Logic: Types of truth
Classical, Intuitionistic, and Paraconsistent Truths
James B. Wilson Professor of Mathematics
True or False or else?
Issues:
- Is it ALL dogs, or ALL trucks?
- Can a dog "love"?
- What would count as evidence?
True/False: Dogs love trucks!
Can this question be answered?
Sentences/Statements
A logical sentence (or statement) is one what we bother to judge as possibly true.
- Must be Well-formed.
- No bad grammar x−=7
- No undefined words or terms: "flubergartten und elevendy"
Classical Truth
A sentence that is exclusively TRUE or FALSE.
- 2+3=5
- (2+3)2=22+32 [normally written (2+3)2=22+32]
- If it rains then the ground will be wet.
Useful in Science/Math to make predictions.
Intuitionistic Truth
A sentence that is exclusively TRUE or FALSE,
but truth comes from evidence.
- f(n)={n/23n+1n evenn odd will end in 1.
- If you visit a bad website then you will receive unwanted emails.
Useful in Computer Science to discuss programs which can't be asked questions without giving them data.
Paraconsistent Truth
A sentence that is TRUE or FALSE or possibly both.
"This sentence is false." (Liar Paradox)
The database of student heights includes [5,6,5,5,50,6].
Claim. The average height is 5.4.
- This is FALSE as the actual average is 12.8.
- This is TRUE, obviously 50 was a mistake entry so it should be dropped in the calculation.
Useful in data science where exceptions to rules are required to address noise and errors.
Classical Logic
Just True,False
Intuitionistic Logic
Truth comes from evidence
Paraconsistent Logic
True/False and some overlaps
Consequences of our truth models
Law of Explosion
- In Classical and Intuitionistic Logic:
- If you find evidence for something that is false
- THEN EVERYTHING IS FALSE
- This is not a fact but a rule.
- So to prove something false you just need ONE counter-example.
- In paraconsistent logic you need more than a single counter-example to prove things false.
Godel's Incompleteness
Godel's "Incompleteness Theorem" says some sentences are
- true and
- have no evidence.
This is true in every form of logic, just stated differently. E.g. in intuitionisitic logic we say the sentence is "neither true nor false"