consistency in logic, ‘consistency’ has two meanings

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Consistency In logic, ‘consistency’ has two meanings.

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Page 1: Consistency In logic, ‘consistency’ has two meanings

Consistency

In logic, ‘consistency’ has two meanings.

Page 2: Consistency In logic, ‘consistency’ has two meanings

Consistency

In logic, ‘consistency’ has two meanings.

Consistency of a system of rules.

Page 3: Consistency In logic, ‘consistency’ has two meanings

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.

Page 4: Consistency In logic, ‘consistency’ has two meanings

Consistency

In logic, ‘consistency’ has two meanings.

Consistency of a group of statements.

Page 5: Consistency In logic, ‘consistency’ has two meanings

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.

Page 6: Consistency In logic, ‘consistency’ has two meanings

Consistency

A group of statements is consistent iff

It is (logically) possible that they be T together.

Page 7: Consistency In logic, ‘consistency’ has two meanings

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 theiroutputs are all T together.

Page 8: Consistency In logic, ‘consistency’ has two meanings

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 theiroutputs are all T together.

Sample: -Q, P>Q is consistent.

Page 9: Consistency In logic, ‘consistency’ has two meanings

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 theiroutputs are all T together.

Sample: -Q, P>Q is consistent.

P Q -Q P>Q T T F TF T F TT F T F F F T T * *

Page 10: Consistency In logic, ‘consistency’ has two meanings

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 theiroutputs are all T together.

Sample: -Q, P>Q is consistent.

P Q -Q P>Q T T F TF T F TT F T F F F T T * *

They can beT together.

Page 11: Consistency In logic, ‘consistency’ has two meanings

Testing for ConsistencyTo show A and B are consistent ... with a table: There is a single row where

A and B are both T.

Page 12: Consistency In logic, ‘consistency’ has two meanings

Testing for ConsistencyTo 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.

Page 13: Consistency In logic, ‘consistency’ has two meanings

Testing for ConsistencyTo 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.

Page 14: Consistency In logic, ‘consistency’ has two meanings

Testing for ConsistencyTo 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.

A tree is essentially a consistency check.

Page 15: Consistency In logic, ‘consistency’ has two meanings

Testing for ConsistencyTo 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.

Sample. This tree shows that the group -Q, P>Q is consistent.

Page 16: Consistency In logic, ‘consistency’ has two meanings

Testing for ConsistencyTo 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.

Sample. This tree shows that the group -Q, P>Q is consistent.

-QP>Q

-P Q *

Page 17: Consistency In logic, ‘consistency’ has two meanings

Validity and Consistency

There is an important relationshipbetween validity and consistency that is

the basis for the tree validity test.

Page 18: Consistency In logic, ‘consistency’ has two meanings

Validity and Consistency

There is an important relationshipbetween 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.

Page 19: Consistency In logic, ‘consistency’ has two meanings

Validity and Consistency

There is an important relationshipbetween 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.

Page 20: Consistency In logic, ‘consistency’ has two meanings

Validity and Consistency

There is an important relationshipbetween 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.

Page 21: Consistency In logic, ‘consistency’ has two meanings

Validity and Consistency

There is an important relationshipbetween 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.

Page 22: Consistency In logic, ‘consistency’ has two meanings

Validity and Consistency

There is an important relationshipbetween 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.

Page 23: Consistency In logic, ‘consistency’ has two meanings

Summary of the TestsTo 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.

For more click here