ADDITIONAL TRUTH TREE RULES FOR IDENTITY AND DIVERSITY

Now we turn to rules for dealing with identity and diversity. The rule for identity is the analogue of our rule of inference SI: if there is an identity a = b between two names of an individual on one line, and a statement Φa (or Φb) involving a or b one or more times on some other line in the same path, then one or more occurrences of the first name can be replaced by the other.

(Note that the = rule can be applied again and again, so that it does not get checked off.) Some examples will suffice to demonstrate how to use these rules. First, let’s take a simple one, a proof that every individual is self-identical:

Now let’s prove that the statement -∣Ξx x = x is a contradiction:

Here’s a slightly harder example, the Harper-Trudeau argument from chapter 21:

The only person who can SPEAK for all Canadians is Pierre Trudeau. Stephen Harper, you are no Pierre Trudeau.

which we symbolized as:

Another proof that looks quite forbidding at first, but turns out to be simple, is exercise

20 from chapter 21.

Hesperus is the EVENING star. Phosphorus is the MORNING star. But the Moming

Star is identical with the Evening Star. Therefore Phosphoms and Hesperus are one and the same heavenly body.

This is symbolized:

Here we see that the definite descriptions aspect of the first two statements was not nec­essary for validity.

For a proof involving our = rule, we can take this sequent as a simple example:

Here having a = b on the first line and a = c on the fourth, our = rule licenses us to sub b for a in the latter, giving b = c. As a final example, consider this argument:

Mrs. Peacock was the only person in possession of a CANDLESTICK. Someone with a candlestick was the MURDERER. Logically, therefore, we must conclude that Mrs. Peacock was the murderer. [UD: people]

Using the truth tree method determine whether each of the following arguments (34-39) is formally valid or invalid:

34. The only EVEN prime is two. Therefore there is an even prime. [UD: prime numbers; Ex := X is even]

35. The Pope is speaking LATIN. Therefore there is someone speaking Latin and he is the Pope.

36. It is not true that no famous AUTHORS like LOGIC. Charles Dodgson certainly likes logic. And Lewis Carroll is a famous author. But Lewis Carroll and Charles Dodgson are one and the same person.

37. The only SURVIVING Beatles are Paul and Ringo. Neither Paul nor Ringo is inter­ested in INDIAN philosophy. So there is no one among the surviving Beatles with an interest in Indian philosophy.

38. The PRINCE of Wales is BALDING. So Charles must be balding, as he is Prince of Wales.

39. Descartes can’t be a SOLIPSIST, because he’s not me. I am one, and there is only one solipsist.

40. (CHALLENGE) Prove that an asymmetric relation cannot be totally connexive.

41. (CHALLENGE) Prove that if two individuals are related by both simple Connexivity and nonconnexivity, then they are identical.

42. (CHALLENGE) The following is a news report from the National Post (October 27, 2000):

On Wednesday, JimFlaherty, the Ontario Attorney-General, contacted officials at the federal immigration and justice departments to see whether Eminem could be barred from the country. However, Derik Hodgson, a spokesman for the Ministry

of Immigration, said officials found no reason to block Eminem, who takes his “Anger Management Tour” to Montreal’s Molson Centre tonight. “We aren’t the thought police,” said Mr. Hodgson, adding “if all people who made bad music were kept out of Canada, we could have stopped disco.”

[Bx := X makes bad music, Kx := x could be kept out of Canada, Dx := x could have stopped disco, m := the Ministry of Immigration, e := Eminem]

What is Hodgson's argument? Supplying the implicit premises “If some people who make bad music could not be kept out of the country, then Eminem could not be kept out,” and “The Ministry could not have stopped disco,” and using the dictionary pro­vided, determine the conclusion of the argument using a truth tree.

Chapter Twenty-Four