, or against the laws of logic. Thus a round square is a logical impossibility, and it is logically impossible to be a tall man without being tall.

But it is not logically impossible to dissolve gold in water. The laws of logic do not tell us that this cannot happen. Rather, the impossibility is due to the laws of physics and chemistry which happen to hold in our universe. If our universe had contained different laws, then perhaps it is possible to dissolve gold in water. Dissolving gold in water is not logically impossible but empirically impossible. Sometimes this is also known as causal or nomologically impossibility.

The sense in which the third statement is true is again different. The laws of physics probably do not prohibit us from travelling from Hong Kong to New York under ten minutes. What is true is that we have no means to achieve this using current technology. Such a trip is therefore technologically impossible, even though it is both logically and empirically possible. Of course, if the technological obstacles can be overcome then such a trip will then become possible.

Finally, visiting the army barracks without a permit is logically, empirically and technologically possible. After all, one might be able to dig a tunnel to enter the barracks without permission. The sense in which entering without a permit is impossible is in the legal sense. What is meant is that it is illegal or against the relevant regulations to enter the barracks without a permit. Here we are talking about legal impossibility.

§1. Different types of necessity and sufficiency

Corresponding to these different notions of possiblity we have different concepts of necessary and sufficient conditions. For example :

• Having four sides is logically necessary for being a square.
• Being a father is logically sufficient for being a parent.
• The presence of oxygen is causally necessary for the proper functioning of the brain.
• Passing current through a resistor is causally sufficient for the generation of heat.
• Being an adult of over 18 years old is legally sufficient for having the right to vote.
• The presence of a witness is legally necessary for a valid marriage.

Note that there might be other types of necessity and possibility as well. For example, a father might advise his son that he must treat his sister well. The sense of "must" here is not legal necessity, as the law does not require us to treat our siblings well. Rather, the sense of necessity might have to do with moral conduct. On the other hand, if the son is told that he must treat his boss well, what is meant might be that being nice to one's boss is required not so much by morality but by rules of prudence.