From 'the' to 'every'

One of the bizarre consequences of Russell’s theory of definite descriptions is that the following argument is perfectly valid:

The monkey in the Bronx Zoo pulls funny faces.
Therefore, every monkey in the Bronx Zoo pulls funny faces.

At first glance, this cannot be right! An average user of the English language reasons like this: I’m thinking of a particular monkey in the Bronx Zoo. It simply doesn’t follow that just because that monkey pulls funny faces, every monkey in the Bronx Zoo does. But the preposterousness is only superficial.

The question, of course, is: how many monkeys are there in the Bronx Zoo? If there is just one, then the conclusion follows trivially. For if there is just one monkey in the Bronx Zoo, then that monkey is every monkey.

And this is how Russell would have liked us to interpret ‘the’. First, that there is a monkey in the Bronx Zoo at all. Second, that all monkeys in the Bronx Zoo are identical to that particular monkey (or, if there are any, then they are identical to it). Finally, that the monkey in question pulls funny faces. The second condition makes the monkey unique, because, in conjunction with the first condition, it simply says that there is at least one monkey and at most one monkey in the Bronx Zoo, that is, that there is exactly one monkey in the Bronx Zoo.

As a thought experiment, the truth of the premise in our argument is conceivable. As a reflection of reality, the situation is rather unlikely. There is bound to be more than one monkey in the Bronx Zoo. A more accurate argument would probably run like this:

A monkey in the Bronx Zoo pulls funny faces.
Therefore, every monkey in the Bronx Zoo pulls funny faces.

Except, of course, that this argument is invalid. The premise says that there is at least one monkey but doesn’t set an upper limit. If it turned out indeed that there was only one monkey, then, yes, the conclusion would follow, but this would take us back to the first example. Our set would have shrunk to a one member set again.

Russellian analysis soon gives the game away. Suppose there is more than one monkey in the Bronx Zoo, but we are thinking of one particular one, the one that habitually twirls its tail. The original argument doesn’t work:

The monkey in the Bronx Zoo that twirls its tail pulls funny faces.
Therefore, every monkey in the Bronx Zoo pulls funny faces.

Here, the sets are not identical; the set in the premise is a subset of the set in the conclusion. For the argument to be valid, we would have to infer:

Therefore, every monkey in the Bronx Zoo that twirls its tail pulls funny faces.

And so on and so forth. Somewhere down the line in conversational English, a sufficient number of distinctive features would have been enumerated to satisfy ourselves that ‘the’ merges with ‘every’, for there would be no other monkey like it to fit the description.

There is another way in which ‘the’ can mean ‘every’, the generic ‘the’, which is not to be confused with a definite description.

The Logic Book, M.Bergmann, J.Moor, J.Nelson, McGraw Hill, 4th edition, problem 10.6E 3b

We are asked to show that the following is a theorem in PDE, the extended predicate derivation system.

(x)(y)(x = x ∨y = y)



I proceed by assuming the negation of the formula, marking the scope of the assumption as I go along. Lines 7 and 8 show contradictions, so the assumption is discharged and shown not to be true (double negation), which reduces to the theorem we were seeking to prove.

1. * ¬ (x)(y)(x = x ∨y = y) / AIP
2. * (∃x) ¬ (y)(x = x ∨y = y) / 1, CQ
3. * (∃x)(∃y) ¬ (x = x ∨y = y) / 2, CQ
4. * (∃y) ¬ (a = a ∨y = y) / 3, EI x/a
5. * ¬ (a = a ∨b = b) / 4, EI y/b
6. * ¬ (a = a) • ¬ (b = b) / 5, DeM
7. * ¬ (a = a) / 6, Simp.
8. * ¬ (b = b) / 7, Simp.
9. ¬ ¬ (x)(y)(x = x ∨y = y) / 1- 7(8) IP
10. (x)(y)(x = x ∨y = y) / 9, DN

Conversion

A quip seen in a film review in last week’s Spectator magazine has given rise to this short logical analysis. The set-up line is:

(1) There is no money in poetry.

The message is that you can’t make money out of poetry, of course. The sentence is true for most people, which itself makes it contingently true, not necessarily true. But whether it is true or false, the pay-off has the same truth value as the original:

(2) There is no poetry in money.

Logic is not too fond of uncountable terms (‘poetry’, ‘money’), but the sentences can be made to conform broadly to the E-type proposition: No P are Q. It follows that if no P are Q, then no Q are P. P and Q are disjoint sets. Yet, despite the sentences having the same truth value, their respective messages seem to be different. Put another way, natural language implies more than can be inferred by logic alone.

There is a well-established distinction (P. Grice, 1967) between logical reasoning and pragmatic reasoning, whereby the latter resorts to conversational conventions, practical considerations, context, and so on. Where in logic (1) says something about money and (2) something about poetry, there is a strong sense that in conversational English both sentences say something about money.

This is reinforced by the only other type of proposition that retains its truth value upon conversion – the I-proposition: There is some money in poetry is equivalent to There is some poetry in money. Again, both sentences seem to be saying something about money.

In propositional logic, the converse of a conditional is obtained by changing around antecedent and consequent. A conditional does not imply its converse:

(3) If you write poetry, you will not make money.
(4) If you haven’t made money, you will have written poetry.

For (3) to be false, we need to find someone who has made money from writing poetry. This combination though makes (4) true, because a conditional is false if and only if the consequent is false while the antecedent is true.

The use of adverbs such as ‘conversely’, ‘on the contrary’, ‘quite the reverse’ and a few others in everyday English betrays only a distant relation to their strict meaning in logic. My Longman Dictionary gives the following example: American consumers prefer white eggs; conversely, the British buyers like brown eggs. Where is the converse relation here? What comes to mind for this example is expressions such as: ‘in contrast’, ‘while’, ‘whereas’. However, it is no use being too dogmatic about it. I understand the sentence the way it was intended, and so do millions.

Deduction, D.Bonevac, Blackwell, 2nd edition, 2003; 8.3 problem 11

The task is to show that the given sequence (line 2) is a consequence of a formula. The formula is a translation of the English sentence: There is one and only one God. The conclusion (the given sequence) reads: Something that is a God has property 'F' if and only if anything that is a God has property 'F'. Perhaps the most valuable hints here are that: a) the universal quantifier on line 6 can be instantiated again, to a constant (here 'a') on line 11, and b) that the universal quantifier can be instantiated to the same constant (here 'a' on line 24 and 25) as the existential quantifier. Then it is just a matter of assuming a = a, without any justification. There is no reason to worry about using the same constant again after line 22 because all previous assumptions have been discharged.

1. (∃x)(y)(y = x ≡ Gy)
2. ∴ (∃x)(Gx • Fx) ≡ (x)(Gx ⊃ Fx)
3. * (∃x)(Gx • Fx) / ACP
4. ** Gx / ACP
5. ** Ga • Fa / 3EI x/a
6. ** (y)(y = m ≡ Gy) / 1EI x/m
7. ** x = m ≡ Gx / 6UI / y/x
8. ** (x = m ⊃ Gx) • (Gx ⊃ x = m) / 7BE
9. ** Gx ⊃ x = m / 8Simp
10. ** x = m / 4,9MP
11. ** a = m ≡ Ga / 6UI y/a
12. ** (a = m ⊃ Ga) • (Ga ⊃ a = m) / 11BE
13. ** Ga ⊃ a = m / 12Simp
14. ** Ga / 5Simp
15. ** a = m / 14,13MP
16. ** m = a / 15Id
17. ** x = a / 10,16Id
18. ** Fa / 5Simp
19. ** Fx / 17,18Id
20. * Gx ⊃Fx / 4 - 19CP
21. * (x)(Gx ⊃Fx) / 20UG
22. (∃x)(Gx • Fx) ⊃(x)(Gx ⊃Fx) / 3 - 21CP
23. * (x)(Gx ⊃Fx) / ACP
24. * (y)(y = a ≡ Gy) / 1EI x/a
25. * a = a ≡ Ga / 24UI y/a
26. * (a = a ⊃Ga) • (Ga ⊃a = a) / 25BE
27. * a = a ⊃Ga / 26Simp
28. * a = a / Id
29. * Ga / 27,28MP
30. * Ga ⊃Fa / 23UI x/a
31. * Fa / 29,30MP
32. * Ga • Fa / 30,31Conj.
33. * (∃x)(Gx • Fx) / 32EG
34. (x)(Gx ⊃Fx) ⊃(∃x)(Gx • Fx) / 23 - 33CP
35. [(∃x)(Gx • Fx) ⊃(x)(Gx ⊃Fx)] • [(x)(Gx ⊃Fx) ⊃(∃x)(Gx • Fx)] / 22,34Conj.
36. (∃x)(Gx • Fx) ≡ (x)(Gx ⊃ Fx) / 35BE

Witches fly on brooms - a truth

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.

St Anselm's ontological argument

In Appendix 4 of Meaning and Argument (Blackwell, 2003), Ernest Lepore sets the task of symbolizing in FOL the following argument:

The perfect being has all positive perfections. Existence is a perfection. So, the perfect being has existence.

This is of course Anselm of Canterbury's ontological argument for the existence of God. The argument has received so much attention and criticism since it was first formulated in the 11th century that no purpose would be served to rehearse any of that here. However, I have only ever seen it proved within the scope of modal logic, so here is my symbolization in first order logic and the subsequent proof. The proof is quite straightforward.

Key:

Px - x is fully perfect ('has all positive perfections')

g - the perfect being

1. (x)(Px ⊃x = g)
2. (∃x)Px
3. ∴(∃x) x = g
4. Pa / 2EI
5. Pa ⊃a = g / 1UI
6. a = g / 4,5MP
7. (∃x) x = g / 6EG