Transcript Logical Truth

Statements
The most important idea in logic:
Validity of an argument.
Statements
The most important idea in logic:
Validity of an argument.
However, there are other important concepts
that concern statements.
Statements
Logical Truths (Tautologies)
Contradictions
Contingent Statements
Logical Truth
A statement is a logic truth (tautology)
iff
it cannot be F.
Logical Truth
A statement is a logic truth (tautology)
iff
it cannot be F.
So its truth table has all Ts
on the output column.
Logical Truth
A statement is a logic truth (tautology)
iff
it cannot be F.
So its truth table has all Ts
on the output column.
Samples:
P>P
Pv-P
Logical Truth
A statement is a logic truth (tautology)
iff
it cannot be F.
Samples:
P>P
Pv-P
Good news:
Logical Truths are always true.
Logical Truth
A statement is a logic truth (tautology)
iff
it cannot be F.
Samples:
P>P
Pv-P
Good news:
Logical Truths are always true.
Bad news:
Logical Truths do not carry information.
Logical Truth
A statement is a logic truth (tautology)
iff
it cannot be F.
Samples:
P>P
Pv-P
Good news:
Logical Truths are always true.
Bad news:
Logical Truths do not carry information.
Desperate Weather Report: If it is raining then it is raining.
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
T
T
*
P
T
F
P v -P
T F
T T
*
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
T
T
*
with a proof:
P
T
F
Prove A.
P v -P
T F
T T
*
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
T
T
*
with a proof:
P>P
P
T
F
Prove A.
GOAL
P v -P
T F
T T
*
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
T
T
*
with a proof:
1)
2)
P
P>P
PA
1-1 >I
P
T
F
Prove A.
P v -P
T F
T T
*
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
T
T
*
with a proof:
1)
2)
P
P>P
P
T
F
Prove A.
P v -P
T F
T T
*
PA
1-1 >I
Pv-P
GOAL
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
T
T
*
with a proof:
1)
2)
P
P>P
PA
1-1 >I
P
T
F
Prove A.
P v -P
T F
T T
*
1)
-(Pv-P)
?&-?
Pv-P
PA
1-? -O
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
T
T
*
with a proof:
1)
2)
P
P>P
PA
1-1 >I
P
T
F
Prove A.
P v -P
T F
T T
*
1)
2)
3)
-(Pv-P)
-P&--P
Pv-P
PA
1 DM
1-2 -O
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
P
T
T
T
F
*
with a proof:
Prove A.
1) P
PA
2) P>P
1-1 >I
with a tree:
P v -P
T F
T T
*
1)
2)
3)
-(Pv-P)
-P&--P
Pv-P
The tree for -A closes.
PA
1 DM
1-2 -O
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
P
T
T
T
F
*
with a proof:
Prove A.
1) P
PA
2) P>P
1-1 >I
with a tree:
P v -P
T F
T T
*
1)
2)
3)
-(Pv-P)
-P&--P
Pv-P
The tree for -A closes.
-(P>P)
PA
1 DM
1-2 -O
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
P
T
T
T
F
*
with a proof:
Prove A.
1) P
PA
2) P>P
1-1 >I
with a tree:
P v -P
T F
T T
*
1)
2)
3)
-(Pv-P)
-P&--P
Pv-P
The tree for -A closes.
-(P>P)
P
-P
*
PA
1 DM
1-2 -O
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
P
T
T
T
F
*
with a proof:
Prove A.
1) P
PA
2) P>P
1-1 >I
with a tree:
P v -P
T F
T T
*
1)
2)
3)
-(Pv-P)
-P&--P
Pv-P
The tree for -A closes.
-(P>P)
-(Pv-P)
P
-P
*
PA
1 DM
1-2 -O
Logical Truth
To show a statement A is a logic truth ...
with a table:
The output row for A has all Ts.
P
T
F
P>P
P
T
T
T
F
*
with a proof:
Prove A.
1) P
PA
2) P>P
1-1 >I
with a tree:
P v -P
T F
T T
*
1)
2)
3)
-(Pv-P)
-P&--P
Pv-P
The tree for -A closes.
-(P>P)
-(Pv-P)
P
-P
-P
--P
*
*
PA
1 DM
1-2 -O
Logical Truth
To show a statement A is a logic truth (tautology) ...
with a table:
The output row for A has all Ts.
with a proof:
Prove A.
with a tree:
The tree for -A closes.
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