Symbolization techniques Identify component propositions

Symbolization techniques Identify component propositions The rent is due, and I have no money.

Symbolization techniques Identify component propositions The rent is due, and I have no money. The rent is due. I have no money.

Symbolization techniques Identify component propositions If the cat’s away, the mice will play.

Symbolization techniques Identify component propositions If the cat’s away, the mice will play. The cat is away (C) The mice will play (M)

Symbolization techniques Identify component propositions If the cat’s away, the mice will play. The cat is away (C) The mice will play (M) C  M

Symbolization techniques Identify component propositions Either the guest of honour wore brown, or he fell in the mud on the way to the party.

Symbolization techniques Identify component propositions Either the guest of honour wore brown, or he fell in the mud on the way to the party. The guest of honour wore brown (B) The guest of honour fell in the mud on the way to the party (M)

Symbolization techniques Identify component propositions Either the guest of honour wore brown, or he fell in the mud on the way to the party. The guest of honour wore brown (B) The guest of honour fell in the mud on the way to the party (M) B  M

Symbolization techniques Identify component propositions Sue and Tim are happy

Symbolization techniques Identify component propositions Sue and Tim are happy Sue is happy (S) Tim is happy (T) S & T

Symbolization techniques Identify component propositions Sue and Tim are happy, and they are throwing a party.

Symbolization techniques Identify component propositions Sue and Tim are happy, and they are throwing a party. Sue is happy (S) Tim is happy (T)

Symbolization techniques Identify component propositions Sue and Tim are happy, and they are throwing a party. Sue is happy (S) Tim is happy (T) Sue is throwing a party. (S 1 ) (?) Tim is throwing a party. (T 1 ) (?) They are throwing a party. (P) (?)

Symbolization techniques Identify component propositions Sue and Tim are happy, and they are throwing a party. Sue is happy (S) Tim is happy (T) Sue is throwing a party. (S 1 ) (?) Tim is throwing a party. (T 1 ) (?) They are throwing a party. (P) (?) (S&T) & (S 1 & T 1 ) (S&T) & P

Symbolization techniques Identify component propositions Sue and Tim are getting married.

Symbolization techniques Identify component propositions Sue and Tim are getting married. M S & T

Symbolization techniques Conjunction An explicit form of a conjunction: Sue is happy and Tim is happy – S & T

Symbolization techniques Conjunction An explicit form of a conjunction: Sue is happy and Tim is happy – S & T These are also conjunctions: A but B A however B A although B A nevertheless B A nonetheless B A moreover B

Symbolization techniques Conjunction An explicit form of a conjunction: Sue is happy and Tim is happy – S & T These are also conjunctions: A but B A however B A although B A nevertheless B A nonetheless B A moreover B She is poor, but honest. (P&H) A disjunction has two components, while a negation has only one. (D&N)

Symbolization techniques She is poor (P) It is true that she is poor (P) She certainly is poor (P)

Symbolization techniques Negation Explicit form: It is not the case that...

Symbolization techniques Negation Explicit form: It is not the case that... It is not the case that the temperature is rising

Symbolization techniques Negation Explicit form: It is not the case that... It is not the case that the temperature is rising The temperature is rising (R)

Symbolization techniques Negation Explicit form: It is not the case that... It is not the case that the temperature is rising (~R) The temperature is rising (R)

Symbolization techniques Negation Explicit form: It is not the case that... It is not the case that the temperature is rising (~R) The temperature is rising (R) It is false that the temperature is rising The temperature is not rising

Symbolization techniques Disjunction Explicit form: Either … or …

Symbolization techniques Disjunction Explicit form: Either … or … She’s either a lunatic or a genius.

Symbolization techniques Disjunction Explicit form: Either … or … She’s either a lunatic or a genius. Either she is a lunatic, or she is a genius L  G

Symbolization techniques Material Conditional Explicit form: If …, then…

Symbolization techniques Material Conditional Explicit form: If …, then… These are also conditionals equivalent to ‘If A, then C’ (A  C): If A, C C, if A C, provided that A C, given A A only if C

Symbolization techniques Material Conditional Explicit form: If …, then… These are also conditionals equivalent to ‘If A, then C’ (A  C): If A, C C, if A C, provided that A C, given A A only if C If these shoes go on sale, I’ll buy two pairs S  B

Symbolization techniques Material Conditional Explicit form: If …, then… These are also conditionals equivalent to ‘If A, then C’ (A  C): If A, C C, if A C, provided that A C, given A A only if C Only ifs I will go camping this weekend if I finish my work I will go camping this weekend only if I finish my work

Symbolization techniques Material Conditional Explicit form: If …, then… These are also conditionals equivalent to ‘If A, then C’ (A  C): If A, C C, if A C, provided that A C, given A A only if C Only ifs I will go camping this weekend if I finish my work I will go camping this weekend only if I finish my work If you buy a ticket, you will win the lottery Only if you buy a ticket, you will win the lottery

Symbolization techniques Material Conditional Explicit form: If …, then… These are also conditionals equivalent to ‘If A, then C’ (A  C): If A, C C, if A C, provided that A C, given A A only if C Only ifs I will go camping this weekend if I finish my work F  C I will go camping this weekend only if I finish my work C  F If you buy a ticket, you will win the lottery T  W Only if you buy a ticket, you will win the lottery W  T

Symbolization techniques Material Biconditional Explicit form: … if and only if …

Symbolization techniques Material Biconditional Explicit form: … if and only if … If a conjunction is true, then both conjuncts are true (C  B) A conjunction is true only if both conjuncts are true (B  C)

Symbolization techniques Material Biconditional Explicit form: … if and only if … If a conjunction is true, then both conjuncts are true (C  B) A conjunction is true only if both conjuncts are true (B  C) If a conjunction is true, then both conjuncts are true, and it is true only if both conjuncts are true ( (C  B) & (B  C) )

Symbolization techniques Material Biconditional Explicit form: … if and only if … If a conjunction is true, then both conjuncts are true (C  B) A conjunction is true only if both conjuncts are true (B  C) If a conjunction is true, then both conjuncts are true, and it is true only if both conjuncts are true ( (C  B) & (B  C) ) Both conjuncts are true if and only if the conjunction is true ( B  C )

Symbolization techniques Material Biconditional Explicit form: … if and only if … Sue is married to Tim if and only if Tim is married to Sue ( S  T )

Symbolization techniques Material Biconditional Explicit form: … if and only if … These are also biconditionals equivalent to A  B: A if but only if B A just in case B

Symbolization techniques Neither… nor… explicit form: both it is not the case that A and it is not the case that B: ~A & ~B

Symbolization techniques Neither… nor… explicit form: both it is not the case that A and it is not the case that B: ~A & ~B Neither Sue not Tim are throwing a party ~S & ~T

Symbolization techniques Neither… nor… explicit form: both it is not the case that A and it is not the case that B: ~A & ~B Neither Sue not Tim are throwing a party ~S & ~T Also correct symbolization is ~ (S  T)

Symbolization techniques Neither… nor… explicit form: both it is not the case that A and it is not the case that B: ~A & ~B Neither Sue not Tim are throwing a party ~S & ~T Also correct symbolization is ~ (S  T) De Morgan’s Laws: ~ (A&B)  (~A  ~B) ~ (A  B)  (~A & ~B)

Symbolization techniques Not both... and... explicit form: it is not the case that both A and B: ~ (A & B)

Symbolization techniques Not both... and... explicit form: it is not the case that both A and B: ~ (A & B) Not both Sue and Tim are throwing a party ~ (S & T)

Symbolization techniques Not both... and... explicit form: it is not the case that both A and B: ~ (A & B) Not both Sue and Tim are throwing a party ~ (S & T) Also correct symbolization is ~S  ~T, because of De Morgan’s Laws: ~ (A&B)  (~A  ~B) ~ (A  B)  (~A & ~B)

Symbolization techniques … unless … explicit form: either A or B if it is not the case that A then B if it is not the case that B then A

Symbolization techniques … unless … explicit form: either A or B if it is not the case that A then B if it is not the case that B then A You and I can be friends unless you take advantage of me F  A ~ A  F ~ F  A

Symbolization techniques some more complex cases 1. I'll go home and watch TV, or I'll think about the election. 2. I'll go home, and I'll watch TV or think about the election.

Symbolization techniques some more complex cases 1. I'll go home and watch TV, or I'll think about the election. 2. I'll go home, and I'll watch TV or think about the election. 1. Either I'll go home and watch TV, or I'll think about the election. 2. I'll go home, and I'll either watch TV or think about the election.

Symbolization techniques some more complex cases 1. I'll go home and watch TV, or I'll think about the election. 2. I'll go home, and I'll watch TV or think about the election. 1. Either I'll go home and watch TV, or I'll think about the election. 2. I'll go home, and I'll either watch TV or think about the election. 1. (H&T)  E 2. H & (T  E)

If I believe in God, then if He exists, I gain, and if He doesn't, then I don't lose. If, on the other hand, I don't believe in God, then if he exists, I lose, and if He doesn't, I don't gain. From this it follows that if I believe, I'll either gain or not lose, while if I don't believe, I'll either lose or fail to gain.

If I believe in God, then if He exists, I gain, and if He doesn't, then I don't lose. If, on the other hand, I don't believe in God, then if he exists, I lose, and if He doesn't, I don't gain. From this it follows that if I believe, I'll either gain or not lose, while if I don't believe, I'll either lose or fail to gain. B: I believe in God E: God exists G: I gain L: I lose (Note: 'lose' is not synonymous with 'not gain'.)

If I believe in God, then if He exists, I gain, and if He doesn't, then I don't lose. If, on the other hand, I don't believe in God, then if he exists, I lose, and if He doesn't, I don't gain. From this it follows that if I believe, I'll either gain or not lose, while if I don't believe, I'll either lose or fail to gain. B: I believe in God E: God exists G: I gain L: I lose B  ( (E  G) & (~E  ~L) )

If I believe in God, then if He exists, I gain, and if He doesn't, then I don't lose. If, on the other hand, I don't believe in God, then if he exists, I lose, and if He doesn't, I don't gain. From this it follows that if I believe, I'll either gain or not lose, while if I don't believe, I'll either lose or fail to gain. B: I believe in God E: God exists G: I gain L: I lose B  ( (E  G) & (~E  ~L) ) ~B  ( (E  L) & (~E  ~G) )

If I believe in God, then if He exists, I gain, and if He doesn't, then I don't lose. If, on the other hand, I don't believe in God, then if he exists, I lose, and if He doesn't, I don't gain. From this it follows that if I believe, I'll either gain or not lose, while if I don't believe, I'll either lose or fail to gain. B: I believe in God E: God exists G: I gain L: I lose B  ( (E  G) & (~E  ~L) ) ~B  ( (E  L) & (~E  ~G) ) ( B  ( G  ~L) ) & ( ~B  (L  ~G) )

George will leave unless Mary doesn't leave. But unless Phil stays, she will leave, and George won't leave provided that it rains. As a result, if it rains, Mary won't leave.

George will leave unless Mary doesn't leave. But unless Phil stays, she will leave, and George won't leave provided that it rains. As a result, if it rains, Mary won't leave. G: George will leave M: Mary does leave P: Phil stays R: It rains

G  ~M

George will leave unless Mary doesn't leave. But unless Phil stays, she will leave, and George won't leave provided that it rains. As a result, if it rains, Mary won't leave. G: George will leave M: Mary does leave P: Phil stays R: It rains G  ~M (P  M) &

George will leave unless Mary doesn't leave. But unless Phil stays, she will leave, and George won't leave provided that it rains. As a result, if it rains, Mary won't leave. G: George will leave M: Mary does leave P: Phil stays R: It rains G  ~M (P  M) & (R  ~G)

George will leave unless Mary doesn't leave. But unless Phil stays, she will leave, and George won't leave provided that it rains. As a result, if it rains, Mary won't leave. G: George will leave M: Mary does leave P: Phil stays R: It rains G  ~M (P  M) & (R  ~G) R  ~M

Try also: If George will leave unless Mary doesn't leave, and unless Phil stays, she will leave, and George won't leave provided that it rains, then, as a result, if it rains, Mary won't leave. Is this sentence tf-true? George will leave unless Mary doesn't leave. But unless Phil stays, she will leave, and George won't leave provided that it rains. As a result, if it rains, Mary won't leave. G: George will leave M: Mary does leave P: Phil stays R: It rains G  ~M (P  M) & (R  ~G) R  ~M

If there is an infinite number of points in a finite line L, then if those points have size, L will be infinitely long, and if they do not have size, L won't have length. Line L is neither infinitely long nor without length. This proves that there is not an infinite number of points in L.

If there is an infinite number of points in a finite line L, then if those points have size, L will be infinitely long, and if they do not have size, L won't have length. Line L is neither infinitely long nor without length. This proves that there is not an infinite number of points in L. I: There is an infinite number of points in the line L S: Points have size L: Line L is infinitely long G: Line L has length

(S  L) & (~S  ~G)

If there is an infinite number of points in a finite line L, then if those points have size, L will be infinitely long, and if they do not have size, L won't have length. Line L is neither infinitely long nor without length. This proves that there is not an infinite number of points in L. I: There is an infinite number of points in the line L S: Points have size L: Line L is infinitely long G: Line L has length I  ( (S  L) & (~S  ~G) )

If there is an infinite number of points in a finite line L, then if those points have size, L will be infinitely long, and if they do not have size, L won't have length. Line L is neither infinitely long nor without length. This proves that there is not an infinite number of points in L. I: There is an infinite number of points in the line L S: Points have size L: Line L is infinitely long G: Line L has length I  ( (S  L) & (~S  ~G) ) ~L & ~~G

If there is an infinite number of points in a finite line L, then if those points have size, L will be infinitely long, and if they do not have size, L won't have length. Line L is neither infinitely long nor without length. This proves that there is not an infinite number of points in L. I: There is an infinite number of points in the line L S: Points have size L: Line L is infinitely long G: Line L has length I  ( (S  L) & (~S  ~G) ) ~L & ~~G ~I

The line L is infinitely divisible, which means that between any two points on the line we can find another point. But the line L can have another point between any two points only if it has an infinite number of points. Therefore L has an infinite number of points.

D: The line L is infinitely divisible B: Between any two points on the line we can find another point I: There is an infinite number of points in the line L

The line L is infinitely divisible, which means that between any two points on the line we can find another point. But the line L can have another point between any two points only if it has an infinite number of points. Therefore L has an infinite number of points. D: The line L is infinitely divisible B: Between any two points on the line we can find another point I: There is an infinite number of points in the line L D

The line L is infinitely divisible, which means that between any two points on the line we can find another point. But the line L can have another point between any two points only if it has an infinite number of points. Therefore L has an infinite number of points. D: The line L is infinitely divisible B: Between any two points on the line we can find another point I: There is an infinite number of points in the line L D D  B

The line L is infinitely divisible, which means that between any two points on the line we can find another point. But the line L can have another point between any two points only if it has an infinite number of points. Therefore L has an infinite number of points. D: The line L is infinitely divisible B: Between any two points on the line we can find another point I: There is an infinite number of points in the line L D D  B B  I

The line L is infinitely divisible, which means that between any two points on the line we can find another point. But the line L can have another point between any two points only if it has an infinite number of points. Therefore L has an infinite number of points. D: The line L is infinitely divisible B: Between any two points on the line we can find another point I: There is an infinite number of points in the line L D D  B B  I I

Lung cancer is not caused by smoking, and this is so for the following reasons. Lung cancer is more common among male smokers than among female smokers. If smoking were the cause of lung cancer, this would not be true. The fact that lung cancer is more common among male smokers implies that it is caused by something in the male makeup. But if it is caused by this, it is not caused by smoking.

Lung cancer is not caused by smoking, and this is so for the following reasons. Lung cancer is more common among male smokers than among female smokers. If smoking were the cause of lung cancer, this would not be true. The fact that lung cancer is more common among male smokers implies that it is caused by something in the male makeup. But if it is caused by this, it is not caused by smoking. L: Lung cancer is caused by smoking M: Lung cancer is more common among male smokers than among female smokers C: Lung cancer is caused by something in the male makeup

M

M L  ~M

Lung cancer is not caused by smoking, and this is so for the following reasons. Lung cancer is more common among male smokers than among female smokers. If smoking were the cause of lung cancer, this would not be true. The fact that lung cancer is more common among male smokers implies that it is caused by something in the male makeup. But if it is caused by this, it is not caused by smoking. L: Lung cancer is caused by smoking M: Lung cancer is more common among male smokers than among female smokers C: Lung cancer is caused by something in the male makeup M L  ~M M  C

Lung cancer is not caused by smoking, and this is so for the following reasons. Lung cancer is more common among male smokers than among female smokers. If smoking were the cause of lung cancer, this would not be true. The fact that lung cancer is more common among male smokers implies that it is caused by something in the male makeup. But if it is caused by this, it is not caused by smoking. L: Lung cancer is caused by smoking M: Lung cancer is more common among male smokers than among female smokers C: Lung cancer is caused by something in the male makeup M L  ~M M  C C  ~L

Lung cancer is not caused by smoking, and this is so for the following reasons. Lung cancer is more common among male smokers than among female smokers. If smoking were the cause of lung cancer, this would not be true. The fact that lung cancer is more common among male smokers implies that it is caused by something in the male makeup. But if it is caused by this, it is not caused by smoking. L: Lung cancer is caused by smoking M: Lung cancer is more common among male smokers than among female smokers C: Lung cancer is caused by something in the male makeup M L  ~M M  C C  ~L ~L