Thursday, 24 February 2011

Which way round?

The ‘which way round?’ questions are never as harmless or as innocent as they look. And they look strange in a language like English that doesn’t do inflections. The answers are no less strange.

Which end do you peel a banana from?
Which way round do you put toilet rolls in the wall holder?
Which way round do you park you car in relation to the curb?

For example, the question: ‘Which way round do you sit in the office in relation to the window?’ can have for an answer:

I sit side-on to the window.
I sit facing the window.
I sit with my back to the window.
I sit at an angle to the window.

We could force all answers to follow one pattern, as I am sure is the case in many languages, but three of the four would sound – well, ‘forced’. The grammar of say: ‘Put the knives and forks in the drawer sharp end first,’ is challenging on account of its simplicity (when I was little, I was taught to hand a knife to another person with the sharp end towards me, that is, with the sharp end away from the person to whom I was handing it).

Negotiating the bends and corners of such grammar causes as much mental fatigue to speakers of little faith in their linguistic capabilities as rotating a triangle in the coordinate planes through 90 degrees about the origin does to a student of literature. But once the grammatical code has been cracked, the serious business begins.

They are Swiftian questions. When Gulliver is shipwrecked on the island of Lilliput, he learns that the empires of Lilliput and Blefuscu had gone to war over which end was the correct end to break a boiled egg at. Inevitably, some of my students say that the big end is the logical way to break an egg at, others, the little end.

From ‘logical’ the debate goes to ‘normal’. Sometimes it is the other way round. There can’t be anything logical about any of this business because bivalent logic admits only ‘true or false’ or ‘valid or invalid’ for an answer, but I accept that the word ‘logical’ is used loosely. ‘Normal’ is a different story. ‘Normal’ promptly leads to ‘the right way’ and ‘the wrong way’ (compare ‘dexterous’ – right-handed but also skillful, adroit, and ‘sinister’ – left-handed but also ominous). By now we are really going places.

As in Swift, we could be talking about whether transubstantiation takes place at an altar or it is just hocus pocus (Swift uses the egg ends to allude to the differences between the Catholics and Protestants). Someone will inevitably chip in with a line that tilting your soup bowl away from you is the posh way of doing it while tilting it towards you is the common way. How about slicing a tomato? Green end nearest you or green end facing down, or holding your tomato green end uppermost and slicing it thus? One way is practical, another positively wasteful. Which is which? Then, there is the trolley in the check-out aisle: narrow end towards you to make loading and unloading easier (naff) or narrow end away from you (to effect an exit in a dignified pose). Put the coat on left arm first or right arm first (is this a gender difference?).

These are silly divisions with strong underlying convictions, and Swift was a sharp observer to notice that answers to questions of the ‘which way round’ or ‘which end first’ type polarize us on many levels. Small cracks conceal big schisms. What we call a force of habit may only be a fig leaf. Speaking of relations and stretching the analogy a little, the mathematical relation congruence modulo 2 partitions the set of integers into two non-overlapping sets: ‘even’ and ‘odd’.

Meaning and Argument, Ernest Lepore, Blackwell, 2003, Problem 16.11(2), p. 289

We are only asked to symbolize this argument, but we may as well prove it here. The layering looks daunting.
The father of the father of Annnette flies. Anyone who has a father who flies fears loss. So, the father of Annette fears loss.
Let Fxy - x is a father of y, a - Annette, Vx - x flies, Lx - x fears loss.
  1. (∃x){Fxa • (z)(Fza ⊃z = x) • (∃y)[Fyx • (w)(Fwx ⊃w = y) • Vy]}
  2. (x)[( ∃y)(Fyx • Vy) ⊃Lx]
  3. ∴(∃x)[Fxa • (y)(Fya ⊃y = x) • Lx]
  4. Fma • (z)(Fza ⊃z = m) • (∃y)[Fym • (w)(Fwm ⊃w = y) • Vy] ......... 1 EI x/m
  5. (∃y)(Fym • Vy) ⊃Lm ......... 2 UI x/m
  6. (∃y)[Fym • (w)(Fwm ⊃w = y) • Vy] ......... 4 Simp.
  7. Frm • (w)(Fwm ⊃w = r) • Vr ......... 6 EI y/r
  8. Frm • Vr ......... 7 Simp.
  9. (∃y)(Fym • Vy) ......... 8 EG
  10. Lm ......... 9,5 MP
  11. (z)(Fza ⊃z = m) ......... 4 Simp.
  12. Fya ⊃y = m ........ 11 UI z/y
  13. (y)(Fya ⊃y = m) ......... 12 UG
  14. Fma ......... 4 Simp.
  15. Fma • (y)(Fya ⊃y = m) • Lm ......... 14,10,13 Conj.
  16. (∃x)[Fxa • (y)(Fya ⊃y = x) • Lx] ......... 15 EG

Thursday, 17 February 2011

Deduction, Daniel Bonevac, Blackwell Publishing 2003, 2nd edition, Problem 8.3(5)

The argument is very simple:

There are exactly two hemispheres. So, there are at least two hemispheres.

In terms of arguments in everyday English, it merits no attention. From a logical point of view, it is worth noting that we can't switch the sentences around. We cannot infer that there are exactly two hemispheres from the fact that there are at least two hemispheres.

We can of course negate the conclusion and derive a contradiction, which would be the stronger of the two methods of proof, but I will simply drop the universal statement which is part of the premise and do existential generalisation on what is left.
  1. (∃x)(∃y){Hx • Hy • ¬ (x = y) • (z)[Hz ⊃(z = x ∨z = y)]}
  2. ∴(∃x)(∃y)[Hx • Hy • ¬ (x = y)]
  3. (∃y){Ha • Hy • ¬ (a = y) • (z)[Hz ⊃(z = a ∨z = y)] ......... 1 EI x/a
  4. Ha • Hm • ¬ (a = m) • (z)[Hz ⊃(z = a ∨z = m) ......... 3 EI y/m
  5. Ha • Hm • ¬ (a = m) ......... 4 Simp.
  6. (∃y)[Ha • Hy • ¬ (a = y)] ......... 5 EG
  7. (∃x)(∃y)[Hx • Hy • ¬ (x = y)] ......... 6 EG

Thursday, 10 February 2011

Six of one, half a dozen of the other

Why do we think that Portia’s line in the Merchant of Venice: ‘All that glitters is not gold’ (‘glisters’ actually) says that there are things that glitter, yet which are not gold? The line does not say this explicitly. This is what we want it to say.

Strictly speaking, what it says is that whatever glitters is not gold. This is not true, because gold does glitter, or at least some gold glitters. As it stands, the sentence conforms to the E-type proposition on the square of oppositions, Aristotelian or modern. E-type propositions are of the kind: No A are B. A paraphrase of Portia’s line then would be: ‘Nothing that glitters is gold.’ Alternatively, ‘It is not the case that there exist things that glitter and are gold.’

What we want is an O-type proposition: ‘There exist things that glitter and are not gold,’ or, which is the same thing, ‘Not everything that glitters is gold,’ with the negation pushed out in front of a universally quantified sentence.

Here is another example which we repeat more often than we think through its implications: ‘Winning is not everything. Winning is the only thing.’ When ‘everything’ is used in the predicate position it usually means ‘is the most important thing’. The expression ‘is the only thing’, again, used as it is in this quote, in its somewhat colloquial or idiomatic garb, means ‘is the most important thing.’ Hence: ‘Winning is not the most important thing. Winning is the most important thing.’ A contradiction!

‘Nothing comes from nothing,’ (Parmenides) and ‘Almost everything comes from nothing,’ (H. Amiel), on the other hand, can happily coexist and give no or little impression of being contradictory. The triumph of pragmatism over logic.

Deduction, Daniel Bonevac, Blackwell, 2nd etition, 2003, 8.3, problem 4, p. 238

Just how convincing is the following argument:

There is at most one God. So, there are at most two Gods.

The pattern for sentences with 'at most n things with a certain property' is to begin with n + 1 universal quantifiers, and then qualify our statement by saying that if all those things have the property in question, then two of them must be the same. So, 'there is at most one God' means that if 'x' is a God and 'y' is a God, then 'x' is identical to 'y'. So they collapse down into one. In the same way, three collapse down into two, and so on.
  1. (x)(y)[(Gx • Gy) ⊃x = y]
  2. ∴(x)(y)(z)[(Gx • Gy • Gz) ⊃(x = y ∨ x = z ∨ y = z)]
  3. * ¬ (x)(y)(z)[(Gx • Gy • Gz) ⊃(x = y ∨ x = z ∨ y = z)] ......... AIP
  4. * (∃x)(∃y)(∃z)[Gx • Gy • Gz • ¬ (x = y) • ¬ (x = z) • ¬ (y = z)] ......... 3 QC
  5. * (∃y)(∃z)[Ga • Gy • Gz • ¬ (a = y) • ¬ (a = z) • ¬ (y = z)] ......... 4 EI x/a
  6. * (∃z)[Ga • Gm • Gz • ¬ (a = m) • ¬ (a = z) • ¬ (m = z)] ......... 5 EI y/m
  7. * Ga • Gm • Gr • ¬ (a = m) • ¬ (a = r) • ¬ (m = r) ......... 6 EI z/r
  8. * (y)[(Ga • Gy) ⊃a = y] ......... 1 UI x/a
  9. * (Ga • Gm) ⊃a = m ......... 8 UI y/m
  10. * Ga • Gm ......... 7 Simp.
  11. * a = m ......... 9,10 MP
  12. * ¬ (a = m) ......... 6 Simp.
  13. * (a = m) • ¬ (a = m) ......... 11,12 Conj.
  14. ¬ ¬ (x)(y)(z)[(Gx • Gy • Gz) ⊃(x = y ∨ x = z ∨ y = z)] ......... 3-13 IP
  15. (x)(y)(z)[(Gx • Gy • Gz) ⊃(x = y ∨ x = z ∨ y = z)] ......... 14 DN

Thursday, 3 February 2011

Understanding Symbolic Logic, Virginia Klenk Prentice Hall, 2008, 5th edition, Unit 20, 1(m), p. 381

The premise is notable for the use of a superlative, but unlike the examples commonly used in logic textbooks, it does not single out a named individual to whom the given quality is attributed. Here, 'a dog', rather than, say, Rover or Fido, is a class of animals that answers to the description of 'canine'. We need to adjust our translation to fit the message which the statement conveys.

The fastest animal on the track is a dog. Therefore, any animal on the track that isn't a dog can be outrun by some dog.
  1. (x)(y){[Ax • Tx • Ay • Ty • Dy • ¬ (x = y)] ⊃Fyx} • (∃y)(Ay • Ty • Dy)
  2. ∴(x)[(Ax • Tx • ¬ Dx) ⊃(∃y)(Dy • Fyx)]
  3. * ¬ (x)[(Ax • Tx • ¬ Dx) ⊃(∃y)(Dy • Fyx)] ......... AIP
  4. * (∃x)[Ax • Tx • ¬ Dx • (y)(Dy ⊃ ¬ Fyx)] ......... 3QC
  5. * Aa • Ta • ¬ Da • (y)(Dy ⊃ ¬ Fya) ......... 4 EI x/a
  6. * (∃y)(Ay • Ty • Dy) ......... 1 Simp.
  7. * Am • Tm • Dm ......... 6 EI y/m
  8. * ¬ Da ......... 5 Simp.
  9. * Dm ......... 7 Simp.
  10. * ¬ (a = m) ......... 8,9 Id
  11. * (y){[Aa • Ta • Ay • Ty • Dy • ¬ (a = y)] ⊃Fya} ......... 1 UI x/a
  12. * [Aa • Ta • Am • Tm • Dm • ¬ (a = m)] ⊃Fma ......... 11 UI y/m
  13. * Aa • Ta ......... 5 Simp.
  14. * Aa • Ta • Am • Tm • Dm • ¬ (a = m) ......... 13,7,10 Conj.
  15. * Fma ......... 14,12 MP
  16. * (y)(Dy ⊃ ¬ Fya) ......... 5 Simp.
  17. * Dm ⊃ ¬ Fma ......... 16 UI y/m
  18. * ¬ Fma ......... 9,17 MP
  19. * Fma • ¬ Fma ......... 15,18 Conj.
  20. ¬ ¬ (x)[(Ax • Tx • ¬ Dx) ⊃(∃y)(Dy • Fyx) ......... 3-19 IP
  21. (x)[(Ax • Tx • ¬ Dx) ⊃(∃y)(Dy • Fyx) ......... 20 DN