Practice ~B  ((B  D)  D) tf-true? (M  ~N) & (M  N)tf-false? (B  H) & (B  ~H)tf-false, true or indeterminate? (C & (B  A)) and ((C&B)  A)tf-equivalent? {B&F, ~(B&G)}  G?

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