Oh, a break! A logic puzzleOh, 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?

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”?

Given what we’ve 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.

What about the third native, who said “the first native is a politician”?

What are the possibilities? He’s telling the truth. He’s lying.

Can we tell which?

Does it matter?

Does it matter?

If he’s lying, he’s a politician and the first native is not.

If he’s telling the truth, then he’s 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.

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.

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.

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.

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.

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.

Derivations Show that {A ~C, A & B} ⊦ ~C

1.1. A ~C A2. A & B A --------------3. A4. ~C E

Derivations Show that {A ~C, A & B} ⊦ ~C

1.1. A ~C A2. A & B A --------------3. A 2 &E4. ~C 1, 3 E

Rules of SDYou have been introduced to R, Reiteration

PP

PP RR

This rule is used in derivations that involve a This rule is used in derivations that involve a subderivation…subderivation…

Rules of SDYou have been introduced to &E

P & QP & Q

PP &E&E

Rules of SDYou have been introduced to &I

PP QQ

P&Q P&Q &I&I

Rules of SDYou have been introduced to E

P P Q Q PP

Q Q E

Rules of SD

Derive CDerive C1.1. (A & B) (A & B) C C AA2.2. AA AA3.3. BB AA ------------------------------------------4.4. A & BA & B5.5. CC EE

Rules of SD

Derive CDerive C1.1. (A & B) (A & B) C C AA2.2. AA AA3.3. BB AA ------------------------------------------4.4. A & BA & B 2, 3 &I2, 3 &I5.5. CC 1, 4 1, 4 E E

Rules of SDHere is the rule I

P P ________

Q Q P P Q Q IIRemember: all subderivations must be discharged Remember: all subderivations must be discharged

in in exactlyexactly the way allowed by a rule! the way allowed by a rule!

Derivation strategies

Derive A Derive A C C1. A 1. A B B AA2. B 2. B C C AA ---- --------- -----3.3. AA AA --------4.4. B B E E5. C5. C E E6. A 6. A C C i i

Derivation strategies

Derive A Derive A C C1. A 1. A B B AA2. B 2. B C C AA ---- --------- -----3.3. AA AA4.4. ------4.4. B B 1, 3 1, 3 E E5. C5. C 2, 4 2, 4 E E6. A 6. A C 3-5 C 3-5 i i

Rules of SD: vI

P or P or PP

P v QP v Q Q v PQ v P

Rules of SD: vIDerive G v H

1.1. B B G G AA2.2. C & B C & B AA ------------------------3.3. BB4.4. GG5.5. G v HG v H vIvI

Rules of SD: vIDerive G v H

1.1. B B G G AA2.2. C & B C & B AA ------------------------3.3. BB 2 &E2 &E4.4. GG 1, 3 1, 3 EE5.5. G v HG v H 4 vI4 vI

Rules of SD: vE

P v QP v Q PP

------RR

QQ------RR

R R

Rules of SD: vE

Derive HDerive H1.1. G v HG v H AA2.2. G G H H AA3.3. H H H H AA

------------------4.4. GG AA

------HH

HH AA------HH

H H vEvE

Rules of SD: vE

Derive HDerive H1.1. G v HG v H AA2.2. G G H H AA3.3. H H H H AA

------------------4.4. GG AA

------5.5. HH 2, 4 2, 4 EE

6.6. HH AA

------7.7. HH 6 R6 R8.8. H H 4-5, 6-7 vE4-5, 6-7 vE

Rules of SD: E

P P Q Q OROR P P Q Q PP QQ

Q Q P P E

Rules of SD: EDerive ~M & B

1.1. C C ~M ~M AA2.2. C & B C & B AA --------------------------3.3. CC &E&E4.4. ~M~M EE5.5. BB &E&E6.6. ~M & B~M & B &I&I

Rules of SD: EDerive ~M & B

1.1. C C ~M ~M AA2.2. C & B C & B AA --------------------------3.3. CC 2 &E2 &E4.4. ~M~M 1, 3 1, 3 EE5.5. BB 2 &E2 &E6.6. ~M & B~M & B 4, 5 &I4, 5 &I

Rules of SD: Rules of SD: i i

PP ------ QQ

QQ ------

PP P P Q Q

Rules of SD: Rules of SD: I IDerive C Derive C D D

1.1. C C D D AA2.2. D D C C AA

----------------------3. C 3. C AA

--------4.4. D D

5. 5. D D AA --------

6.6. C C7.7. C C D D II

Rules of SD: Rules of SD: I IDerive C Derive C D D

1.1. C C D D AA2.2. D D C C AA

----------------------3. C 3. C AA

--------4.4. D D 1, 3 1, 3 EE

5. 5. D D AA --------

6.6. C C 2, 5 2, 5 EE7. 7. C C D D 3-4, 5-6 I3-4, 5-6 I

Rules of SD: ~IRules of SD: ~I

PP --------

QQ~Q~Q

~P ~P ~I~I

Both ~ rules make use of Both ~ rules make use of recductio ad recductio ad absurdumabsurdum

Rules of SD: ~IRules of SD: ~IDerive ~GDerive ~G

1. G 1. G C C AA2. ~C & B2. ~C & B AA

--------------------3. G3. G AA

----------------4.4. C C5.5. ~C~C6.6. ~G~G ~I~I

Rules of SD: ~IRules of SD: ~IDerive ~GDerive ~G

1. G 1. G C C AA2. ~C & B2. ~C & B AA

--------------------3. G3. G AA

----------------4.4. C C 1, 3 1, 3 EE5.5. ~C~C 2 &E2 &E6.6. ~G~G 3-5 ~I3-5 ~I

Rules of SD: ~ERules of SD: ~E

~P~P --------

QQ~Q~Q

P P ~E~E

Rules of SD: ~ERules of SD: ~EDerive A & BDerive A & B1. ~(A & B) 1. ~(A & B) C C AA2.2. A A ~C ~C AA3.3. AA AA

------------------------------4.4. ~(A & B)~(A & B)

------------------------

5.5. CC6.6. ~C ~C 7.7. A & BA & B ~E~E

Rules of SD: ~ERules of SD: ~EDerive A & BDerive A & B1. ~(A & B) 1. ~(A & B) C C AA2.2. A A ~C ~C AA3.3. AA AA

------------------------------4.4. ~(A & B)~(A & B) AA

------------------------

5.5. CC 1, 4, 1, 4, E E6.6. ~C ~C 2, 3 2, 3 E E7.7. A & BA & B 4-6 ~E4-6 ~E