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