Symbolization techniques Identify component propositions

Symbolization techniques

Identify component propositions



The rent is due, and I have no money.



The rent is due, and I have no money.

The rent is due.I have no money.



If the cat’s away, the mice will play.




The cat is away (C)The mice will play (M)




The cat is away (C)The mice will play (M)

C M



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)




The guest of honour wore brown (B)The guest of honour fell in the mud on the way to the party (M)

B M



Sue and Tim are happy



Sue and Tim are happy

Sue is happy (S)Tim is happy (T)

S & T



Sue and Tim are happy, and they are throwing a party.









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

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





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



Sue and Tim are getting married.



Sue and Tim are getting married.

MS & T


Conjunction

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


Conjunction


These are also conjunctions:A but BA however BA although BA nevertheless BA nonetheless BA moreover B


Conjunction


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)


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)


Negation

Explicit form:It is not the case that ...


Negation


It is not the case that the temperature is rising


Negation


It is not the case that the temperature is rising

The temperature is rising (R)


Negation


It is not the case that the temperature is rising (~R)



Negation


It is not the case that the temperature is rising (~R)


It is false that the temperature is risingThe temperature is not rising


Disjunction

Explicit form:Either … or …


Disjunction


She’s either a lunatic or a genius.


Disjunction


She’s either a lunatic or a genius.

Either she is a lunatic, or she is a geniusL G


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






Only ifs

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






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






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


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 (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) )




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 )




Sue is married to Tim if and only if Tim is married to Sue ( S T )




These are also biconditionals equivalent to A B:

A if but only if BA just in case B


Neither… nor…

explicit form: both it is not the case that A and it is not the case that B:

~A & ~B


Neither… nor…


~A & ~B

Neither Sue not Tim are throwing a party~S & ~T


Neither… nor…


~A & ~B


Also correct symbolization is ~ (S T)


Neither… nor…


~A & ~B


Also correct symbolization is ~ (S T)

De Morgan’s Laws:~ (A&B) (~A ~B)~ (AB) (~A & ~B)


Not both ... and ...

explicit form: it is not the case that both A and B:

~ (A & B)




~ (A & B)

Not both Sue and Tim are throwing a party~ (S & T)




~ (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)


… 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


… 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


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. 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) E2. H & (TE)

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 GodE: God existsG: I gainL: 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 GodE: God existsG: I gainL: I lose

B ( (E G) & (~E ~L) )


B ( (E G) & (~E ~L) )

~B ( (E L) & (~E ~G) )


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 leaveM: Mary does leaveP: Phil staysR: It rains

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 leaveM: Mary does leaveP: Phil staysR: It rains G ~M


(P M) &


(P M) & (R ~G)


(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?


(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 LS: Points have sizeL: Line L is infinitely longG: Line L has length


(S L) & (~S ~G)


I ( (S L) & (~S ~G) )


I ( (S L) & (~S ~G) )

~L & ~~G


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 divisibleB: Between any two points on the line we can find another pointI: There is an infinite number of points in the line L



D



D

D B



D

D B

B I



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 smokingM: Lung cancer is more common among male smokers than among female smokersC: Lung cancer is caused by something in the male makeup


M


ML ~M


ML ~MM C


ML ~MM CC ~L


ML ~MM CC ~L~L

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Symbolization techniques Identify component propositions