Week 7

Logic

“Logic deals with meanings in a language system, not with actual behaviour of any sort. Logic deals centrally with PROPOSITIONS” (H&H, p131)

Task 1:

Do the following statements use the words logic, logical, and illogical in this narrow sense?

  1. It’s not logical to want to kill oneself.
  2. Harry is so illogical: first he says he doesn’t want to come, then he changes his mind.
  3. The truth of the proposition that Socrates is mortal follows logically from the fact that Socrates is a man and the fact that all men are mortal.
  4. Max is not coming is, logically, the negation of Max is coming.
  5. The logic of Churchill’s tactics in the Eastern Mediterranean is quite baffling.

Notations for simple propositions

The predicator is represented by CAPITAL LETTERS

The arguments are represented by single lower-case letters

e.g. Chomsky teaches philosophy:                 c TEACH p

Task 2:

Translate the following into simple notation:

  1. Arthur dreamed
  2. Ali helped John
  3. Alit introduced John to Arthur
  4. Ahmad is my brother
  5. Ahmad is a hero
  6. Water is beside Billy
  7. Margaret is looking for Billy
  8. Margaret is looking after Billy
  9. Sidney put his hat on the table
  10. Sidney put his hat beside the table

Notations for complex propositions

We connectives such as and and or to join simple propositions to form complex propositions.

e.g.:                 c COME g                              (Caeser came to Gaul)

                        c SEE g                                   (Caeser saw Gaul)

                        c CONQUER g                      (Caeser conquered Gaul)

            From these, a single complex formula can be formed:

                                    (c COME g) & (c SEE g) & (c CONQUER g)

            Translation:     Caesar came, saw and conquered Gaul.

Task 3:

  1. Suggest an English translation for: j TALL & m SMALL

      …………………………………………………………………………

  1. Translate the following into logical notation:
    1. Andy entered and Mary left

…………………………………………………………………………

  1. John loves Mary and Mary loves Bill

…………………………………………………………………………

  1. John and Mary are Irish

…………………………………………………………………………

  1. Can the following sentence be unpacked in the same way as (2c)?

                  Adam and Eve are a happy couple.

Rules of inference

  1. The teacher punished Ahmad and Zaid                      t PUNISH a & z
  2. The teacher punished Zaid and Ahmad                      t PUNISH z & a

            Commutativity of conjuction:

                                                                        p & q               (premiss)

                                                                        q & p               (conclusion)

Task 4:

Say whether the following rules of inference are correct or incorrect.

1.   p                      (premises)

      q

      p & q               (conclusion)

2.   p & q               (premiss)

      p                      (conclusion)

3.   p                      (premiss)

      p & q               (conclusion)

Task 5:

Given below are pairs consisting of a sentence and rule of inference. Using the given sentence as a premiss, write down the conclusion warranted by the given rule of inference.

1. Melanie is pregnant and Mike is in Belgium                                             p & q

……………………………………………………………………….           q & p

2. Lorna left and Bill stayed                                                                           p & q

……………………………………………………………………….           P

3. Frances sang and Harry danced                                                                  p & q

……………………………………………………………………….           q

4. Now, what is the sense relation that holds between each of the sentences given above and the answer you have written under?

Logic and Truth

            Truth value:         a sentence being true or false

            Truth conditions: the facts that would have to obtain in reality to make a
                                         sentence true or false.

1.         Negative:

            a. Your car has been stolen                               p

            b. Your can has not been stolen                      ~p

Truth table:

      P          ~p

            T          F

            F          T

2.         Conjunction &

            a. The house is on fire.

            b. The fire brigade are on the way.

            c. The house is on fire and the fire brigade are on the way

Truth table:

      P          q          p & q

            T          T             T

            T          F             F

            F          T             F

            F          F             F

3.         Inclusive or

            a. I’ll see you today or tomorrow

Truth table:

      P          q          p v q

            T          T             T

            T          F             T

            F          T             T

            F          F             F  

4.         Exclusive or

            You will pay the fine or you will go to jail

Truth table:

      P          q          p ve q

            T          T             F

            T          F             T

            F          T             T

            F          F             F

5.         The material implication  (if …. then)

            a. If it rains, then I’ll go to the movies

Truth table:

            P          q          p → q

            T          T             T

            T          F             F

            F          T             T

            F          F             T

6.         Biconditional

            a. We’ll leave if and only if we’re forced to

Truth table:

      P          q          p q

            T          T             T

            T          F             F

            F          T             F

            F          F             T

Task 6:

Take three sentences, p, q, and r as follows:

p: The sun is shining

q: The day is warm

r: The sun is shining and the day is warm

Let’s make the working assumption that we can represent sentence r by the logical formula p & q. Draw a truth table to show the truth value of r when:

  1. p is true; q is false
  2. p is true; q is true
  3. p is false; q is true

Entailment

A sentence p entails a sentence q when

(i) the truth of the first (p) guarantees the truth of the second

(ii) and the falsity of the second (q) guarantees the falsity of the first (p).

6.         a) The anarchist assassinate the emperor

      b) The emperor died

Truth Table:

            P           →         q

            T            →         T

            F            →         T or F

            F            ←         F

            T or F    ←         T

The source of entailment can be lexical:

7.         a) I bought a dog

            b) I bought an animal

or syntactic:

8.         a) King Arthur built the round table

            b) The round table was built by King Arthur

Task 7:

Use the truth table above for entailment to decide whether the (a) sentence entails its (b) partner.

1. a) Margarita passed her driving test.

    b) Margarita didn’t fail her driving test

2. a) Cassidy inherited a farm

    b) Cassidy owned a farm

3. a) Cassidy inherited a farm

    b) Cassidy owns a farm

4. a) Arnold poisoned his wife

    b) Arnold killed his wife

5. a) We brought this meal

    b) This meal was brought by us

6. a) Not everyone will like the show

    b) Someone will like the show

The relationship of entailment allows us to define:

            A) Paraphrases which mutually entail each other

9.         a) Alice owns this book

            b) This book belongs to Alice

Truth Table of Synonymy:

            P                         q

            T            →         T

            F            →         F

            F            ←         F

            T            ←         T

            B) Contradiction which is the opposite of synonymy.

10.       a) Mary Jones stole my car

            b) Mary Jones did not steal my car

 Truth Table of Contradiction:

            P                         q

            T            →         F

            F            →         T

            T            ←         F

            F            ←         T

Presupposition

A sentence p presupposes a sentence q if

(i) p (the presupposing sentence) is true then q (the presupposed sentence) is true

(ii) p is false then q is still true

(iii) q is true then p could be either true or false.

11.       a) Her husband is a fool

            b) She has a husband

Truth Table:

            P                         q

            T            →         T

            F            →         T

            T or F    ←         T

Task 8:

Decide the (a) sentences below entail or presuppose their (b) counterparts

1. a) Dave is angry because Jim crashed the car

   b) Jim crashed the car

2. a) Zaire is bigger than Alaska

    b) Alaska is smaller than Zaire

3. a) The minister blames her secretary for leaking the memo to the press

   b) The memo was leaked to the press

4. a) Everyone passed the examination

    b) No one failed the examination

5. a) Ali has resumed his habit of sleeping late

    b) Ali has a habit of sleeping late