Consistency In logic, ‘consistency’ has two meanings

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 theiroutputs 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 theiroutputs 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 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 * *

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.

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

A and B are both T.

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.

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.

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.

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.

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 *

Validity and Consistency

There is an important relationshipbetween validity and consistency that is

the basis for the tree validity test.

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.

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.

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.

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.

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.

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.

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