Thursday 26 August 2010

'in' and 'at'

The preposition ‘at’ in English is a funny kind of preposition. I can make an equally strong case in my head for not having it in the language as for keeping it firmly where it belongs. If I give too much thought to one side of the argument, that’s the way the pendulum swings at that particular time.

A friend once told me that a local newspaper printed her father’s obituary saying he had died of a heart attack in the Whitstable harbour. The information was only partly true. He died of a heart attack at Whitstable Harbour. He was walking along the quayside when the fatal attack struck and he fell over. Crucially, he didn’t fall into the water – he fell on the ground.

I was reminded of this story the other day while listening to a report about miners trapped in a coalmine in Chile. The news reader said, as one would expect, that the BBC reporter was at the time ‘at the coalmine’ from where he sent the report. Had he been ‘in the coalmine’, he wouldn’t have been able to file the report, of course.

The distinction between ‘in’ and ‘at’ is necessary, but, pointless as it is, it is interesting to speculate by what reasoning our ancestors arrived at it. Were they thinking: ‘in’ for things being inside, and ‘at’ for things being … well, precisely, inside, outside, beside, neither here nor there, kind of generally in the vicinity? Or perhaps: we’ve got most relations covered by now, but where no preposition seems to fit, ‘at’ is the default option. Or maybe, after the fashion of Donald Rumsfeld: there are named and unnamed relations, and of the unnamed ones there are those we know to exist and those we don’t know to exist, and those that we don’t know we don’t know to exist, and them perhaps are best left to ‘at’.

It would seem that ‘on the top rung of the ladder’ was prior to ‘at the top of the ladder’ if we adopt the Darwinian view of language evolution: more complex concepts come after simpler concepts. The concept ‘at the top of the ladder’ is more complex because it is more ambiguous: we can mean the top rung or the second rung from the top, or perhaps even the third rung down from the top one, if the ladder is sufficiently long.

It is easy to see that ‘at’ cannot be a favourite of logicians and mathematicians. It is not precise enough. In a world consisting of objects in a plane, with no value attached to the objects (cones, cubes, spheres, etc), there is no place for relations involving ‘at’. All relations can be handled by: next to, to the left of, to the right of, over, under, etc. However, having said that, what about two non-parallel lines which meet ‘at’ point A? Well, then we come to what a point is in geometry. The point is usually left undefined. But if we try to define it, say, a set of coordinates in the Cartesian system or a circle of radius 0, then we get very precise indeed, and that in turn conflicts with our intuitive understanding of ‘at’, which is used for imprecise descriptions. A vicious circle.

Prepositions have fixed opposites: in / out, on / off, over / under, to / from, or context-dependent opposites: across, through, along, around, etc. What is the opposite of ‘at’? I can’t really say that the opposite of ‘I’ll meet you at the theatre’ is ‘I’ll meet you inside the theatre’.

There is a sense in which ‘in’ meets ‘at’ and a sense in which it doesn’t. Some distinctions seem to me to have very little basis indeed, like ‘arrive at’ and ‘arrive in’. The most satisfying test in choosing between ‘in’ and ‘at’ is to look at the implications: the Whitstable and Chile examples above, or ‘spent a lot of time in the sea’, so suffers from hypothermia; ‘spent a lot of time at sea’, so feels homesick, and so on, but that doesn’t change the fact that I feel kind of rudderless with ‘at’ and at sea without it.

Symbolic Logic, Dale Jacquette, Wadwsworth, 2001, Chpt. 8, Ex. III, problem 17, p. 434

Demonstrate that the following is a tautology:├ ¬ (z)[(∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy)] ⊃¬(x)(Fx ⊃Gx). We proceed by showing that we can derive the implication without the use of any premises other than what we assume.
  1. ├ ¬ (z)[(∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy)] ⊃¬(x)(Fx ⊃Gx)
  2. (x)(Fx ⊃Gx) ⊃(z)[(∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy)] ......... 1Contrapositive
  3. * (x)(Fx ⊃Gx) ......... ACP
  4. * * (∃y)(Fy • Hzy) ......... ACP
  5. * * Fa C Hza ......... 4EI y/a
  6. * * Fa ......... 5Simp.
  7. * * Fa ⊃Ga ......... 3UI x/a
  8. * * Ga ......... 6,7MP
  9. * * Hza ......... 5Simp.
  10. * * Ga • Hza ......... 8,9Conj.
  11. * * (∃y)(Gy • Hzy) ......... 10EG
  12. * (∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy) ......... 4-11CP
  13. (x)(Fx ⊃Gx) ⊃(z)[(∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy)] ......... 2-12CP
  14. ¬ (z)[(∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy)] ⊃¬(x)(Fx ⊃Gx) ......... 13Contrap.

Thursday 19 August 2010

Symbolic Logic, Dale Jacquette, Wadsworth, 2001, Chpt. 8, Ex. III, problem 16, p. 434

The task: give a natural deduction proof for the following tautology, ├ (x)(Fx ⊃Gx) ⊃(y)[(∃z)(Fz • Hyz) ⊃(∃z)(Gz • Hyz)].

├ (x)(Fx ⊃Gx) ⊃(y)[(∃z)(Fz • Hyz) ⊃(∃z)(Gz • Hyz)]
  1. * (x)(Fx ⊃Gx) ......... ACP
  2. * * (∃z)(Fz • Hyz) ......... ACP
  3. * * Fa • Hya ......... 2EI z/a
  4. * * Fa ⊃Ga ......... 1UI x/a
  5. * * Fa ......... 3Simp.
  6. * * Ga ......... 5,4MP
  7. * * Hya ......... 3Simp.
  8. * * Ga • Hya ......... 6,7Conj.
  9. * * (∃z)(Gz • Hyz)] ......... 8EG
  10. * (∃z)(Fz • Hyz) ⊃(∃z)(Gz • Hyz) ......... 2-9CP
  11. (x)(Fx ⊃Gx) ⊃(y)[(∃z)(Fz • Hyz) ⊃(∃z)(Gz • Hyz)] ......... 1-10CP

Thursday 12 August 2010

Deduction, Daniel Bonevac, Blackwell Publishing, 2nd Edition, 2003, chpt. 8.3, problem 15

Show that (∃x)[Gx • (Fx ⊃Ha)] ≡ [(∃x)(Gx • Fx) ⊃Ha] is a consequence of (∃x)(y)(y = x ≡ Gy).


  1. (∃x)(y)(y = x ≡ Gy)
  2. ∴(∃x)[Gx • (Fx ⊃Ha)] ≡ [(∃x)(Gx • Fx) ⊃Ha]
  3. * (∃x)[Gx • (Fx ⊃Ha)] ... ACP
  4. * * (∃x)(Gx • Fx) ... ACP
  5. * * Gm • Fm ... 4EI x/m
  6. * * (y)(y = h ≡ Gy) ... 1EI x/h
  7. * * m = h ≡ Gm ... 6UI y/m
  8. * * (m = h ⊃Gm) •(Gm ⊃m = h) ... 7BE
  9. * * Gm ... 5Simp.
  10. * * Gm ⊃m = h ... 8Simp.
  11. * * m = h ... 9,10MP
  12. * * Gr • (Fr ⊃Ha) ... 3EI x/r
  13. * * r = h ≡ Gr ... 6UI y/r
  14. * * (r = h ⊃Gr) •(Gr ⊃r = h) ... 13BE
  15. * * Gr ... 12Simp.
  16. * * Gr ⊃r = h ... 14Simp.
  17. * * r = h ... 15,16MP
  18. * * h = m ... 11Id.
  19. * * r = m ... 17,18Id.
  20. * * Fr ⊃Ha ... 12Simp.
  21. * * Fm ... 5Simp.
  22. * * Fr ... 19,21Id.
  23. * * Ha ... 22,20MP
  24. * (∃x)(Gx • Fx) ⊃Ha ... 4-23CP
  25. [(∃x)[Gx • (Fx ⊃Ha)] ⊃[(∃x)(Gx • Fx) ⊃Ha] ... 3-24CP
  26. * (∃x)(Gx • Fx) ⊃Ha ... ACP
  27. * * ¬ (∃x)[Gx • (Fx ⊃Ha) ... AIP
  28. * * (x)[Gx ⊃(Fx • ¬ Ha) ... 27QC
  29. * * (y)(y = m ≡ Gy) ... 1EI x/m
  30. * * m = m ≡ Gm ... 29UI y/m
  31. * * (m = m ⊃Gm) •(Gm ⊃m = m) ... 30BE
  32. * * m = m ⊃Gm ... 31Simp.
  33. * * m = m ... Id.
  34. * * Gm ... 33,32MP
  35. * * Gm ⊃(Fm • ¬ Ha) ... 28UI x/m
  36. * * Fm • ¬ Ha ... 34,35MP
  37. * * ¬ Ha ... 36Simp.
  38. * * Fm ... 36Simp.
  39. * * Gm • Fm ... 34,38Conj.
  40. * * (∃x)(Gx • Fx) ... 39EG
  41. * * Ha ... 26,40MP
  42. * * Ha • ¬ Ha ... 41,37Conj.
  43. * ¬ ¬ (∃x)[Gx • (Fx ⊃Ha) ... 27-42IP
  44. * (∃x)[Gx • (Fx ⊃Ha) ... 43DN
  45. [(∃x)(Gx • Fx) ⊃Ha] ⊃[(∃x)[Gx • (Fx ⊃Ha)] ... 26-44CP


  46. [(∃x)[Gx • (Fx ⊃Ha)] ⊃[(∃x)(Gx • Fx) ⊃Ha] • [(∃x)(Gx • Fx) ⊃Ha] ⊃[(∃x)[Gx • (Fx ⊃Ha)] ... 25,45Conj.


  47. (∃x)[Gx • (Fx ⊃Ha)] ≡ [(∃x)(Gx • Fx) ⊃Ha] ... 46BE

Thursday 5 August 2010

The Sting: a cerebral movie

The Sting (directed by Roy Hill, 1973), besides its many other endearing qualities, is a two hour documentary of what happens in the mind, in a fraction of a second, when we are trying to solve a mathematical or logical puzzle. I had an opportunity to convince myself of this recently while watching it again 25 years after I first saw it.

The overall effect of the movie on the viewer is like the moment when the solution presents itself to you after you’ve been staring at an equation or a proof. It is neat and elegant, and seemingly effortless, but between the moment you think you don’t know and the moment you do the brain will have performed dozens of operations. Just how? Well, I can only know retrospectively by writing each step out on paper, and even then I can’t be sure my account is really a reflection of what has happened in my head, or only some kind of simplification.

Those steps, which I write on the right hand side of a proof, are like the title cards in the movie. And, uncannily, the title cards actually correspond to the sort of operations required in a logical proof. The Players (step 1) is a set of premises which I must list, together with the conclusion. The Set-Up (step 2) is the planning: the laying out of a strategy which I will follow. This involves thinking many steps ahead, predicting what will happen, say, on line 17 while I’m still on line 7. It’s a process which requires that many items of information be held in the head simultaneously, without letting go of any of them, while the brain is mapping out the route.

The Hook (step 3) is very much like the assumption that I must make to break up an intractable string of symbols. Like in the movie, what I do at this stage will pay off much later, more handsomely than anything I would have achieved without the assumption. But also like in the movie, I have to discharge the assumption, that is, I can’t have a sentence starting with ‘then’, ‘therefore’, ‘thus’ or ‘so’, without being bound by the ‘if’ of the assumption in the same sentence. In the movie, once Gondorff pulls the first stunt on Lonnegan at the poker table, there is no turning back. He and the boys are now committed to working towards closing the scam.

The Tale (step 4) is basically working restlessly through the material now at hand applying the various rules of inference until a workable formula crystallizes. In the movie, it is the putting together of various parts of the elaborate off-track betting den hoax.

The Wire (step 5) and The Shut-Out (step 6) are unconventional or counterintuitive moves, which may surprise the problem solver himself – something such as bringing an entirely new expression into the proof by means of a disjunction (known as addition) or considering two alternatives (proof by cases) whereby you can assure yourself of the correctness of your reasoning by getting the same result from both alternatives.

Finally, there is The Sting (step 7) – the last title card in the movie – when all the strands come together and all the loose ends are tied up. Beats me.

The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.5E, problem 3b, p. 562

Show that the sentence: (x)[Ax ⊃(Ax ⊃Bx)] ⊃(x)(Ax ⊃Bx), is a theorem of predicate logic.
  1. (x)[Ax ⊃(Ax ⊃Bx)] ⊃(x)(Ax ⊃Bx)
  2. * (x)[Ax ⊃(Ax ⊃Bx)] ... ACP
  3. * * Ax ... ACP
  4. * * Ax ⊃(Ax ⊃Bx) ... 2 UI x/x
  5. * * Ax ⊃Bx ... 3,4 MP
  6. * * Bx ... 3,5 MP
  7. * Ax ⊃Bx ... 3-6 CP
  8. * (x)(Ax ⊃Bx) ... 7UG
  9. (x)[Ax ⊃(Ax ⊃Bx)] ⊃(x)(Ax ⊃Bx) ... 2-8CP

Sunday 1 August 2010

Predicate logic semantics: conditional v universally quantified statements

It is important in logic and mathematics to be able to distinguish between statements which are implications and statements which are universally quantified, because their truth value analysis can produce different results. I claim no special insight into the problem other than an observation that the otherwise flawless explanations that I often come across miss some key element which could help grasp the idea faster.

First, it is a lot easier to work with mathematical concepts in truth value analysis than with everyday English sentences. The reason: fish are aquatic animals, but suppose you were asked the question: ‘Are all fish aquatic animals?’ during a TV game show. Should you suspect a trap? Some people might. We live in an age of knowledge fragmentation where one never sees the full picture, and so can’t be sure of objective truths. The ancients and the medievals, by contrast, may have dwelled on fiction but they sought to consolidate knowledge and, conceivably, were less prone to relativist dilemmas.

Let our universe of discourse, UD, be integers then. Let Fx stand for ‘x is even’ and Gx for ‘x is evenly divisible by 2’. Our two sentences are:

(1) If a number is an even integer, then every integer is evenly divisible by 2.

(x)Fx ⊃ (x)Gx

(2) If a number is an even integer, then it is evenly divisible by 2.

(x)(Fx ⊃ Gx)

Sentence (1) is an example of material implication. Material implication is an example of a compound statement, with its truth value dependent on the logical operator ‘⊃’. In plain English, the two halves of the sentence are independent, or each is capable of taking on its own truth value. We know that sentences like this are false if and only if the consequent, here (x)Gx, is false while the antecedent, here (x)Fx, is true. Sentence (1) is patently false, because not every integer is evenly divisible by 2.

Sentence (2) can be translated to look like a conditional sentence, and in the ordinary sense of the English grammar it certainly is a conditional: For all x, if x is an even integer, then x is evenly divisible by 2. But the scope of the universal quantifier extends over the whole sentence, which means that the ‘x’ in the second half of the sentence has the same reference as the ‘x’ in the first half of the sentence, thus the sentence says in fact: All even integers are evenly divisible by 2. For this to be false, we would need to find only one counterexample, that is, some even integer that is not evenly divisible by 2. I can’t find such a counterexample. The sentence is true.

Sentences (1) and (2) are not equivalent then. However, where a conditional statement and a universally quantified statement are equivalent for some A and B, as in:

(3) (∃x)Ax ⊃ B

(4) (x)(Ax ⊃ B)

the truth values will of course be the same. The difference is in how we choose to express ourselves: (3) is false when B is false and (∃x) Ax is true, and (4) is false when for at least one A, B does not hold, but the fact remains that when (3) is false (4) will be false, and when (3) is true (4) will be true too.