No stranger language

On a light note before the festive season: a sentence heard on a travel report update for the UK the other day has left me wondering about the threshold of entry into English by anyone who takes things too literally. It must be set very high indeed. A member of a gritting crew interviewed for the programme said:

We are putting down salt to help ensure the roads stay free of snow and ice.

Is there a language in the world, I wonder, where this translates word for word? How much of what is in this sentence is eliminable or perversely complicated? Starting from left to right one could proceed to obtain:

We are spreading salt to help ensure the roads stay free of snow and ice.
We are spreading salt to ensure the roads stay free of snow and ice.
We are spreading salt so that the roads stay free of snow and ice.
We are spreading salt so that there is no snow and ice on the roads.

Why do we need four verbs in the original sentence, on a conservative count, where there are, by any measure of reality, only two activities: ‘gritting’ and ‘being’ (or more specifically ‘there being not’)? The English love their verbs and like to confuse the foreigners at every turn.

And why is an object lacking a quality said to be ‘staying’, the which verb is then modified to suit the purpose of the speaker (‘free’), rather than saying that that particular quality is absent from the object (‘no snow on the roads’)? Not to mention that ‘putting down salt’ is a funny way of saying that the roads are being covered by it.

Symbolic Logic, Dale Jacquette, Wadsworth, 2001, Chapter 8, Ex. III, problem 8

We are to show that the following formula is a tautology. A useful hint here is that before we instantiate line 2 by universal instantiation, we should first instantiate our assumption by egistential instantiation on line 3, into the constant 'a', or any other. This ensures that the procedure is valid. We assume on line 1 a non-negated disjunct in anticipation of the fact that once we discharge the assumption we will be easily able to turn a conditional into a disjunct by conditional exchange (or material implication) on the last line of the proof.

├ ¬ (x)(Fx ≡ Gx) ∨[(∃x)Fx ≡ (∃x)Gx]

1. * (x)(Fx ≡ Gx) / ACP
2. * (x)[(Fx ⊃Gx) • (Gx ⊃Fx)] / 1BE
3. * * (∃x)Fx / ACP
4. * * Fa / 3EI
5. * * (Fa ⊃Ga) • (Ga ⊃Fa) / 2UI
6. * * Fa ⊃Ga / 5Simp.
7. * * Ga / 4,6MP
8. * * (∃x)Gx / 7EG
9. * (∃x)Fx ⊃(∃x)Gx / 3-8CP
10. * * (∃x)Gx / ACP
11. * * Gm / 10EI
12. * * (Fm ⊃Gm) • (Gm ⊃Fm) / 2UI
13. * * Gm ⊃Fm / 12Simp.
14. * * Fm / 11,13MP
15. * * (∃x)Fx / 14EG
16. * (∃x)Gx ⊃(∃x)Fx / 10-15CP
17. * [(∃x)Fx ⊃(∃x)Gx] • [(∃x)Gx ⊃(∃x)Fx] / 9,16Conj.
18. * (∃x)Fx ≡ (∃x)Gx / 17BE
19. (x)(Fx ≡ Gx) ⊃[(∃x)Fx ≡ (∃x)Gx ] / 1-18CP
20. ¬ (x)(Fx ≡ Gx) ∨[(∃x)Fx ≡ (∃x)Gx] / 19CE

English and quantifiers

Take the pair of sentences:

(1) If a gentleman is in the room, Becky turns on her charm.
(2) If every gentleman is in the room, Becky turns on her charm.

Most English speakers would probably see enough clear water between these two sentences to consider them distinct. Most people would also agree that:

(3) If any gentleman is in the room, Becky turns on her charm.

is more like (1) than (2). But ‘any’ behaves rather differently if we take the following sentences:

(4) Becky will flirt with any gentleman who is worth five thousand pounds per year.
(5) Becky will flirt with every gentleman who is worth five thousand pounds per year.

Here, unmistakably, ‘any’ is more like ‘every’. The subtle differences between (4) and (5) may involve perhaps Becky flirting with one gentleman of such means at a time (4), as opposed to all of them at once (5), or as many of them as come within her grasp at any given time as opposed to, again, all who have that sort of income. Quantificationally, (4) and (5) are equivalent.

More interestingly, in:

(6) Becky will flirt with a gentleman who is worth five thousand pounds per year.

the article ‘a’ has the force of ‘every’ as well. Alternatively, we could say that ‘a’ is like ‘any’, but not the ‘any’ of (3) but the ‘any’ of (4). If, in (6), we were to capture the sense of ‘a’ in (1), we would have to say:

(7) There is a gentleman worth five thousand pounds per year with whom Becky will flirt.

The conclusion which follows from these observations is that quantification in English is only nominally assigned to certain words. Both ‘a’ and ‘any’ can mean ‘one’ and ‘many’ – a rhyme which should make it easy to remember. The quantification of ‘the’ can also shift between ‘one’ and ‘many’, as in:

(8) The tiger caught an antelope.
(9) The tiger is a fierce animal.

Compare the respective paraphrases:

(10) There is a tiger such that if any tiger is identical to it then it caught an antelope. (thus there is only one such tiger)
(11) Any tiger is fierce.

Trying to impose any order on quantifiers in English is like herding cats, or tigers. Logic has only two quantifiers: universal and existential, and all we are left with is trying to fit the material to the tools.

Strong and weak readings

If pressed to explain my way out of a tight corner in English, I find that the distinction between the strong and weak reading of some English structures is often a very handy tool. Once in that mode, I further discover that its application can be far wider than is usually suggested in logic courses.

The textbook example is that of a disjunction in propositional logic. The inclusive sense of ‘or’ traditionally gets a weak reading; the exclusive sense of ‘or’ – a strong reading. The standard reading in the inference mechanism is that of the inclusive ‘or’. Reasoning with the exclusive ‘or’ leads to fallacies.

p or q
p
Therefore, not q.

If the argument is: Either the coin comes up heads or it comes up tails. It has come up heads. Therefore, it hasn’t come up tails; the reasoning works. But if the argument is: Either a cat or a fox has been getting at the chickens. A cat has been getting at the chickens. Therefore, a fox hasn't; we cannot be absolutely sure. Exclusive disjunction is non-validating.

Another example is the choice of quantifier in predicate logic. If we choose to turn the sentence: If something is good, it is forbidden, into the language of FOL, we get:

(x)(x is good ⊃ x is forbidden)

and the reading is that if ‘anything’ is good, it is forbidden. If we choose the existential quantifier:

(∃x)(x is good ⊃ x is forbidden)

the reading is that either there is something that is not good, or it is forbidden – a much weaker reading by all accounts.

But aside from these examples, there are sentences which seem to hover on the border between well-formed and ill-formed, as the second one in this pair:

There will be complaints until someone does something about it.
There will be complaints before someone does something about it.

The first sentence clearly gets a strong reading. It seems to say that someone doing something about it, whatever that ‘it’ is, is a necessary condition for the complaints ceasing. The second sentence is merely a temporal sequence of events: complaints first, someone doing something about it later.

Or, take for example the pair:

I have nearly finished.
I’m not far off finishing.

In neither case do we actually indicate completion, nor is it possible to say that in one case we are further along in getting to the end than in the other, yet the first sentence seems to have more force.

A classic case is double negation, which needs little explanation:

The result was to be expected.
The result was not unexpected.

The difficulty is in defining ‘strong’ and ‘weak’ without getting entangled in the semantics. If we assume that ‘strong’ means definitive, categorical or conclusive, then I should be able to explain the difference between the active and passive structures in such terms, but I can’t decide which of these two sentences gets a strong reading and which weak:

Someone will do it.
It will get done.

A separate issue involves instances where some speakers of English give a weak reading to certain conjunctions, such as as well as, where the conjunction in fact suggests a strong reading.

Power plants trade in energy as well as in emission credits.

The purpose of as well as is to introduce ‘known information’ rather than ‘new information’. On current knowledge, it would make more sense to say:

Power plants trade in emission credits as well as in energy.

Propositional Logic, Howard Pospesel, Prentice Hall, 2000, 3rd edition, Chapter 9, ex. 22

The task is to prove that the biconditional is associative. The hardest thing is to think of the correct assumptions to get us started. Eventually, two assumptions for conditional proof and two for indirect proof are used before we can begin to unpick line 4.

1. C ≡ (D ≡ E)
2. ∴(C ≡ D) ≡ E
3. [C ⊃(D ≡ E)] [(D ≡ E) ⊃C] / 2BE
4. {C ⊃[(D ⊃E) • (E⊃D)]} • {[(D⊃E) • (E⊃D)] ⊃C} / 3BE
5. * C ⊃D / ACP
6. * * D ⊃C / ACP
7. * * * ¬ E / AIP
8. * * * * ¬ D / AIP
9. * * * * ¬ C / 8,5MT
10. * * * * [(D⊃E) • (E⊃D)] ⊃C / 4Simp.
11. * * * * ¬ [(D⊃E) • (E⊃D)] / 9,10MT
12. * * * * [ ¬ (D⊃E) ∨¬ (E⊃D)] / 11DeM
13. * * * * ¬ D ∨E / 8Add
14. * * * * ¬ (D • ¬ E) / 13DeM
15. * * * * (D • ¬ E) ∨(E • ¬ D) / 12DeM
16. * * * * E • ¬ D / 14,15DS
17. * * * * E / 16Simp.
18. * * * * E • ¬ E / 7,17Conj.
19. * * * ¬ ¬ D / 8-18IP
20. * * * D / 19DN
21. * * * C / 6,20MP
22. * * * C ⊃[(D ⊃E) • (E⊃D)] / 4Simp.
23. * * * (D ⊃E) • (E⊃D) / 21,22MP
24. * * * D ⊃E / 23Simp.
25. * * * ¬ D / 7,24MT
26. * * * ¬ D • D / 20,25Conj.
27. * * ¬ ¬ E / 7,26IP
28. * * E / 27DN
29. * (D ⊃C) ⊃E / 6-28CP
30. (C ⊃D) ⊃[(D ⊃C) ⊃E] / 5-29CP
31. [(C ⊃D) • (D ⊃C)] ⊃E / 30Exp
32. * E / ACP
33. * * C / ACP
34. * * C ⊃[(D ⊃E) • (E⊃D)] /4Simp.
35. * * (D ⊃E) • (E⊃D) / 33,34MP
36. * * E⊃D / 35Simp.
37. * * D / 32,36MP
38. * C ⊃D / 33-37CP
39. * * D / ACP
40. * * D ∨¬ E / 37Add
41. * * E ⊃D / 40CE or MI
42. * * E ∨¬ D / 32Add
43. * * D ⊃E / 42CE or MI
44. * * (D ⊃E) • (E ⊃D) / 41,43Conj.
45. * * [(D⊃E) • (E⊃D)] ⊃C / 4Simp.
46. * * C / 44,45MP
47. * D ⊃C / 39-46CP
48. * (C ⊃D) • (D ⊃C) /38,47Conj.
49. E ⊃[(C ⊃D) • (D ⊃C)] / 32-48CP
50. {[(C ⊃D) • (D ⊃C)] ⊃E} • {E ⊃[(C ⊃D) • (D ⊃C)]} / 31,49Conj.
51. [(C ≡ D) ⊃E] • [E⊃(C ≡ D)] / 50BE
52. (C ≡ D) ≡ E / 51BE

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

For Kev, Understanding Symbolic Logic, V. Klenk, Unit 8, problem 7m

Try Indirect Proof. This helps to unpackage the first two premises. Then the trick is to use addition to set up propositions which will help break down the other two premises.

1. G ⊃(H • I)


2. J ⊃(H • K)


3. [(L ⊃¬ G) • M] ⊃N


4. (M ⊃N) ⊃(L • J)


5. ∴ I ∨K


6. * ¬ (I ∨K) / AIP


7. * ¬ I • ¬ K / 6DeM


8. * ¬ I / 7Simp


9. * ¬ K / 7Simp


10. * ¬ G ∨(H • I) / 1CE


11. * (¬ G ∨H) • (¬ G ∨I) /10Dist


12. * ¬ G ∨I / 11Simp


13. * ¬ G / 8,12DS


14. * ¬ J ∨(H • K) / 2CE


15. * (¬ J ∨H) • (¬ J ∨K) / 14Dist


16. * ¬ J ∨K / 15Simp


17. * ¬ J / 9,16DS


18. * ¬ G ∨¬ L /13Add


19. * ¬ L ∨¬ G / 18Comm


20. * (L ⊃¬ G) / 19CE


21. * ¬ J ∨¬ L / 17Add


22. * ¬ L ∨¬ J / 21Comm


23. * ¬ (L • J) / 22DeM


24. * ¬ (M ⊃N) / 4,23MT


25. * ¬ (¬ M ∨N) / 24CE


26. * M • ¬ N / 25DeM


27. * M / 26Simp


28. * ¬ N / 26Simp


29. * ¬ [(L ⊃¬ G) • M] / 3,28MT


30. * ¬ (L ⊃¬ G) ∨¬ M / 29DeM


31. * ¬ (L ⊃¬ G) / 27,30DS


32. * (L ⊃¬ G) • ¬ (L ⊃¬ G) / 20,31Conj


33. ¬ ¬ (I ∨K) / 6-32IP


34. I ∨K / 33DN

Contraposition

If a sentence, especially one with two or more negations, does not yield up its secrets on first reading, try contraposition.

In syllogistic logic contraposition is an operation where the subject and predicate terms are changed around and replaced by their term complements (read “negation”). For example:

All even numbers are divisible by 2. (True)
All numbers not divisible by 2 are non-even. (True)

Some solids are non-hexahedra. (True)
Some hexahedra are non-solids. (False)

Here, contraposition does not always preserve the truth of the original statement (as can be seen). In propositional and predicate logic though, the contrapositive is equivalent to the original statement, and that statement is a conditional. The mechanism is rather similar: we negate the antecedent and the consequent and swap them round.

If the accountant sees the figures, he will quit.
If the accountant hasn’t quit, then he hasn’t seen the figures yet.

Statements such as this where a pronoun in the consequent is anaphoric on the antecedent of the conditional can be easily turned into statements with relative clauses.

The accountant who has seen the figures has quit.

This is but a step away from complex sentences with relative clauses and multiple negations, such as lawyers are fond of. The additional complication is that parts of the sentence need not be in an order which makes it immediately obvious what is what. Here is an example from a limitations of liability clause:

"Nothing herein shall be taken to exclude or limit the Council’s liability for: (iii) any liability which cannot be excluded or limited by applicable law."

We work backwards.

First, restate the sentence identifying the relative clause:

The Council’s liability which cannot be excluded or limited by applicable law shall not be taken to be excluded or limited in this agreement (‘herein’).

Then, turn this into a conditional:

If the Council’s liability is not excluded or limited by applicable law, then it is not taken to be excluded or limited in this agreement.

Restate by contraposition and simplify:

If this agreement excludes or limits the Council’s liability, then so does the applicable law.

The second part of the sentence (the consequent) is in fact a kind of supposition. What it says is that if the agreement in question exludes or limits the Council's liability, then the liability will have been exluded or limited by the applicable law too, but students often find it hard to get their heads around the future perfect tense.

The material we work with can vary in complexity, and the patterns may not be apparent at first sight. Sometimes we may find it convenient to contrapose by implicit negations, as in this example:

If an accountant makes a mistake, the calculations are out.
If the calculations are true and correct, the accountants have been very scrupulous.

One way or another, contraposition tends to produce reliable and satisfying outcomes.

Reasoning about articles

The argument often runs like this:

Student:

I know I didn’t get all the articles right in my sentences but you understood what I said, didn’t you? So, the articles are not necessary.

There are two premises and a conclusion in this argument. It is an inductive argument, and it is uncogent. That is, neither the argument is strong nor the premises are exactly true, although only one of these conditions would have been enough to disqualify it. In short, it is fallacious.

How many fallacies does it commit? Typically, we pick the one that is most obvious, but the louder the arguer protests that articles are for ornament only, the stronger the urge to lay as many charges as will stick. There are many classifications of informal fallacies but, according to Hurley (2006), these charges at least can be thrown at the culprit:

INFORMAL FALLACIES

Fallacies of Relevance

· Missing the Point

Fallacies of Weak Induction

· Hasty Generalization
· False Cause

Fallacies of presumption

· Begging the Question
· False Dichotomy
· Suppressed Evidence

Missing the Point (Ignoratio Elenchi) occurs when the premises support one conclusion but a different conclusion is drawn. When we suspect this fallacy has been committed, we are often in a position to identify the correct conclusion that suggests itself. Here the conclusion would have been: The listener relied on non-grammatical information for understanding.

Hasty Generalisation extends evidence that pertains to a selected sample to all members of a group. More immediately, just because I understood this particular message, if I did indeed, does not mean that I would understand all and any sentences where articles have been used sloppily.

False Cause occurs whenever the link between the premises and the conclusion depends on a causal connection that probably does not exist. A variety of the false cause fallacy is a simplified cause, where otherwise a number of causes are responsible for a certain effect. Here: my understanding of, let us stress, this particular sentence, uttered by this particular speaker in this particular situation, makes it seem as if articles are redundant. In actual fact, my understanding alone would not be enough to render articles unnecessary.

Begging the Question (Petitio Principii) involves leaving out a dodgy premise while creating the illusion of completeness. In our case, the argument begs the questions: ‘What makes you think that I understood what you said in the way you intended it, even if I say so myself?’ or ‘Why do you think it is my independent understanding rather than my willing cooperation and benefit of the doubt that makes articles dispensable?’

False Dichotomy turns on presenting two unlikely alternatives as if they were the only ones available. The arguer then eliminates the undesirable alternative, leaving the desirable one as the conclusion. An illusion is created that the two alternatives are jointly exhaustive.

Either articles are unnecessary or you didn’t understand what I said. But you did understand what I said. Therefore, articles are unnecessary.

Interestingly, this argument has a valid form:

~ A ∨ ~ B
B
________

~ A

The problem is that both disjuncts are in fact false, or probably false, which makes the argument unsound.

Suppressed Evidence is a case of leaving out some vital piece of information which, had it been stated, would have led to a different conclusion. The missing information in our case is that the original conversation took place in a controlled environment (classroom), that the listener did a better job of saving the message than the speaker did of ruining it, that the context was abundantly clear, and so on.

Articles are a curious thing, and it is my contention that even languages that don’t have them have ways of alerting the listener to a change of meaning where in English the articles would serve the purpose.

Deduction, Daniel Bonevac, Blackwell 2003, 7.3 problem 19

I have first established that the argument is valid by means of a truth tree. The deduction follows. On line 8 'z' gets instantiated by UI to 'x', which we are within the rules to do.

1. (x)(y)(z)[(Fxy • Fyz) ⊃Fxz]
2. ¬ (∃x)Fxx
3. ∴(x)(y)(Fxy ⊃¬ Fyx)
4. (x) ¬ Fxx / 2CQ
5. ¬ Fxx / 4UI
6. (y)(z)[(Fxy • Fyz) ⊃Fxz] / 1UI
7. (z)[(Fxy • Fyz) ⊃Fxz] / 6UI
8. (Fxy • Fyx) ⊃Fxx / 7UI
9. ¬ (Fxy • Fyx) / 5,8MT
10. ¬ Fxy ∨¬ Fyx / 9DeM
11. * Fxy / ACP
12. * ¬ Fyx / 11,10DS
13. Fxy ⊃¬ Fyx / 11-12CP
14. (y)(Fxy ⊃¬ Fyx) / 13UG
15. (x)(y)(Fxy ⊃¬ Fyx) / 14UG

Express lane - ten items or less

A sign in the supermarket says, ‘Express lane – ten items or less’. An antisocial git at the front of the queue with a trolley full of trophies is locked in a battle of wills with the cashier while telling everyone else to mind their business and get their logic right. ‘Express lane – ten items or less’ is not a law. Is it logic?

The sign can be phrased in terms of a conditional: You can get in the express lane only if you have ten items or less in your basket or trolley. On the first order logic reading of the sentence, there are two issues at play here: the necessary versus sufficient condition, and the truth value of the conditional. To see how they interact, it is best, perhaps, to turn the sentence into a simple material conditional:

(1) If you are in an express lane, you have ten items or less.

By all accounts, having ten items or less is a necessary condition for getting in an express lane, while being in an express lane is only a sufficient condition for having ten items or less. You could after all go to a regular check-out with ten items or less and nobody would mind. They might even thank you.

The standard theory turns on just this symmetry: the consequent of a conditional is the necessary condition for the antecedent, while the antecedent is a sufficient condition for the consequent.

On the truth-value reading of conditionals, a conditional sentence is false when the consequent is false while the antecedent is true. If it is indeed the case that you are in an express lane but have more than ten items in the trolley, (1) is false as a reflection of reality. It is true under all other circumstances, including when you are not in an express lane but have ten items or less, which is just as it should be.

What would you make of (1) if antecedent and consequent were flipped around:

(2) If you have ten items or less, you are in an express lane.

Assuming the facts have not changed: you have more than ten items and are in an express lane, the antecedent in (2) is now false while the consequent is true. On the truth-value reading of conditionals, (2) is true.

The truth of (2) may appear somewhat counterintuitive at first but it is less so once you realize what it means to assert the truth of: If you have ten items or less, you are in an express lane. Nothing in this sentence implies that having more than ten items while being in an express lane is not true. It may be, and, from the logical point of view, is true, too.

Interpreting the sign in terms of (2) will please the antisocial git, as he can shrug his shoulders and point out that (2), as it stands, is true of reality – him having more items and being in an express lane.

But in what sense is being in an express lane a necessary condition for having ten items or less in the trolley?

You could win the argument with the antisocial git if you got him to admit that his interpretation of the sign is: You can have ten items or less in your trolley only if you get in the express lane. I rate your chances of a successful argument though along with the chances of the antisocial git holding the chair of the faculty of ethics in town.

Understanding Symbolic Logic, Virginia Klenk, Prentice Hall, 2008, Unit 20, Problem 1l

Working backwards from the conclusion here gets us only so far, which is why the conclusion is best reached by Indirect Proof after instantiating the first two premises. The tricky part was getting to step 44 from the information on line 35, 36 and 37. Instinctively, we could argue:

Either Samuel Clemens was Mark Twain or Clement Samuelson was Mark Twain.

Neither Samuel Clemens nor Clement Samuelson was Dick Tracy.

Therefore, Dick Tracy was not Mark Twain.

However, the reasoning appears more convincing to me if we argue by another Indirect Proof, on line 38. If we can prove that 'Dick Tracy was Mark Twain' is a contradiction, given our premises, we can be sure that the original conclusion 'Dick Tracy was not Mark Twain' is supported by the premises. Once this hurdle is cleared, it only remains to follow up with multiple instantiations of the main conclusion until we obtain another contradiction on line 88.

1. (∃x)(∃y)(∃z){Cx • Cy • Cz • Rx • Ry • Rz • ¬ (x = y) • ¬ (x = z) • ¬ (z = y) • (w){[(Cw • Rw) ⊃[(w = x) ∨(w = y) ∨(w = z)]}}
   
2. (∃x){Cx • Rx • Px • (y)[(Cy • Ry • Py) ⊃y = x]}


3. (x)[(Cx • Rx • ¬ Px) ⊃Ox]


4. ∴(∃x)(∃y)[Ox • Oy • Rx • Ry • ¬ (x = y)]


5. (∃y)(∃z){Ca • Cy • Cz • Ra • Ry • Rz • ¬ (a = y) • ¬ (a = z) • ¬ (z = y) • (w){[(Cw • Rw) ⊃[(w = a) ∨(w = y) ∨(w = z)]}} / 1EI


6. (∃z){Ca • Cm • Cz • Ra • Rm • Rz • ¬ (a = m) • ¬ (a = z) • ¬ (z = m) • (w){[(Cw • Rw) ⊃[(w = a) ∨(w = m) ∨(w = z)]}} / 5EI


7. Ca • Cm • Ch • Ra • Rm • Rh • ¬ (a = m) • ¬ (a = h) • ¬ (h = m) • (w){[(Cw • Rw) ⊃[(w = a) ∨(w = m) ∨(w = h)]}} / 6EI


8. Ci • Ri • Pi • (y)[(Cy • Ry • Py) ⊃y = i] / 2EI


9. * ¬ (∃x)(∃y)[Ox • Oy • Rx • Ry • ¬ (x = y)] / AIP


10. * (x)(y)[(Ox • Oy • Rx • Ry) ⊃x = y] / 9CQ


11. * (y)[(Oa • Oy • Ra • Ry) ⊃a = y] / 10UI


12. * (Oa • Om • Ra • Rm) ⊃a = m / 11UI


13. * ¬ (a = m) / 7Simp


14. * ¬ (Oa • Om • Ra • Rm) / 13,12MT


15. * ¬ Oa ∨ ¬ Om ∨ ¬ Ra ∨ ¬ Rm / 14DeM


16. * Ra • Rm / 7Simp


17. * ¬ Oa ∨ ¬ Om ∨/ 16,15DS


18. * (Ca • Ra • ¬ Pa) ⊃Oa / 3UI


19. * (Cm • Rm • ¬ Pm) ⊃Om / 3UI


20. * [(Ca • Ra • ¬ Pa) ⊃Oa] • [(Cm • Rm • ¬ Pm) ⊃Om] /18,19Conj


21. * ¬ (Ca • Ra • ¬ Pa) ∨¬ (Cm • Rm • ¬ Pm) / 17,20DD


22. * ¬ Ca ∨¬ Ra ∨ Pa ∨¬ Cm ∨¬ Rm ∨Pm / 21DeM


23. * Ca • Ra • Cm • Rm / 7Simp


24. * Pa ∨Pm / 22,23DS


25. * (y)[(Cy • Ry • Py) ⊃y = i] / 8Simp


26. * (Ca • Ra • Pa) ⊃a = i / 25UI


27. * (Ca • Ra) ⊃(Pa ⊃a = i) / 26Exp


28. * Ca • Ra / 23Simp


29. * Pa ⊃a = i / 28,27MP


30. * (Cm • Rm • Pm) ⊃m = i / 25UI


31. * Cm • Rm / 23Simp


32. * (Cm • Rm) ⊃(Pm ⊃m = i) / 30Exp


33. * Pm ⊃m = i / 31,32MP


34. * (Pa ⊃a = i) • (Pm ⊃m = i) / 29,33Conj


35. * a = i ∨ m = i / 24,34CD


36. * ¬ (a = h) / 7Simp


37. * ¬ (h = m) / 8Simp


38. * * i = h / AIP


39. * * ¬ (m = i) / 37,38Id


40. * * a = i / 39,35DS


41. * * i = a / 40Id


42. * * ¬ (i = h) / 41,36Id


43. * (i = h) • ¬ (i = h) / 38,42Contr


44. * ¬ (i = h) / 38-43IP


45. * (Ch • Rh • Ph) ⊃h = i / 25UI


46. * ¬ (Ch • Rh • Ph) / 44,45MT


47. * ¬ Ch ∨¬ Rh ∨ ¬ Ph / 46DeM


48. * Ch • Rh / 7Simp


49. * ¬ Ph / 48,47DS


50. * (Ch • Rh • ¬ Ph) ⊃Oh / 3UI


51. * Ch • Rh • ¬ Ph / 48,49Conj


52. * Oh / 51,50MP


53. * (y)[(Oa • Oy • Ra • Ry) ⊃a = y] / 10UI


54. * (Oa • Oh • Ra • Rh) ⊃a = h / 53UI


55. * ¬ (a = h) / 7Simp


56. * ¬ (Oa • Oh • Ra • Rh) / 55,54MT


57. * ¬ Oa ∨¬ Oh ∨¬ Ra ∨¬ Rh / 56DeM


58. * Ra • Rh / 7Simp


59. * ¬ Oa∨¬ Oh / 58,57DS


60. * ¬ Oa / 52,59DS


61. * (Ca • Ra • ¬ Pa) ⊃Oa / 3UI


62. * ¬ (Ca • Ra • ¬ Pa) / 60,61MT


63. * ¬ Ca ∨¬ Ra ∨Pa / 62DeM


64. * Pa / 23,63DS


65. * (Ca • Ra • Pa) ⊃a = i / 25UI


66. * Ca • Ra • Pa / 28,64Conj


67. * a = i / 66,65MP


68. * ¬ (m = a) / 13Id


69. * ¬ (m = i) / 67,68Id


70. * (y)[(Oh • Oy • Rh • Ry) ⊃h = y] / 10UI


71. * (Oh • Om • Rh • Rm) ⊃h = m / 70UI


72. * ¬ (h = m) /7Simp


73. * ¬ (Oh • Om • Rh • Rm) / 72,71MT


74. * ¬ Oh ∨¬ Om ∨¬ Rh ∨¬ Rm / 73DeM


75. * Rh • Rm / 7Simp


76. * ¬ Oh ∨¬ Om / 75,74DS


77. * (Ch • Rh • ¬ Ph) ⊃Oh / 3UI


78. * (Cm • Rm • ¬ Pm) ⊃Om / 3UI


79. * [(Ch • Rh • ¬ Ph) ⊃Oh] • [(Cm • Rm • ¬ Pm) ⊃Om] / 77,78Conj


80. * ¬ (Ch • Rh • ¬ Ph) ∨¬ (Cm • Rm • ¬ Pm) / 76,79DD


81. * ¬ Ch ∨¬ Rh ∨Ph ∨¬ Cm ∨¬ Rm ∨Pm / 80DeM


82. * Ch • Cm • Rh • Rm / 7Simp


83. * Ph ∨Pm / 82,81DS


84. * Pm / 49,83DS


85. * (Cm • Rm • Pm) ⊃m = i / 25UI


86. * Cm • Rm • Pm / 31,84Conj


87. * m = i / 86,85MP


88. * m = i • ¬ (m = i) / 87,69Contr


89. (∃x)(∃y)[Ox • Oy • Rx • Ry • ¬ (x = y)] / 9-88IP

Symbolic Logic, Irving M. Copi, Prentice Hall, 1979, 5th edition, Part II, exercise 8, p.134

Old man Copi has left a number of quirky brain teasers in his Symbolic Logic (1979), which has been the core of college level logic courses in some universities. Professor Peter Suber, from Earlham College, posted answers to many of them in his course hand-outs for the academic year 1996-97: http://www.earlham.edu/~peters/courses/log/loghome.htm At one point though, he gives up saying: 'I haven't had time to finish this set of exercises. I hope to do soon ...' (Polyadic Predicate Logic). When I last emailed him, he said he had retired. He left off at what is in my edition Part II, exercise 8, p. 134. Symbolize the following argument and construct a formal proof of validity:

There is a professor who is liked by every student who likes at least one professor. Every student likes some professor or other. Therefore, there is a professor who is liked by all students.

1. (∃x){Px • (y){[Sy • (∃z)(Pz • Lyz)] ⊃Lyx]}
2. (y)[Sy ⊃(∃x)(Px • Lyx)]
3. ∴(∃x)[Px • (y)(Sy ⊃Lyx)]
4. Pa • (y){[Sy • (∃z)(Pz • Lyz)] ⊃Lya] / 1EI
5. Pa / 4Simp
6. (y){[Sy • (∃z)(Pz • Lyz)] ⊃Lya] / 4Simp
7. * Sy / ACP
8. * Sy ⊃(∃x)(Px • Lyx) / 2UI
9. * (∃x)(Px • Lyx) / 7,8MP
10. * Pm • Lym / 9EI
11. * (∃z)(Pz • Lyz) / 10EG
12. * Sy • (∃z)(Pz • Lyz) / 7,11Conj
13. * [Sy • (∃z)(Pz • Lyz)] ⊃Lya / 6UI
14. * Lya / 12,13MP
15. Sy ⊃Lya / 7-14CP
16. (y)(Sy ⊃Lya) / 15UG
17. Pa • (y)(Sy ⊃Lya) /5,16Conj
18. (∃x)[Px • (y)(Sy ⊃Lyx)] / 17EG

The least obvious strategy here perhaps involves a change of variable, from 'x' on line 9 to 'z' on line 11 by first instantiating it to a constant, other than 'a', by Existential Instantiation and then generalising it back to a variable by Existential Generalisation.

A Concise Introduction to Logic, 9th edition, P. Hurley, 8.7, III (10)

This problem is solved in the book but I thought to incude it in my answers because my translation of the last premise is slightly different than Hurley's and because the conclusion can be derived without having to use Indirect Proof. The problem:

The tallest building in North America is the Sears Tower. The tallest building in North America is located in Chicago. If one thing is taller than another, then the latter is not taller than the former. Therefore, the Sears Tower is located in Chicago. (Bx: x is a building in North America; Txy: x is taller than y; Cx: x is located in Chicago; s: the Sears Tower)

Hurley translates the last premise as (x)(y)(Txy ⊃¬ Tyx). I show that we will get the same result if we make explicit the fact that x and y are not the same things: (x)(y){[¬(x=y) • Txy] ⊃¬ Tyx}.

1. (x){[Bx • ¬ (x=s)] ⊃Tsx} • Bs
2. (∃x){Bx • (y){[By • ¬ (y=x)] ⊃Txy} • Cx}
3. (x)(y){[¬(x=y) • Txy] ⊃¬ Tyx}
4. ∴Cs
5. Ba • (y){[By • ¬ (y=a)] ⊃Tay} • Ca / 2 EI
6. (y){[By • ¬ (y=a)] ⊃Tay / 5 Simp
7. [Bs • ¬ (s=a)] ⊃Tas / 6 UI
8. Bs ⊃[¬ (s=a) ⊃Tas] / 7 Exp
9. Bx / 1 Simp
10. ¬ (s=a) ⊃Tas / 9, 8 MP
11. [Ba • ¬ (a=s)] ⊃Tsa / 1 UI
12. (y){[¬(a=y) • Tay] ⊃¬ Tya / 3 UI
13. [¬(a=s) • Tas] ⊃¬ Tsa / 12 UI
14. Tsa ⊃¬ [¬(a=s) • Tas] / 13 Contrap
15. [Ba • ¬ (a=s)] ⊃¬ [¬(a=s) • Tas] / 11, 14 HS
16. ¬ [Ba • ¬ (a=s)] ∨¬ [¬(a=s) • Tas] / 15 Impl
17. [¬ Ba ∨ (a=s)] ∨[(a=s) ∨¬ Tas] / 16 DeM
18. ¬ Ba ∨ [(a=s) ∨(a=s) ∨¬ Tas] / 17 Assoc
19. Ba / 5 Simp
20. (a=s) ∨(a=s) ∨¬ Tas / 18, 19 DS
21. (a=s) ∨¬ Tas / 20 Taut
22. ¬ Tas ∨(a=s) / 21 Comm
23. Tas ⊃ (a=s) / 22 Impl
24. ¬ (s=a) ⊃(a=s) / 10, 23 HS
25. (s=a) ∨ (a=s) / 24 Impl
26. (a=s) ∨(a=s) / Id
27. a=s / Taut
28. Ca / 5 Simp
29. Cs / Id

Comma or no comma?

What can be an unfathomable point of the English syntax – use of commas with relative clauses – is actually quite straightforward if you apply a bit of logic. Relative clauses are something of a speciality of predicate logic, in fact. The problem touches on the question of anaphora – more about which another time.

There is a correlation between the scope of the quantifier and the use of commas, which looks like this:

narrow scope of quantifier – comma required
wide scope of quantifier – no comma

Some examples:

(1) Merchants flatter politicians, who are corrupt.

(2) Merchants flatter any politician who is corrupt.

(3) A merchant will flatter any politician who can help him.

(1.1) (x)[Mx ⊃ (∃y)(Py • Fxy)] • (y)(Py ⊃ Cy)

(2.1) (x){Mx ⊃ (y)[(Py • Cy) ⊃ Fxy)]}

(3.1) (x){Mx ⊃ (y)[(Py • Hyx) ⊃ Fxy)]}


Sentence (1.1) – narrow scope of quantifier over ‘merchant’ – says that each merchant flatters some politician and that all politicians are corrupt. The main connective is a conjunction, which means that if there is at least one politician who is not corrupt, the sentence is false.

Sentence (2.1) – wide scope of quantifier over ‘merchant’ – says that each merchant flatters corrupt politicians. The main connective is a conditional, which means that the fact that there may be some politicians who are not corrupt and whom merchants flatter does not falsify the sentence.

Sentence (3.1) – wide scope of quantifier over ‘merchant’ – leaves us no option really, because if the pronoun ‘him’ is to be a bound variable, the scope of quantification over ‘a merchant’ must extend to ‘him’.

The same applies to commas around relative clauses modifying the subject of the sentence:

(4) Merchants, who buy off politicians, are corrupt.

(5) Merchants who buy off politicians are corrupt.

(4.1) (x)(Mx ⊃ Cx) • (x)[Mx ⊃ (∃y)(Py • Bxy)]

(5.1) (x){[Mx • (∃y)(Py • Bxy)] ⊃ Cx}

Understanding Symbolic Logic, V. Klenk, Prentice Hall 2008, Unit 18, Ex.1(o)

1. (x)[(Px • ¬ Rxx) ⊃(y)(Py ⊃¬ Ryx)]
2. (x){Px ⊃ (y)[(Py • ¬ Rxy) ⊃¬ Hxy]}
3. ∴(x){[Px • (y)(Py ⊃¬ Rxy)] ⊃¬ (∃z)(Pz • Hzx)}
4. * Px • (y)(Py ⊃¬ Rxy) / ACP
5. * Px / 4 Simp
6. * (y)(Py ⊃¬ Rxy) / 4 Simp
7. * Px ⊃¬ Rxx / 6 UI
8. * ¬ Rxx / 5, 7 MP
9. * Px • ¬ Rxx / 5, 8 Conj
10. * (Px • ¬ Rxx) ⊃(y)(Py ⊃¬ Ryx) / 1 UI
11. * (y)(Py ⊃¬ Ryx) / 9, 10 MP
12. * * Pz / ACP
13. * * Pz ⊃¬ Rzx / 11 UI
14. * * ¬ Rzx / 12, 13 MP
15. * * Pz ⊃(y)[(Py • ¬ Rzy) ⊃¬ Hzy] / 2 UI
16. * * (y)[(Py • ¬ Rzy) ⊃¬ Hzy] / 12, 15 MP
17. * * (Px • ¬ Rzx) ⊃¬ Hzx / 16 UI
18. * * Px • ¬ Rzx / 5, 14 Conj
19. * * ¬ Hzx / 17, 18 MP
20. * Pz ⊃¬ Hzx / 12 - 19 CP
21. * (z)(Pz ⊃¬ Hzx) / 20 UG
22. * ¬ (∃z)(Pz • Hzx) / CQ
23. [Px • (y)(Py ⊃¬ Rxy)] ⊃¬ (∃z)(Pz • Hzx) / 4 - 22 CP
24. (x){[Px • (y)(Py ⊃¬ Rxy)] ⊃¬ (∃z)(Pz • Hzx)} / 23 UG

Comment

This deduction calls for multiple variable rewrite, including on lines 7, 13, and 17.

This is the first in my series of worked solutions to deduction problems left unanswered in logic course books. The aim is strictly educational and the answers are meant to help those who study on their own.

I haven’t worked out yet how to use the blogger tools to show the scope of assumptions in the way I find most clear – by indentation, or how to show step descriptions to the right of my work. So until I do that, I mark the scope of each new assumption with a star (*) to the left of the assumption, and separate the step descriptions with a forward stroke. It looks messy but I'd rather explain what I am doing on the line I'm doing it than add the comments below or not at all. When an assumption is discharged, the star is taken off. Most other notation is pretty standard. CP stands for discharge of conditional proof, CQ - change of quantifier (justified by one of the four equivalences).

Tutorial on existence

I am constantly reminded what a challenge the basic building blocks of English are for students who study it. After all, nothing is more basic and few things are more complex than the verb ‘to be’. What sentence can you make up from members of the set A:

A = {lion, white, is}

Trivially:

(1) A lion is white.

(2) There is a white lion.

(3) The lion is white.


Sentence (2) we are told implies existence. I agree fully, but such explanation is misleading without qualification. Students and teachers may be led to believe that (1) and (3) do not imply existence, which is false. They do! OK, (1) is not much of a sentence, and (2) is only a slight improvement on (1), but what is a matter of elegance in grammar may sometimes be an irrelevance in logic.

The question is, what does it mean to say that (2) implies existence? And what about (1) and (3)?

One way to tackle the problem is by responding with ‘So what?’ to someone uttering (2) to you, out of the blue. If your response is indeed ‘So what?’, or ‘Why are you telling me this?’, you already know what it means for ‘there is’ to express existence. The point is existence is so basic we don’t need to talk about it. If there is a white lion, let’im be.

Language, however, wrongfoots you. It gives you (2) and (3), never mind (1), with a verb ‘to be’ sitting squarely in the middle. Some philosophers call it referential tautology. The implication of existence is already present in my uttering the word ‘lion’. The verb ‘to be’ or ‘to exist’ adds nothing.

In first-order logic (2), as (1), can be represented as:

(2.1) (∃x)(Lx • Wx)


where Lx – ‘x’ is a lion, and Wx – ‘x’ is white, and where ∃ is an existential quantifier. The sentence thus reads: there exists an ‘x’ such that ‘x’ is a lion and ‘x’ is white. We have in fact used the word ‘is’ in our paraphrase, but nothing in our symbolic translation corresponds to it! To say that ‘Lions exist’, ‘Some lions exist’, ‘There is at least one lion,’ or, if you can bear it, that ‘Lions are,’ or else that ‘There are lions,’ is simply: (∃x)Lx.

Note that the plural in the preceding paragraph is no more than a variant of ‘There is a lion’ or ‘A lion is,’, which is true. It is a kind of minimal or limiting plural. It definitely does not suggest that ‘All lions exist,’ for when put like this the sentence is meaningless in English. The universal sentence (x)Lx, which can be paraphrased: for any ‘x’, ‘x’ is a lion, or everything is a lion, is clearly false. The existential and the universal would have the same truth value if and only if there was only one object in the universe, and that object was not necessarily a lion either.

(3.1) (∃x)[Lx • (y)(Ly ⊃ y = x) • Wx]


Translation (3.1) is just like (2.1) except for the bit in the middle, and is a representation of sentence (3): The lion is white. Just like (2.1) it is says that there exists an ‘x’ such that ‘x’ is a lion and ‘x’ is white, but it also adds (the bit in the middle) that if any ‘y’ is a lion then that ‘y’ is identical to our ‘x’, and this is what makes our lion ‘unique’, ‘the only lion’, or simply ‘the lion’.

Summing up, all three sentences assert existence, it is just that (1) and (2) assert nothing but the existence of white lions. Sentence (3) tells us more: it tells us that there is a unique lion. Logical notation thus makes explicit to a non-English speaker what the English language doesn’t by merely replacing ‘a’ with ‘the’: it adds one extra piece of information – the uniqueness claim.

And if (1) and (2) do not sound interesting, that is precisely because, when taken out of context, they tell you about nothing other than that somewhere a white lion exits – something you didn’t expect to be told without first asking about it. Sentences about existence become interesting only when we have reason to doubt existence: There is life on Mars; There is a God; and so on.

Getting started

Welcome to If English then Logic, a blog aimed at promoting English through logic and logic through English. I will share my interests in the English language, my fascination with language in general, logic and philosophy. Specifically:

- students of English as a foreign language may find an alternative in my back-to-first-principles approach to the tyranny of Cambridge exams, business English, and learn-quick schemes (called ‘methods’). I will expand on this more, share examples and, time permitting, exercises.
- students of first-order logic will find answers to exercises that are notoriously omitted from logic textbooks for reasons of space. They can take up rather a lot of it. I will trawl through the books used in university logic courses and post those answers which I find more difficult or more interesting for some reason.
- I will tackle some formal logic symbolization problems and look at how logic can improve – though never take over – your understanding of a language like English.
- Finally, I will post my thoughts and observations prompted by current work, marrying my principle interests and looking beyond for inspiration.

Wherever I know the source of my information, I credit it accordingly. I do the same with opinions, unless they are mine. Any errors and omissions are entirely mine. Feel free to point them out.