Practice

~B ((BD) D) tf-true?(M ~N) & (M N) tf-false?(BH) & (B~H) tf-false, true or indeterminate?(C & (BA)) 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. (S1) (?)Tim is throwing a party. (T1) (?)

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. (S1) (?)Tim is throwing a party. (T1) (?)

They are throwing a party. (P) (?)

(S&T) & (S1 & T1)(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.

MS & 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 BA however BA although BA nevertheless BA nonetheless BA 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 BA however BA although BA nevertheless BA nonetheless BA 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)She certainly 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 risingThe 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 geniusL 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’ (AC):

If A, CC, if AC, provided that AC, given AA only if C

Symbolization techniques

Material Conditional

Explicit form:If …, then…

These are also conditionals equivalent to ‘If A, then C’ (AC):

If A, CC, if AC, provided that AC, given AA only if C

If these shoes go on sale, I’ll buy two pairsS B

Symbolization techniques

Material Conditional

Explicit form:If …, then…

These are also conditionals equivalent to ‘If A, then C’ (AC):

If A, CC, if AC, provided that AC, given AA only if C

Only ifs

I will go camping this weekend if I finish my workI 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’ (AC):

If A, CC, if AC, provided that AC, given AA only if C

Only ifs

I will go camping this weekend if I finish my workI will go camping this weekend only if I finish my work

If you buy a ticket, you will win the lotteryOnly 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’ (AC):

If A, CC, if AC, provided that AC, given AA only if C

Only ifs

I will go camping this weekend if I finish my workF C

I will go camping this weekend only if I finish my workC F

If you buy a ticket, you will win the lotteryT W

Only if you buy a ticket, you will win the lotteryW 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 BA 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)~ (AB) (~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)~ (AB) (~A & ~B)

Symbolization techniques

… unless …

explicit form: either A or B

if it is not the case that A then Bif 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 Bif it is not the case that B then A

You and I can be friends unless you take advantage of meF A

~ A F~ F A