Thursday, 29 October 2009

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.

Saturday, 24 October 2009

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

Sunday, 11 October 2009

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.

Wednesday, 7 October 2009

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