Only P if Q v P only if Q

Am I thick or are they doing it on purpose? Here is a sentence from a case brief summarizing a court’s ruling which I was challenged to explain this week:

The recent Court of Appeal case of Sagal v Atelier Bunz GmbH held that an agent with the authority to contract (as opposed to simply the authority to negotiate) is only a commercial agent for the purposes of the Commercial Agents Regulations 1993 if they contract in the name of the principal.

The ambiguity is captured by the following pair of sentences:

(1) A person is only a grease monkey if he has failed school.
(2) A person is a grease monkey only if he has failed school.

Sentences (1) and (2) present two non-equivalent readings. Sentence (1) is false just in case a person is not a grease monkey having earlier failed school. Sentence (2) is false just in case a person is a grease monkey having earlier finished school, and true otherwise.

The position of the adverb ‘only’ in a sentence does make a difference then. In the first sentence ‘only’ modifies the noun, suggesting that being a grease monkey is a lowly job. In the second sentence ‘only’ is part of the expression ‘only if’, which introduces a necessary condition. This means that one doesn’t become a grease monkey without failing school. Another way of looking at sentence (2) is this: A person is not a grease monkey unless he has failed school. Paraphrased again, the sentence says: If a person has not failed school, he is not a grease monkey. Stripped down to Boolean grammar, sentence (2) is equivalent to: Either a person has failed school or he is not a grease monkey. The last paraphrase may sound too radical too many speakers, but the meaning is preserved.

The original sentence then seems to imply that, if one contracts in the name of the principal, one is a commercial agent as opposed to some ‘other’ kind of agent who enjoys a higher status or more privileges.

This is not what was intended though, as we learn from the rest of the case summary. Mr Sagal was seeking to fall within the definition of a commercial agent precisely because commercial agents can claim compensation or indemnity on termination of the agency. The court looked at the evidence and found that, on balance, Mr Sagal was trading in his own name, denying him the status of a commercial agent.

To round off the point, let us return to the sentences about a grease monkey, but let us rewrite them like this instead:

(3) You will be a grease monkey if you fail school.
(4) You will have failed school if you are a grease monkey.

Again, sentence (3) is false when the antecedent (you fail school) is true and the consequent (you will be a grease monkey) is false. Under the same circumstances sentence (4) is true (false antecedent, true consequent). Suppose ‘you are a grease monkey’ is true and ‘you failed school’ is true. Suppose also that ‘you failed school’ is false and it is equally false that ‘you are a grease monkey’. What then? Then we have a biconditional, or material equivalence:

(5) You will be a grease monkey if and only if you fail school.

- a statement which is true when antecedent and consequent have the same truth value: both true or both false.

Symbolic Logic, Irving M. Copi, Prentice Hall, 1979, 5th edition, p.150, problem 3

Task: Prove the validity of the following argument. In the glossary that Copi provides I have changed the predicate letter Sxy - 'x is smaller than y' into Lxy, just to avoid confusion with Sx - 'x is a state'. The other letters remain unchanged: Nx - 'x is in New England', Ix - 'x is primarily industrial'. The argument is:

The smallest state is in New England. All states in New England are primarily industrial. Therefore, the smallest state is primarily industrial.

1. (∃x){Sx • (y){[Sy • ¬ (y = x)] ⊃Lxy} • Nx}
2. (x)[(Sx • Nx) ⊃Ix]
3. ∴(∃x){Sx • (y){[Sy • ¬ (y = x)] ⊃Lxy} • Ix}
4. Sa • (y){[Sy • ¬ (y = a)] ⊃Lay} • Na / 1EI x/a
5. Sa / 4Simp.
6. Na / 4Simp.
7. (y){[Sy • ¬ (y = a)] ⊃Lay} / 4Simp.
8. (Sa • Na) ⊃Ia / 2UI x/a
9. Sa • Na / 5,6Conj.
10. Ia / 9,8MP
11. Sa • (y){[Sy • ¬ (y = a)] ⊃Lay} • Ia / 5,7,10Conj.
12. (∃x){Sx • (y){[Sy • ¬ (y = x)] ⊃Lxy} • Ix} / 11EG

Symbolic Logic, Irving M. Copi, Prentice Hall, 5th edition, 1979, p. 149, problem 1

We are asked to prove the validity of the following argument:

The architect who designed Tappan Hall designs only office buildings. Therefore, Tappan Hall is an office building.

The proof is straightforward. The thing to watch out for in symbolizing the premise is 'only'. 'he designs only office buildings' translates as 'if z is not an office building then he did not desing it' or, by contraposition, 'if he designs it then it is an office building'.

1. (∃x){Ax • Dxt • (y)[(Ay • Dyt) ⊃y = x] • (z)(Dxz ⊃Oz)}
2. ∴Ot
3. Am • Dmt • (y)[(Ay • Dyt) ⊃y = m] • (z)(Dmz ⊃Oz)} / 1EI x/m
4. Dmt / 3Simp.
5. (z)(Dmz ⊃Oz) / 3Simp.
6. Dmt ⊃Ot / 5UI z/t
7. Ot / 4,6MP

Deduction, Daniel Bonevac, Blackwell Publishing, 2nd edition, 2003, Ex. 8.3, problem 13

As elsewhere in this set of problems, the task is to show that the formula (∃x)(y)(x = y ≡ Gy) implies the conclusion of the argument reproduced below. Since the conclusion itself is a biconditional, we show first that the formula on the left implies the formula on the right, and then the reverse. We are free to use the same constants from line 18 on as in the first assumption because that assumption has been discharged on line 17.

1. (∃x)(y)(x = y ≡ Gy)
2. ∴(∃x)(Gx • Fxx) ≡ (∃x)(∃y)(Gx • Gy • Fxy)
3. * (∃x)(Gx • Fxx) / ACP
4. * Ga • Faa / 3EI x/a
5. * (y)(h = y ≡ Gy) / 1EI x/h
6. * h = a ≡ Ga / 5UI y/a
7. * (h = a ⊃ Ga) • (Ga ⊃h = a) / 6BE
8. * Ga / 4Simp.
9. * Ga ⊃h = a / 8Simp.
10. * h = a / 8,9MP
11. * Faa / 4Simp.
12. * Fah / 10,11Id
13. * Gh / 10,8Id
14. * Ga • Gh •Fah / 9,12,13Conj.
15. * (∃y)(Ga • Gy • Fay) / 14EG
16. * (∃x)(∃y)(Gx • Gy • Fxy) / 15EG
17. (∃x)(Gx • Fxx) ⊃(∃x)(∃y)(Gx • Gy • Fxy) / 3-16CP
18. * (∃x)(∃y)(Gx • Gy • Fxy) / ACP
19. * (∃y)(Ga • Gy • Fay) / 18EI x/a
20. * Ga • Gm • Fam / 19EI y/m
21. * (y)(h = y ≡ Gy) / 1EI x/h
22. * h = a ≡ Ga / 21UI y/a
23. * (h = a ⊃ Ga) • (Ga ⊃h = a) / 22BE
24. * Ga ⊃h = a / 23Simp.
25. * Ga / 20Simp.
26. * h = a / 24,25MP
27. * h = m ≡ Gm / 21UI y/m
28. * (h = m ⊃ Gm) • (Gm ⊃h = m) / 27BE
29. * Gm / 20Simp.
30. * Gm ⊃h = m / 28Simp.
31. * h = m / 29,30MP
32. * m = h / 31Id
33. * m = a / 32,26Id
34. * Fam / 20Simp.
35. * Faa / 33,34Id
36. * Ga • Faa / 25,35Conj.
37. * (∃x)(Gx • Fxx) / 36EG
38. (∃x)(∃y)(Gx • Gy • Fxy) ⊃ (∃x)(Gx • Fxx) / 18-37CP
39. [(∃x)(Gx • Fxx) ⊃(∃x)(∃y)(Gx • Gy • Fxy)] • [(∃x)(∃y)(Gx • Gy • Fxy) ⊃ (∃x)(Gx • Fxx)] / 17,38Conj.
40. (∃x)(Gx • Fxx) ≡ (∃x)(∃y)(Gx • Gy • Fxy) / 39BE

Which way is forward?

When the clocks change I always entertain myself for a few days with the idea of consistency as we apply it in our efforts to describe what is actually happening to time. Consistency is at the heart of logic. Strictly speaking, a set of statements is consistent if and only if it is possible for all the statements to be true.

Assigning values of true or false takes place against a set of criteria, and in order to make sense of our comparisons we assign the same criteria each time we compare similar things. If my criterion for ‘bright light’ is ‘sufficient to illuminate my desk to work at’, then it is false that a candle is bright and true that a 100W bulb is bright. Change the criteria by substituting ‘football pitch’ for ‘my desk’, and both sentences are false.

What are we to make of the following English sentences then?

The clocks go forward in the spring by an hour.
We’ve brought the meeting forward by an hour.

Does ‘forward’ suggest a movement from an earlier to a later time or from a later time to an earlier time? How important is our perceived position relative to the change? I can best describe what I do when I advance the hands of my clocks as ‘turning them forward’, so that what was 7am is now 8am. By this logic the first sentence is true. By the same logic the second sentence is false, because in bringing the meeting forward we are moving from a later time to an earlier time.

There are a number of inconsistencies like this in English, many of them only apparent. This problem, for example, is only superficially similar to that of ‘an alarm going off’, where, if one were to choose a sense of termination or completion as a criterion for ‘off’, or the shutting off of operation of electric devices, one would be led to believe that the alarm has just stopped ringing. The facts are that the verb ‘to go off’ is much older than commercial electricity and that it was traditionally used to describe bursts of energy, explosions, and so on.

Coming back to ‘forward’ and ‘back’ (consider that to move a meeting back by an hour is to postpone it by that amount of time), I have aged by an hour by putting my clocks forward. Fortunately this loss will be reversed in the autumn, but is aging itself a movement forward or back? On McTaggart’s time line, whereby an event which is now present, was future, and will be past, aging is a backward movement. Or is it something to do with another one of our dearly held conventions, namely that the forward movement is a movement from left to right?

In an item heard last week scientists have tested our intuitions again by endowing ‘forward’ with the meaning it has in the sentence about rescheduling meetings rather than changing the clocks:

The seasons are moving forward across the northern hemisphere.

Deduction, Daniel Bonevac, Blackwell Publishing, 2nd edition, 2003, Ex. 8.3, problem 12

Show that the given statement is a consequence of a formula. To show that it is so, we construct an argument with the first statement as a premise and the second as a conclusion. Two things worth noting about this proof. One: 'y' from line 4 is instantiated to 'a' on line 5 by universal instantiation even though, or perhaps because, we have already instantiated into 'a' by existential instantiation on line 4. Two: this allows us to set up a biconditional which can be easily simplified and the remainder dispatched by Modus Ponens after having first introduced the identity a = a.

1. (∃x)(y)(y = x ≡ Gy)
2. ∴(x)Fx ⊃(∃x)(Gx • Fx)
3. * (x)Fx / ACP
4. * (y)(y = a ≡ Gy) / 1EI x/a
5. * a = a ≡ Ga / 4UI y/a
6. * (a = a ⊃Ga) • (Ga ⊃a = a) / 5BE
7. * a = a ⊃Ga / 6Simp.
8. * a = a / Id
9. * Ga / 8,7MP
10. * Fa / 3UI x/a
11. * Ga • Fa / 9,10Conj.
12. * (∃x)(Gx • Fx) / 11EG
13. (x)Fx ⊃(∃x)(Gx • Fx) / 3-12CP