Thursday 24 June 2010

The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, Ex. 10.5E, 4(f), p. 563

Show that the following sentences are equivalent. Solution: if they are equivalent, then we can derive one from the other.
(x)[Ax • (∃y) ¬ Bxy] is equivalent to ¬ (∃x)[¬ Ax ∨(y)(Bxy • Bxy)]
  1. (x)[Ax • (∃y) ¬ Bxy]
  2. * ¬ ¬ (∃x)[¬ Ax ∨(y)(Bxy • Bxy)] / AIP
  3. * (∃x)[¬ Ax ∨(y)(Bxy • Bxy)] / 2DN
  4. * ¬ Aa ∨(y)(Bay • Bay) / 3EI x/a
  5. * Aa • (∃y) ¬ Bay / 1UI x/a
  6. * Aa / 5Simp.
  7. * (y)(Bay • Bay) / 4,6DS
  8. * (∃y) ¬ Bay / 5Simp.
  9. * ¬ Bam / 8EI y/m
  10. * Bam • Bam / 7UI y/m
  11. * Bam / 10Taut.
  12. * ¬ Bam • Bam / 9,11Conj.
  13. * ¬ ¬ ¬ (∃x)[¬ Ax ∨(y)(Bxy • Bxy)] / 2-12IP
  14. * ¬ (∃x)[¬ Ax ∨(y)(Bxy • Bxy)] / 13DN

We have shown that the second sentence is derivable from the first by indirect proof.

  1. ¬ (∃x)[¬ Ax ∨(y)(Bxy • Bxy)]
  2. (x)[Ax • ¬ (y)(Bxy • Bxy)] / 1QC
  3. Ax • ¬ (y)(Bxy • Bxy) / 2UI x/x
  4. Ax / 3Simp.
  5. ¬ (y)(Bxy • Bxy) / 3Simp.
  6. (∃y)(¬ Bxy ∨¬ Bxy) / 5DeM
  7. ¬ Bxm ∨¬ Bxm / 6EI y/m
  8. ¬ Bxm / 7Taut.
  9. (∃y)¬ Bxy / 8EG
  10. Ax • (∃y)¬ Bxy / 4,9Conj.
  11. (x)[Ax • (∃y)¬ Bxy ] / 10UG

The first sentence is also derivable from the second. So, the two sentences are equivalent.

Thursday 17 June 2010

True negatives

We routinely deal in logic with sentences which are true because they are negations of sentences which are false and sentences which are false because they affirm the negation of sentences which are true. In other words: not p is true if and only if p is false, and p is false if and only if not p is true.

Natural language also offers this possibility, but I haven’t made up my mind yet whether the motives for doing so are deviousness, showing off one’s verbal dexterity, or plain sloppiness. Unlike logic, English allows us to say things the way we want to, not the way we need to.

Something called an infoscreen beamed a public service announcement to weary commuters at an underground station today: ‘Latest scientific research shows that sweets negatively affect cancer of the pancreas.’ Are we to make of this that sweets halt the growth of pancreatic cancer or promote it?

Making a few replacements, we get this sentence: ‘Drought negatively affects plant growth.’ Clearly this means the slowing down of plant growth. On this logic, the announcement would seem to be encouraging people to eat sweets. But we know different, don’t we? So the reasoning should be this: plant growth is generally a good thing; cancer is a bad thing. Thus, ‘to negatively affect cancer’ means ‘to stimulate the growth of cancerous cells.’ But would you trust this kind of language if your life depended on it?

The context where these constructions are prevalent is announcements of changes in legislation and voting in general. It is common to hear about MPs approving the dropping of a proposal to send more troops to Afghanistan or, closer to home, voting in favour of rejecting the motion to withhold approval for a member of the board of my commonholders’ association. Speak in such terms to flat owners, and you will have a lot of abstentions in the room.

From another quarter, if you have the stated medical condition, your test comes back positive. If you are clear, the test is negative. Strictly speaking, we should say ‘if you test positive, you have the infection,’ because the infection is a necessary condition for the test being positive, not the other way round, but this is beside the point. The point is it is good to test negative and bad to test positive.

The logic here is easy enough to follow: if a lab technician is testing for a particular condition and fails to find it, the test result is negative, and it is good news for us. Trouble is our everyday language is elliptical and we are just as likely to say: ‘If you test negative, you don’t have the stated medical condition.’ This takes us away from the original sentence and is bound to make some people confused.

A professor I once knew would have put it down to postmodernism. Like many other endeavours in our life, our communication is also postmodern. The tape in a tape player rolls in the opposite direction to the direction of the arrows on the fastforward / rewind buttons, we turn left on a motorway in order to go right, and ticking a box means you choose not to receive advertising material rather than asking for it.

The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, Ex. 10E, 5(f), p. 563

The following set of sentences is inconsistent. Show that it is so. Strategy: we show that the original assumptions lead to a contradiction on line 16.
  1. (x)[(Sx • Bxx) ⊃ Kax] / Assumption
  2. (x)(Hx ⊃Bxx) / Assumption
  3. (∃x)(Sx • Hx) / Assumption
  4. (x) ¬ (Kax • Hx) / Assumption
  5. Sm • Hm / 3EI x/m
  6. Hm / 5Simp.
  7. Hm ⊃Bmm / 2UI x/m
  8. Bmm / 6,7MP
  9. Sm / 5Simp.
  10. Sm • Bmm / 8,9Conj.
  11. (Sm • Bmm) ⊃ Kam / 1UI x/m
  12. Kam / 10,11MP
  13. ¬ (Kam • Hm) / 4UI x/m
  14. ¬ Kam ∨ ¬ Hm / 13DeM
  15. ¬ Hm / 12,14DS
  16. Hm • ¬ Hm / 6,15Conj.

Thursday 10 June 2010

Understanding Symbolic Logic, Virginia Klenk, Pearson Prentice Hall, 2008, 5th edition, Unit 18, Ex. 1(p), p. 354

Construct a proof for the following argument. Tip: we can't generalize by universal generalization within the scope of an assumption if the assumption contains a free variable on the first line, but we are within our rights to do so by existential generalization (line 20). Then, it is just the question of pushing the negation out, which changes the quantifier to a universal one (line 21).


  1. (∃x)[Px • (y)(Sy ⊃Txy)]
  2. (x){[Px • ¬ (y)(Wy ⊃Ayx)] ⊃(z)(Bxz ⊃Sz)}
  3. (x){Px ⊃¬ (∃y)(Wy • (Txy ∨ Ayx)]}
  4. ∴(∃x){Px • (y)[Wy ⊃¬ (Ayx ∨Bxy)]}
  5. Pa • (y)(Sy ⊃Tay) / 1EI x/a
  6. Pa ⊃¬ (∃y)(Wy • (Tay ∨ Aya) / 3UI x/a
  7. Pa / 5Simp.
  8. ¬ (∃y)(Wy • (Tay ∨ Aya) / 7,6MP
  9. (y)[Wy ⊃¬ (Tay ∨ Aya)] / 8QC
  10. * Wy / ACP
  11. * Wy ⊃¬ (Tay ∨ Aya) / 9UI y/y
  12. * ¬ (Tay ∨ Aya) / 10,11MP
  13. * ¬ Tay • ¬ Aya / 12DeM
  14. * (y)(Sy ⊃Tay) / 5Simp.
  15. * Sy ⊃Tay / 14UI y/y
  16. * ¬ Tay / 13Simp.
  17. * ¬ Sy / 16,15MT
  18. * ¬ Aya / 13Simp.
  19. * Wy • ¬ Aya / 10,18Conj.
  20. * (∃y)(Wy • ¬ Aya) / 19EG
  21. * ¬ (y)(Wy ⊃Aya) / 20QC
  22. * Pa • ¬ (y)(Wy ⊃Aya) / 7,21Conj.
  23. * [Pa • ¬ (y)(Wy ⊃Aya)] ⊃(z)(Baz ⊃Sz) / 2UI x/a
  24. * (z)(Baz ⊃Sz) / 22,23MP
  25. * Bay ⊃Sy / 24UI z/y
  26. * ¬ Bay / 17,25MT
  27. * ¬ Bay • ¬ Aya / 26,18Conj.
  28. * ¬ (Bay ∨ Aya) / 27DeM
  29. Wy ⊃ ¬ (Bay ∨ Aya) / 10-28CP
  30. (y)[Wy ⊃¬ (Aya ∨Bay)] / 29UG
  31. Pa • (y)[Wy ⊃¬ (Aya ∨Bay)] / 7,30Conj.
  32. (∃x){Px • (y)[Wy ⊃¬ (Ayx ∨Bxy)]} / 31EG

Saturday 5 June 2010

I'm a zip learner

It is sometimes claimed that we can hold up to 9 objects or numbers in our mind’s eye without the sequence coming unraveled at either end. Nine is really pushing the limits for me, but I can just about get there if I arrange the numbers or objects three across by three down. Then, the addition of an extra item on the fourth line draws the eye to it and the mind lets go of the rest.

This ability is clearly not the same as memory although people who claim to be visual learners might have an advantage over the rest of us. Are there any auditory learners by admission? I haven’t met any. Nor have I met anyone who claimed to be a kinesthetic learner.

I have long been skeptical about the self-professed ability to remember information more easily according as we do so by means of our eyes or by means of our ears. The visual auditory distinction may have more to do with the kinds of habits we’ve been cultivating than with memory itself. And the kind of habit that the western man has been cultivating for the last few hundred years is that of finding things on a piece of paper, lately on a screen.

The ability to remember varies directly as the ability to process. Suppose a voice recorder from a fatal flight was found. That voice recorder would be rightly called a cockpit voice recorder.

A cockpit voice recorder matters to the extent it contains some data about the last minutes of the flight. Hence, data from the cockpit voice recorder.

Suppose that data was analysed. We could then talk about the analysis of data from the cockpit voice recorder but not data analysis from the cockpit voice recorder. We could lump the words together and have: cockpit voice recorder data analysis.

Suppose the analysis was mishandled, manipulated or otherwise skewed. While mishandling of analysis of data from the cockpit voice recorder sounds OK, a headline would scream: ‘Cockpit voice recorder data analysis bungle’. Where does the string of nouns stop?

Suppose the bungling of the analysis of data from the cockpit voice recorder was then covered up by people who had decided long before the analysis was carried out whom to blame and whom to exonerate. The headline would then run: ‘Cockpit voice recorder data analysis bungle cover-up’. I can hold this information in my mind without much effort, not because I remember the order in which the words appear in the sequence but because I can process it into something that makes sense, though, admittedly, I’d rather have it written than spoken to me. The important thing is that these are not random words. We rely on the relations between the atomic concepts for our understanding.

Suppose the cover-up was investigated. Might it not be the case that our headline would now say: ‘Cockpit voice recorder data analysis bungle cover-up probe’?

And what if the investigation into the cover-up of the whole affair itself was a failure? We could compose a suitable line: ‘Cockpit voice recorder data analysis bungle cover-up probe fiasco.’ English is compositional that way, and if understanding doesn’t break down, what is to stop me?

Suppose an independent enquiry was ordered into what was now a failed attempt to discover the truth. That could be heralded as: ‘Cockpit voice recorder data analysis bungle cover-up probe fiasco enquiry.’ Who said that English is a verbal language?

And suppose that that enquiry didn’t get anywhere. We’d see a headline: ‘Cockpit voice recorder data analysis bungle cover-up probe fiasco enquiry botched.’

I can’t vouch for others, but what I feel happens in my mind is that the whole description before ‘enquiry’ clusters around ‘enquiry’ in a shapeless mass, leaving no more than a trace of its content in my memory. It has been zipped up into a tidy bundle. I can’t easily repeat the words nor form a seamless sentence but if someone was to quiz me on the cabin voice recorder or unbiased analysis, I’d know I was being contradicted. I zip things up and file them away.

Wednesday 2 June 2010

The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 4th edition, Exercise 10.5E, 5(d)

Show that the following set of sentences is inconsistent. And to do so, we show that we can derive a sentence and its negation within the set of primary assumptions.
  1. (∃x)(y)[Hxy ⊃ (w)Jww] / Assumption
  2. (∃x) ¬ Jxx • ¬ (∃x) ¬ Hxm / Assumption
  3. (∃x) ¬ Jxx / 2Simp.
  4. ¬ Jaa / 3EI x/a
  5. (∃w) ¬ Jww / 4EG
  6. ¬ (w)Jww / 5CQ
  7. (y)[Hcm ⊃ (w)Jww] / 1EI x/c
  8. Hcm ⊃ (w)Jww / UI y/s
  9. ¬ Hcm / 6,8MT
  10. (∃x) ¬ Hxm / 8EG
  11. ¬ (x)Hxm / 10CQ
  12. ¬ (∃x) ¬ Hxm / 2Simp.
  13. ¬ ¬ (x) Hxm / 12 CQ
  14. (x)Hxm / 13DN
  15. ¬ (x) Hxm / 10CQ
  16. ¬ (x) Hxm • (x)Hxm / 14,15Conj.