Oh, a break! A logic puzzle

1
Oh, a break! A logic puzzle

• In a mythical (?) community, politicians always
  lie and non-politicians always tell the truth. A
  stranger meets 3 natives.
• She asks the first native if he is a politician.
  He answers.
• The second native states that the first denied
  being a politician.
• The third native says that the first native is a
  politician.
• How many of these natives are politicians?

2

• Possible solutions?
• None, one, two, or three.
• She asks the first native if he is a politician.
  He answers.
• What might he have answered?
• Could he answer No?
• Could he answer Yes?

3

• Given what weve found out, have we learned
  anything about the second native, who said
• The first native denied being a politician.
• Yes she is telling the truth, and thus not a
  politician.
• So far, then, we know there are at most two
  politicians.

4

• What about the third native, who said the first
  native is a politician?
• What are the possibilities?
• Hes telling the truth.
• Hes lying.
• Can we tell which?
• Does it matter?

5

• Does it matter?
• If hes lying, hes a politician and the first
  native is not.
• If hes telling the truth, then hes not a
  politician and the first native is.
• So, what we know is that either the first native
  or the third is lying, and that the other is
  telling the truth.
• So, we know that there is one, and only one,
  politician.

6
The syntax of SL

• Defining logical notions (validity, logical
  equivalence, and so forth) in terms of
  derivability
• A derivation a finite number of steps, based on
  the rules of SD, that demonstrates that some
  sentence of SL can be derived from some other
  sentence of SL or set of sentences of SL
  (including the empty set), using the derivation
  rules of SD.
• Like the truth table method, derivations are an
  effective method for demonstrating logical
  status.
• SD the derivation system.
• 11 rules for each connective, one rule to
  introduce it and one rule to eliminate it, plus
  reiteration.

7
The syntax of SL

• Defining logical notions (validity, logical
  equivalence, and so forth) in terms of
  derivability (and in this case in the system SD.
• Examples
• An argument is valid in SD IFF the conclusion can
  be derived from the premises in SD.
• A sentence is a theorem in SD IFF it can be
  derived from the empty set.
• The only notions not carried over are logical
  falsehood and logical indeterminacy.

8
Derivation conventions and rules

• Derivations always include one scope line (a
  vertical line). This indicates what follows (what
  sentence is derivable using SD) from another
  because each falls within the scope of that line.
• Each line in a derivation is numbered.
• If the derivation includes primary assumptions,
  these form the first rows and are followed by a
  horizontal line.

9
Derivation conventions and rules

• Every line of a derivation must be justified it
  must either be a primary assumption (and noted as
  such) an auxiliary assumption when the rule calls
  for one, and noted as such, and/or a sentence for
  which the rule and line numbers from which it is
  derived must be cited. Justifications are noted
  to the right of each line.
• The single turnstile ? is used to symbolize
  derivability.

10
Derivation conventions and rules

• Every line of a derivation must be justified it
  must either be a primary assumption (and noted as
  such) or an auxiliary assumption when the rule
  calls for one
• 4 rules require subderivations, which in turn
  require a new scope line and an auxiliary
  assumption.
• All subderivations must be discharged in the way
  they dictate and at the main scope line.

11
Derivations Show that A ? C, A B ? C

• A ? C A
• A B A
• --------------
• A
• C ? E

12
Derivations Show that A ? C, A B ? C

• A ? C A
• A B A
• --------------
• A 2 E
• C 1, 3 ? E

13
Rules of SDYou have been introduced to R,
Reiteration

• P
• P R
• This rule is used in derivations that involve a
  subderivation

14
Rules of SDYou have been introduced to E

• P Q
• P E

15
Rules of SDYou have been introduced to I

• P
• Q
• PQ I

16
Rules of SDYou have been introduced to ? E

• P ? Q
• P
• Q ? E

17
Rules of SD

• Derive C
• (A B) ? C A
• A A
• B A
• ---------------------
• A B
• C ?E

18
Rules of SD

• Derive C
• (A B) ? C A
• A A
• B A
• ---------------------
• A B 2, 3 I
• C 1, 4 ? E

19
Rules of SDHere is the rule ? I

• P
• ____
• Q
• P ? Q ?I
• Remember all subderivations must be discharged
  in exactly the way allowed by a rule!

20
Derivation strategies

• Derive A ? C
• 1. A ? B A
• 2. B ? C A
• ---- -----
• A A
• ----
• 4. B ? E
• 5. C ? E
• 6. A ? C ? i

21
Derivation strategies

• Derive A ? C
• 1. A ? B A
• 2. B ? C A
• ---- -----
• A A
• ---
• 4. B 1, 3 ? E
• 5. C 2, 4 ? E
• 6. A ? C 3-5 ? i

22
Rules of SD vI

• P or P
• P v Q Q v P

23
Rules of SD vIDerive G v H

• B ? G A
• C B A
• ------------
• B
• G
• G v H vI

24
Rules of SD vIDerive G v H

• B ? G A
• C B A
• ------------
• B 2 E
• G 1, 3 ?E
• G v H 4 vI

25
Rules of SD vE

• P v Q
• P
• ---
• R
• Q
• ---
• R
• R

26
Rules of SD vE

• Derive H
• G v H A
• G ? H A
• H ? H A
• ---------
• G A
• ---
• H
• H A
• ---
• H
• H vE

27
Rules of SD vE

• Derive H
• G v H A
• G ? H A
• H ? H A
• ---------
• G A
• ---
• H 2, 4 ?E
• H A
• ---
• H 6 R
• H 4-5, 6-7 vE

28
Rules of SD ?E

• P ? Q OR P ? Q
• P Q
• Q P ? E

29
Rules of SD ? EDerive M B

• C ? M A
• C B A
• -------------
• C E
• M ?E
• B E
• M B I

30
Rules of SD ? EDerive M B

• C ? M A
• C B A
• -------------
• C 2 E
• M 1, 3 ?E
• B 2 E
• M B 4, 5 I

31
Rules of SD ? i

• P
• ---
• Q
• Q
• ---
• P
• P ? Q

32
Rules of SD ? I

• Derive C ? D
• C ? D A
• D ? C A
• -----------
• 3. C A
• ----
• 4. D
• 5. D A
• ----
• 6. C
• C ? D ?I

33
Rules of SD ? I

• Derive C ? D
• C ? D A
• D ? C A
• -----------
• 3. C A
• ----
• 4. D 1, 3 ?E
• 5. D A
• ----
• 6. C 2, 5 ?E
• 7. C ? D 3-4, 5-6 I

34
Rules of SD I

• P
• ----
• Q
• Q
• P I
• Both rules make use of recductio ad absurdum

35
Rules of SD I

• Derive G
• 1. G ? C A
• 2. C B A
• ----------
• 3. G A
• --------
• 4. C
• C
• G I

36
Rules of SD I

• Derive G
• 1. G ? C A
• 2. C B A
• ----------
• 3. G A
• --------
• 4. C 1, 3 ?E
• C 2 E
• G 3-5 I

37
Rules of SD E

• P
• ----
• Q
• Q
• P E

38
Rules of SD E

• Derive A B
• 1. (A B) ? C A
• A ? C A
• A A
• ---------------
• (A B)
• ------------
• C
• C
• A B E

39
Rules of SD E

• Derive A B
• 1. (A B) ? C A
• A ? C A
• A A
• ---------------
• (A B) A
• ------------
• C 1, 4, ? E
• C 2, 3 ? E
• A B 4-6 E