SL: Basic syntax alphabet

1
SL Basic syntax -alphabet

• SL alphabet
• - A, B, C, ...
• - , , ?, ?, ?
• - (, )

2
SL Basic syntax - propositions
A, B, C, ... propositions. have value T or F
3
SL Basic syntax - negation

• negation
• A is read not A
• ?A

4
SL Basic syntax - conjunction
conjunction AB is read A and B A is left
conjunct, B is right conjunct A?B, A?B
5
SL Basic syntax - disjunction
? disjunction A?B is read A or B A is left
disjunct, B is right disjunct
6
SL Basic syntax - implication
? material implication A?B is read A implies
B, or If A, then B A is the antecedent B is
the consequent A?B, A?B
7
SL Basic syntax - biconditional
? material biconditional or equivalence A?B is
read A is equivalent to B A?B, A?B
8
SL Basic syntax - ???

• A, B, C, ... propositions. have value T or F
• negation
• A is read not A
• conjunction
• AB is read A and B
• A is left conjunct, B is right conjunct
• ? disjunction
• A?B is read A or B
• A is left disjunct, B is right disjunct
• ? material implication
• A?B is read A implies B, or If A, then B
• A is the antecedent
• B is the consequent
• ? material biconditional or equivalence
• A?B is read A is equivalent to B

9
SL Basic syntax - formulae

• Atomic formulae have no connectives

10
SL Basic syntax - formulae

• Atomic formulae have no connectives
• If A is a formula, so is A

11
SL Basic syntax - formulae

• Atomic formulae have no connectives
• If A is a formula, so is A
• If A and B are formulae, so are
• AB, A?B, A?B, A?B

12
SL Basic syntax - formulae

• Atomic formulae have no connectives
• If A is a formula, so is A
• If A and B are formulae, so are
• AB, A?B, A?B, A?B
• nothing else is a formula

13
SL Basic syntax main connective
For more complex formulae brackets tell as what
is the main connective
14
SL Basic syntax main connective
For more complex formulae brackets tell as what
is the main connective (-------) is negation
15
SL Basic syntax main connective
For more complex formulae brackets tell as what
is the main connective (-------) is
negation (---------) (---------) is a
conjunction (---------) ? (----------) is an
implication, etc.
16
SL Basic syntax main connective
For more complex formulae brackets tell as what
is the main connective (-------) is
negation (---------) (---------) is a
conjunction (---------) ? (----------) is an
implication, etc. Formula (((AB) ? C) ? D) ? B
is an implication. (If A and B or not C is
equivalent to D, then not B.)
17
SL Basic syntax main connective
For more complex formulae brackets tell as what
is the main connective (-------) is
negation (---------) (---------) is a
conjunction (---------) ? (----------) is an
implication, etc. Formula (((AB) ? C) ? D) ? B
is an implication. (If A and B or not C is
equivalent to D, then not B.) The antecedent is
a biconditional.
18
SL Basic syntax main connective
For more complex formulae brackets tell as what
is the main connective (-------) is
negation (---------) (---------) is a
conjunction (---------) ? (----------) is an
implication, etc. Formula (((AB) ? C) ? D) ? B
is an implication. (If A and B or not C is
equivalent to D, then not B.) The antecedent is
a biconditional. The consequent is a negation.
19
SL Truth table for

• negation
• A is read not A

20
SL Truth table for
conjunction AB is read A and B A is left
conjunct, B is right conjunct
21
SL Truth table for ?
? disjunction A?B is read A or B A is left
disjunct, B is right disjunct
22
SL Truth table for ?
? material implication A?B is read A implies B,
or If A, then B A is the antecedent. B is the
consequent.
23
SL Truth table for ?
? material biconditional or equivalence A?B is
read A is equivalent to B
24
Symbolization techniques
Identify component propositions
25
Symbolization techniques
Identify component propositions
The rent is due, and I have no money.
26
Symbolization techniques
Identify component propositions
The rent is due, and I have no money.
The rent is due. (R) I have no money. (M)
27
Symbolization techniques
Identify component propositions
The rent is due, and I have no money.
The rent is due. (R) I have no money. (M)
RM
28
Symbolization techniques
Identify component propositions
If the cats away, the mice will play.
29
Symbolization techniques
Identify component propositions
If the cats away, the mice will play.
The cat is away (C) The mice will play (M)
30
Symbolization techniques
Identify component propositions
If the cats away, the mice will play.
The cat is away (C) The mice will play (M)
C ? M
31
Symbolization techniques
Identify component propositions
Either the guest of honour wore brown, or he fell
in the mud on the way to the party.
32
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)
33
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
34
Symbolization techniques
Conjunction
An explicit form of a conjunction Sue is happy
and Tim is happy S T
35
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
36
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. (PH) A disjunction has
two components, while a negation has only one.
(DN)
37
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) All these
sentences are translated as P
38
Symbolization techniques
Negation
Explicit form It is not the case that ...
39
Symbolization techniques
Negation
Explicit form It is not the case that ...
It is not the case that the temperature is rising
40
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)
41
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)
42
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
43
Symbolization techniques
Disjunction
Explicit form Either or
44
Symbolization techniques
Disjunction
Explicit form Either or
Shes either a lunatic or a genius.
45
Symbolization techniques
Disjunction
Explicit form Either or
Shes either a lunatic or a genius.
Either she is a lunatic, or she is a genius L ? G
46
Symbolization techniques
Material Conditional
Explicit form If , then
47
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
48
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, Ill buy two pairs S ?
B
49
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
50
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
51
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
52
Symbolization techniques
Material Biconditional
Explicit form if and only if
53
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)
54
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) )
55
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 )
56
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 )
57
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
58
Symbolization techniques
Neither nor
explicit form both it is not the case that A
and it is not the case that B A B
59
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
60
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)
61
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 Morgans Laws (AB) ? (A ? B) (A?B) ?
(A B)
62
Symbolization techniques
Not both ... and ...
explicit form it is not the case that both A
and B (A B)
63
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)
64
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 Morgans Laws (AB) ? (A ? B) (A?B) ?
(A B)
65
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
66
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
67
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.
68
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.
69
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. (HT) ? E 2. H (T?E)
70
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.
71
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'.)
72
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) )
73
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 ? L) (E
? G) )
74
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 ? L) (E
? G) ) ( B ? ( G ? L) ) ( B ? (L ? G) )
75
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.
76
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
77
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
78
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)
79
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) (R ? G)
80
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) (R ? G) R ? M
81
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) (R ? G) R ? M
Try alsoIf 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?
82
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.
83
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
84
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)
85
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
I ? ( (S ? L) (S ? G) )
86
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
I ? ( (S ? L) (S ? G) ) L G
87
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
I ? ( (S ? L) (S ? G) ) L G I
88
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.
89
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
90
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
91
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 ? B
92
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 ? B B ? I
93
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 ? B B ? I I
94
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.
95
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
96
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
97
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 L ? M
98
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 L ? M M ? C
99
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 L ? M M ? C C ? L
100
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 L ? M M ? C C ? L L