Clear Language, Clear Mind

Examples:

1. I cannot find my socks. / I couldn’t find my socks.

2. I cannot find a counter-example to the theory. / I couldn’t find a counter-example to the theory.

The word “cannot” is usually taken to express impossibility and be somewhat equivalent in meaning with the word “impossible”, thought they cannot be used in the same way because they are different word types.

However, if we rewrite the above sentences to use the other word, then the meaning changes. Consider:

1a. It is impossible to find my socks. / It was impossible for me to find my socks.

2a. It is impossible to find a counter-example to the theory. / It was impossible to find a counter-example to the theory.

Why does (1) and (1a) not mean the same? It is because, I think, that we use “cannot” to express a different kind of lack of possibility than we use with “impossible”. This possibility I call willful possibility. It is defined like this: It is willfully possible for agent S to A iff that S wills to bring about A materially implies that A would happen.

This is applicable to the above examples like this:

1b. S wanted to find his socks but did not. Thus, it was not willfully possible to for S to find his socks.

2b. S wanted to find a counter-example to the theory but did not. Thus, it was not willfully possible to for S to find a counter-example.

The meanings of (1) and (2) are also expressible in another way:

1c. I tried to find my socks but I failed.

2c. I tried to find a counter-example to the theory but I failed.

Though there are also some relevant considerations about time. Willful possibilities change from time to time. Suppose for instance that at some later time were S to look for his socks, then he would find them. Similarly with counter-examples to the theory. I did not find a way to easily incorporate this into the definition above.

One should not confuse willful possibility with logical, physical or some other kind of possibility. It would be odd to interpret (1) and (2) as logical etc. possibility, but it would not be odd to interpret (1a) and (2a) that way. Also the sentences in (1a) are odd and would probably not be used very often. However (2a) are not particularly odd since that if some theory is true, then there are no counter-examples to that theory. And since one cannot (willfully) find a counter-example that does not exist, then it is (willfully) impossible to find a counter-example to such a theory.

When interpreting sentences like (1) and (2) we should not be misled to interpret them as something along the lines of (1a) and (2a) but should (probably) interpret them as something like (1c) and (2c).

Joyce does a rather strange interpretation in The Myth of Morality p. 121. He writes:

However, I doubt we even need concede that much. These “conditional reasons” are very shady customers. Take what seems to be a straightforward one mentioned above: one’s reason to save a drowning child if one exists. There are two readings:

(i) If there exists a drowning child, then S has a reason to save him/her.

(ii) S has a reason to save a drowning child if one exists.”

The absence of a comma after “child” in (ii) makes the difference. (ii) is saying that S has a reason all along: when there are no drowning children, when S is asleep, while S is witching TV, etc.”

Normally conditional sentences can be written in two ways in english (and other languages that I am familiar with), forwards and backwards. Forwards being what is similar to their logical structure and backwards what is not similar. Take the conditional sentence “If I don’t have a job, then I will not get money”. It is forwards for it is similar to its logical structure P→Q (with the obvious interpretation keys). The sentence “I will not get money if I don’t have a job” seems to express exactly the same conditional (i.e. proposition), but Joyce apparently thinks that it does not (if he accepted an analogy with his own example).

When I read the passage above I spent some time thinking about how to properly formalize his two interpretations. I came up with this:

I. (∃x)(Dx)→(∃y)(Ryx)

That there exists an x such that x is a drowning child materially implies that there exists an y such that y is a reason for S to save x.

II. (∃y)∧(∀x)(Dx)→(Ryx)

There exists an y and for all all x, that x is a drowning child materially implies that y is a reason for S to save x.

It seems to capture what he meant.

How Joyce made up these interpretations I don’t know. I note that (I) does not imply that (if there is no drowning child, then S does not have a reason to save one) [¬(∃x)(Dx)→¬(∃y)(Ryx)].

Joyce goes on to make distinction in a strange way:

“This observation is entirely generalizable, to the conclusion that there are not really any “conditional reasons.” Anything true of the form “S has a reason to Ø if C obtains” should be read as “If C obtains, then S has a reason to Ø,” not “S has a reason to Ø-if-C-obtains.”

Joyce would have benefited from using logic here for clarification instead of this. It’s not the case that Joyce wanted to completely avoid using logical symbols in his book anyway, for just two pages earlier we find some simplistic predicate logic formalizations of sentences. (Assuming that he would not have them there if he did not want logical symbols in the book.)

Sometimes it is clear that “always” should be interpreted as various temporal logics suggest. Other times it should not be interpreted as anything that has to do with time.

Consider this fictive conversation:

“Generally women prefer men with high social above men without.”

“Not always.”

But the first utterance has nothing to do with time. When “not always” is uttered in such contexts it means the same as the phrase “not in all cases [¬(∀x)(Fx)/(∃x)(¬Fx)]”. Similarly with other temporal words such as “often” (in many cases) and “never” (¬(∃x)(Fx)/(∀x)(¬Fx)).