Translating into L1
| University of Edinburgh | PHIL08004 |
1 / 33
If your card has a red back, then your card is an Ace.
Now flip your card over. Raise your hand if I lied to you.
2 / 33
True
Your card has red back → Your card is an Ace.
True
True
Back
Front
3 / 33
→
Red
T
T
F
F
Ace (Red → Ace)
T
T
F
T
F
4 / 33
True
Your card has red back → Your card is an Ace.
False
False
Back
Front
5 / 33
→
Red
T
T
F
F
Ace (Red → Ace)
T
T
F
T
F
T
6 / 33
True
Your card has red back → Your card is an Ace.
False
True
Back
Front
7 / 33
→
Red
T
T
F
F
Ace (Red → Ace)
T
T
F
T
T
F
T
8 / 33
False!
Your card has red back → Your card is an Ace.
True
False
Back
Front
9 / 33
→
Red
T
T
F
F
Ace (Red → Ace)
T
T
F
F
T
T
F
T
10 / 33
→
I
Truth table for conditional
2
T
T
F
F
# (2 → #)
T
T
F
F
T
T
F
T
11 / 33
Am I lying?
I
If your card has a blue back, then your card has a blue back.
I
If your card has a green back, then your card has a green back.
I
If your card is an Ace, then your card has a red back.
I
If you are a robot, you have a playing card.
12 / 33
Intended interpretations of ‘~’ and ‘→’
I
And we needn’t say what ‘˜’ and ‘→’ mean to define the
formal language and system
I
But the formal system is a model of our logical reasoning
I
There is an intended interpretation of the symbols
I
The formal system is set up in a way to capture certain
aspects of our reasoning
13 / 33
‘→’:Conditionals
I
“If Fido is a canine, then Fido is carnivorous”
I
“if . . . , then. . . ” is used to combine two propositions
I
I
I
if P, then Q
I
I
I
A conditional expresses hypothetical relationships between two
propositions
Conditionals in computer programming:
IF (x)(y)
P is the antecedent
Q is the consequent
(P → Q)
14 / 33
The King
Puzzle 6. Assume that there is a King of the Island of Knights
and Knaves. Who is the King? You come across two inhabitants A
and B and A makes the following statement: “If I’m a knight, then
B is the King”.
Is B the King?
15 / 33
16 / 33
A: If I’m a knight, then B is the King
I
Assume A is a knave
I
So what A said is false
I
So: A is a knight but B is not the King
I
Contradiction!
I
So A is a knight
I
And, so B is the King
17 / 33
Synonyms of ‘if’
I
‘if’: If Maria sings, Xavier will leave
I
‘provided that’: Provided that Maria sings, Xavier will leave
I
‘assuming that’: Assuming that Maria sings, Xavier will leave
I
‘given that’: Given that Maria sings, Xavier will leave
I
‘in case’: In case Maria sings, Xavier will leave
I
‘on the condition that’: On the condition that Maria sings,
Xavier will leave
18 / 33
If Maria sings, Xavier will leave
I
S: Maria sings
I
X: Xavier will leave
(S → X )
Provided that Maria sings, Xavier will leaveAssuming that Maria
sings, Xavier will leaveGiven that Maria sings, Xavier will leave In
case Maria sings, Xavier will leave On the condition that Maria
sings, Xavier will leave
19 / 33
Synonyms of ‘if’
I
Xavier will leave if Maria sings
I
Xavier will leave provided that Maria sings
I
Xavier will leave assuming that Maria sings
I
Xavier will leave given that Maria sings
I
Xavier will leave in case Maria sings
I
Xavier will leave on the condition that Maria sings
20 / 33
‘only if’
I
Xavier will leave only if Maria sings
I
S: Maria sings
I
X: Xavier will leave
(X → S)
21 / 33
From English to (2 → #)
I
“If P, Q”
P→Q
I
“P only if Q”
P→Q
I
“P if Q”
Q→P
I
“Only if P, Q”
Q→P
22 / 33
‘~’:
Negation
I
“Grass is not green”
I
‘not’ is used to deny a proposition
I
P: Grass is green
I
‘˜P’ symbolises ‘Grass is not green’
I
‘˜˜P” symbolises ‘It is not the case that Grass is not green’
23 / 33
∼
I
Truth table for negation
2 ∼2
T F
F T
24 / 33
Synonyms of ‘not’
I
The salad was not poisonous
I
The salad wasn’t poisonous
I
It is not the case that the salad was poisonous
I
The salad failed to be poisonous
25 / 33
Alfred is not tired.
I
T: Alfred is tired
˜T
26 / 33
If Alfred sleeps, then it is not the case that he is tired.
I
I
S: Alfred sleeps
T: Alfred is tired
(S → ˜T )
27 / 33
If Alfred loves tautologies, then Alfred loves
tautologies.
I
Z: Alfred loves tautologies.
(Z → Z )
28 / 33
If Alfred studies, then Alfred will pass if he
concentrates.
I
I
I
S: Alfred studies
P: Alfred passes
R: Alfred concentrates
(S → (R → P))
29 / 33
If Alfred studies then Alfred will pass, if he
concentrates.
I
I
I
S: Alfred studies
P: Alfred passes
R: Alfred concentrates
(R → (S → P))
30 / 33
If David buys wine, Jane won’t be happy.
I
I
W: David buys wine.
P: Jane will be happy.
(W → ˜P)
31 / 33
Jane is happy only if David isn’t.
I
I
P: Jane is happy.
Q: David is happy.
(P → ˜Q)
32 / 33
It is not the case that if David buys wine, Jane will be
happy.
I
I
W: David buys wine.
P: Jane will be happy.
˜(W → P)
33 / 33