November. Witches. Goes without saying. But is the sentence:
(1) Witches fly on brooms.
which is a variant of: All witches are creatures that fly on brooms, true then? YES. Why is it true? Because witches do not exist. A universal statement about anything that does not exist is trivially true. Here are a number of reasons.
Consider the negation of (1). We ought to be able to say any of the following:
(2) Some witches do not fly on brooms.
(3) Some witch does not fly on a broom.
(4) There is a witch that does not fly on a broom.
(5) There is at least one witch that does not fly on a broom.
(6) There are witches that do not fly on brooms.
All of these are complete negations of (1), as is the statement: It is not the case that all witches fly on brooms. (The sentence: No witches fly on brooms, is not, by the way, a complete negation of (1), just as No sheep are black is not a complete negation of All sheep are black. Both sentences can be false, and are.) What of it? Well, if (1) was false, then each of (2) – (6) would have to be true, for the simple reason that given a sentence and its negation, both cannot be true (or false). This is a fundamental law of logic – in a contradictory pair of sentences, one sentence is false, one is true.
Now, sentences (2) – (6) say a rather peculiar thing. On closer inspection they say that witches exist – it is just that they don’t fly on brooms. This is most evident in sentences (4) – (6), but can also be intuited from (2) and (3).
Witches do not exist, of course, so (2) – (6) are all false. This makes sentence (1) true. Put another way, try as you might you will not find any falsifying instances of (1).
This is the modern approach. The old Aristotelian approach was ill-equipped to deal with sentences which talked of non-existing entities. It would have classified (1) as false, which would please those who profess common sense above all, but it would make logic unworkable. This is because it would have made: Some witches fly on brooms and Some witches do not fly on brooms both false, since both assert that witches exist. In so doing it would have undone itself, because the statements: Some S are P, and Some S are not P, in a universe populated with entities of the S kind, can both be true, but not both false. Alternatively, if we accepted that Some witches fly on brooms is true, it would be an offence to reason, coming from the falsity of All witches fly on brooms.
The modern Boolean approach simply abolishes any implication of existence going from: All witches fly on brooms to Some witches fly on brooms, but preserves the relationship of contradiction: All S are P as against Some S are not P.
Another reason for (1) being true is this. Consider this dialogue:
A: Is it your aunt that has just taken off on a broom?
B: All witches fly on brooms, you know.
What do you make of B’s answer to A’s question? Is it a lie? It certainly is not. It can at worst be viewed as evasive, but not a lie. Clearly, to say that all witches fly on brooms is not a lie because there are no witches.
If witches existed, then establishing the truth of (1) would be a matter of checking whether every single one of them did fly on brooms. If they did, the sentence would be true. If at least one didn’t fly on brooms, then (1) would be false, and sentences (2) – (6) would be true.
Again, since we know that witches don’t exist, it is far more palatable to affirm the truth of an assertion that if there are any witches at all, then they fly on brooms, than to affirm that there are witches that do not fly on brooms.
(1) Witches fly on brooms.
which is a variant of: All witches are creatures that fly on brooms, true then? YES. Why is it true? Because witches do not exist. A universal statement about anything that does not exist is trivially true. Here are a number of reasons.
Consider the negation of (1). We ought to be able to say any of the following:
(2) Some witches do not fly on brooms.
(3) Some witch does not fly on a broom.
(4) There is a witch that does not fly on a broom.
(5) There is at least one witch that does not fly on a broom.
(6) There are witches that do not fly on brooms.
All of these are complete negations of (1), as is the statement: It is not the case that all witches fly on brooms. (The sentence: No witches fly on brooms, is not, by the way, a complete negation of (1), just as No sheep are black is not a complete negation of All sheep are black. Both sentences can be false, and are.) What of it? Well, if (1) was false, then each of (2) – (6) would have to be true, for the simple reason that given a sentence and its negation, both cannot be true (or false). This is a fundamental law of logic – in a contradictory pair of sentences, one sentence is false, one is true.
Now, sentences (2) – (6) say a rather peculiar thing. On closer inspection they say that witches exist – it is just that they don’t fly on brooms. This is most evident in sentences (4) – (6), but can also be intuited from (2) and (3).
Witches do not exist, of course, so (2) – (6) are all false. This makes sentence (1) true. Put another way, try as you might you will not find any falsifying instances of (1).
This is the modern approach. The old Aristotelian approach was ill-equipped to deal with sentences which talked of non-existing entities. It would have classified (1) as false, which would please those who profess common sense above all, but it would make logic unworkable. This is because it would have made: Some witches fly on brooms and Some witches do not fly on brooms both false, since both assert that witches exist. In so doing it would have undone itself, because the statements: Some S are P, and Some S are not P, in a universe populated with entities of the S kind, can both be true, but not both false. Alternatively, if we accepted that Some witches fly on brooms is true, it would be an offence to reason, coming from the falsity of All witches fly on brooms.
The modern Boolean approach simply abolishes any implication of existence going from: All witches fly on brooms to Some witches fly on brooms, but preserves the relationship of contradiction: All S are P as against Some S are not P.
Another reason for (1) being true is this. Consider this dialogue:
A: Is it your aunt that has just taken off on a broom?
B: All witches fly on brooms, you know.
What do you make of B’s answer to A’s question? Is it a lie? It certainly is not. It can at worst be viewed as evasive, but not a lie. Clearly, to say that all witches fly on brooms is not a lie because there are no witches.
If witches existed, then establishing the truth of (1) would be a matter of checking whether every single one of them did fly on brooms. If they did, the sentence would be true. If at least one didn’t fly on brooms, then (1) would be false, and sentences (2) – (6) would be true.
Again, since we know that witches don’t exist, it is far more palatable to affirm the truth of an assertion that if there are any witches at all, then they fly on brooms, than to affirm that there are witches that do not fly on brooms.
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