Completeness and Consistency
So far we have 11 rules.
PA, and an In and an Out rule
for each of 5 connectives.
Call this system of 11 rules: P
Completeness and Consistency
So far we have 11 rules.
PA, and an In and an Out rule
for each of 5 connectives.
Call this system of 11 rules: P
Worry: Do we have enough rules?
Are there valid arguments that have no proofs in P?
Completeness and Consistency
So far we have 11 rules.
PA, and an In and an Out rule
for each of 5 connectives.
Call this system of 11 rules: P
Worry: Do we have enough rules?
Are there valid arguments that have no proofs in P?
There is a reason to worry.
Irvin Copi’s first book had 25 rules,
but there are valid arguments that
can’t be proven in it.
Completeness and Consistency
So far we have 11 rules.
PA, and an In and an Out rule
for each of 5 connectives.
Call this system of 11 rules: P
Worry: Do we have enough rules?
Are there valid arguments that have no proofs in P?
Luckily we can prove that P is has all the rules its needs.
If an argument is valid then it is provable in P.
Completeness and Consistency
Definition of Completeness:
A system is complete
iff
If an argument is valid then it can be proven in the system.
Completeness and Consistency
Definition of Completeness:
A system is complete
iff
If an argument is valid then it can be proven in the system.
Luckily we can prove that P is complete:
If an argument is valid then it is provable in P.
Unfortunately Copi’s 25 rule system is not complete,
despite its large number of rules.
Completeness and Consistency
A second worry:
Can you prove any invalid arguments in P?
(That would be a disaster!)
Completeness and Consistency
A second worry:
Can you prove any invalid arguments in P?
(That would be a disaster!)
You might think we are safe:
Each rule is clearly valid, and
proofs use only valid arguments.
Completeness and Consistency
A second worry:
Can you prove any invalid arguments in P?
(That would be a disaster!)
You might think we are safe:
Each rule is clearly valid, and
proofs use only valid arguments.
However more than once in history
logicians have developed systems of rules that
allowed one to prove everything!
Completeness and Consistency
Definition of Consistency
A system is consistent
iff
If an argument can be proven in the system, then it is valid.
Completeness and Consistency
Definition of Consistency
A system is consistent
iff
If an argument can be proven in the system, then it is valid.
Fortunately it has been verified that P
never proves invalid arguments.
Completeness and Consistency
A system is complete
iff
If an argument is VALID then it can be PROVEN in the system.
A system is consistent
iff
If an argument can be PROVEN in the system, then it is VALID.
Completeness and Consistency
A system is complete
iff
If an argument is VALID then it can be PROVEN in the system.
V>P
A system is consistent
iff
If an argument can be PROVEN in the system, then it is VALID.
Completeness and Consistency
A system is complete
iff
If an argument is VALID then it can be PROVEN in the system.
V>P
A system is consistent
iff
If an argument can be PROVEN in the system, then it is VALID.
P>V
Completeness and Consistency
A system is complete
iff
If an argument is VALID then it can be PROVEN in the system.
V>P
A system is consistent
iff
If an argument can be PROVEN in the system, then it is VALID.
P>V
V<>P
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