Transcript Consistency

Consistency
In logic, ‘consistency’ has two meanings.
Consistency
In logic, ‘consistency’ has two meanings.
Consistency of a system of rules.
Consistency
In logic, ‘consistency’ has two meanings.
Consistency of a system of rules.
If an argument is provable in the system,
then that argument is valid.
Consistency
In logic, ‘consistency’ has two meanings.
Consistency of a group of statements.
Consistency
In logic, ‘consistency’ has two meanings.
Consistency of a group of statements.
This is closer to the ordinary use of the word.
John may be wrong. but at least he is consistent.
Consistency
A group of statements is consistent
iff
It is (logically) possible that they be T together.
Consistency
A group of statements is consistent
iff
It is (logically) possible that they be T together.
iff
There is a row of their truth table where their
outputs are all T together.
Consistency
A group of statements is consistent
iff
It is (logically) possible that they be T together.
iff
There is a row of their truth table where their
outputs are all T together.
Sample: -Q, P>Q is consistent.
Consistency
A group of statements is consistent
iff
It is (logically) possible that they be T together.
iff
There is a row of their truth table where their
outputs are all T together.
Sample: -Q, P>Q is consistent.
P
T
F
T
F
Q
T
T
F
F
-Q P>Q
F
T
F
T
T
F
T
T
*
*
Consistency
A group of statements is consistent
iff
It is (logically) possible that they be T together.
iff
There is a row of their truth table where their
outputs are all T together.
Sample: -Q, P>Q is consistent.
P
T
F
T
F
Q
T
T
F
F
-Q P>Q
F
T
F
T
T
F
T
T
*
*
They can be
T together.
Testing for Consistency
To show A and B are consistent ...
with a table:
There is a single row where
A and B are both T.
Testing for Consistency
To show A and B are consistent ...
with a table:
There is a single row where
A and B are both T.
with a proof:
No proof test is possible.
Testing for Consistency
To show A and B are consistent ...
with a table:
with a proof:
There is a single row where
A and B are both T.
No proof test is possible.
with a tree:
The tree for A, B is open.
Testing for Consistency
To show A and B are consistent ...
with a table:
with a proof:
There is a single row where
A and B are both T.
No proof test is possible.
with a tree:
The tree for A, B is open.
A tree is essentially a consistency check.
Testing for Consistency
To show A and B are consistent ...
with a table:
with a proof:
There is a single row where
A and B are both T.
No proof test is possible.
with a tree:
The tree for A, B is open.
Sample. This tree shows that the group
-Q, P>Q is consistent.
Testing for Consistency
To show A and B are consistent ...
with a table:
with a proof:
There is a single row where
A and B are both T.
No proof test is possible.
with a tree:
The tree for A, B is open.
Sample. This tree shows that the group
-Q, P>Q is consistent.
-Q
P>Q
-P
Q
*
Validity and Consistency
There is an important relationship
between validity and consistency that is
the basis for the tree validity test.
Validity and Consistency
There is an important relationship
between validity and consistency that is
the basis for the tree validity test.
A, B | C is valid
iff
it is impossible that A=T, B=T and C=F.
Validity and Consistency
There is an important relationship
between validity and consistency that is
the basis for the tree validity test.
A, B | C is valid
iff
it is impossible that A=T, B=T and C=F.
iff
it is impossible that A=T, B=T and -C=T.
Validity and Consistency
There is an important relationship
between validity and consistency that is
the basis for the tree validity test.
A, B | C is valid
iff
it is impossible that A=T, B=T and C=F.
iff
it is impossible that A=T, B=T and -C=T.
iff
the group A, B, -C is inconsistent.
Validity and Consistency
There is an important relationship
between validity and consistency that is
the basis for the tree validity test.
A, B | C is valid
iff
it is impossible that A=T, B=T and C=F.
iff
it is impossible that A=T, B=T and -C=T.
iff
the group A, B, -C is inconsistent.
iff
the tree for A, B, -C is closed.
Validity and Consistency
There is an important relationship
between validity and consistency that is
the basis for the tree validity test.
A, B | C is valid
iff
the tree for A, B, -C is closed.
Summary of the Tests
To show A and B are consistent ...
with a table:
There is a single row where
A and B are both T.
with a proof:
No proof test is possible.
with a tree:
The tree for A, B is open.
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