Intuitionistic Logic
An Alternative to Classical Logic
What is Intuitionistic
Logic?
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Related to “intuitionism”
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Q: How do we know something is true?
A: It is proven!
Intuitionistic logic (IL) is the logic of
intuitionists
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A thing is true iff it is provable
Provability
Classical
Logic
Intuitionistic
Logic
A
A is true
A is provable
~A
A is false
A is disprovable
~(~A)
“A is false” is false “A is disprovable”
(A is true)
is disprovable
Law of the Excluded Middle
(LEM)
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Under classical logic, ~(~A) → A (LEM)
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If A is not false, A is true
Under intuitionistic logic, LEM is not
always true!
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Gödel’s Theorem
Gödel’s Theorem
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Let S be the statement: “S is not
provable.”
Is S provable?
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No!
Is S disprovable?
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No!
S is neither provable nor disprovable!
The third state
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Not A, but not ~A. So what is it?
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“Unknown”
Something is either provable,
disprovable, or unknown
Reductio ad Absurdum
• Proving “A” with Reducio ad
Absurdum:
1. Assume ~A.
2. Show ~A is impossible.
3. ~(~A), therefore A.
• Problem: in intuitionistic logic, step 3
is invalid!
Classical Logic
Example
• Prove there are an infinity of primes.
1. Let a, b, c, ..., p stand for all of the
(finite number of) primes where p is the
biggest.
2. But then a ⋅b⋅...⋅p + 1 is prime (not
divisible by any primes below it) but it is
bigger even than p. Contradiction!
3. There is not a finite number of primes.
Therefore, there are an infinite number
of primes.
Why Intuitionist Logic
is Annoying
3. There is not a finite number of primes.
Therefore, there are an infinite number of
primes.
This is just Reducio ad Absurdum, and we
can’t use Reducio ad Absurdum in
intuitionist logic!
A Summary
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Law of the excluded middle: gone
Reductio ad absurdum: gone
Aggravation: record heights
Why do we suffer?!
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Arises from philosophical questions:
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How do we know anything?
Troubling:
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How can we do anything without
Reductio ad Absurdum?
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More complex routes; better?
Maybe intuitionist logic isn’t so bad after
all
The Advantages
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Not prone to paradoxes
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“Consistent”
As prone to paradoxes as classical
logic
“Will it rain tomorrow?”
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Classical answer: yes or no
Intuitionistic answer: unknown!