BHK interpretation
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In mathematical logic, the BHK interpretation of intuitionistic logic was proposed by L. E. J. Brouwer, Arend Heyting and independently by Kolmogorov. It is also sometimes called the realizability interpretation, because of the connection with the realizability theory of Stephen Kleene.
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[edit] The interpretation
The interpretation states exactly what is intended to be a proof of a given formula. This is specified by induction on the structure of that formula:
- A proof of
is a pair <a,b> where a is a proof of P and b is a proof of Q. - A proof of
is a pair <a,b> where a is 0 and b is a proof of P, or a is 1 and b is a proof of Q. - A proof of
is a function f which converts a proof of P into a proof of Q. - A proof of
is a pair <a,b> where a is an element of S, and b is a proof of φ(a). - A proof of
is a function f which converts an element a of S into a proof of φ(a). - The formula
is defined as 
, so a proof of it is a function f which converts a proof of P into a proof of 
. - is absurdity. There ought not to be a proof of it.
The interpretation of a primitive proposition is supposed to be known from context.
[edit] Examples
The identity function is a proof of the formula 
, no matter what P is.
The law of non-contradiction 
expands to 
:
- A proof of is a function f which converts a proof of
to a proof of . - A proof of is a pair of proofs <a,b>, where a is a proof of P, and b is a proof of .
- A proof of is a function which converts a proof of P to a proof of .
Putting it all together, a proof of is a function f which converts a pair <a,b> -- where a is a proof of P, and b is a function which converts a proof of P to a proof of -- into a proof of . The function 
fits the bill, proving the law of non-contradiction, no matter what P is.
On the other hand, the law of excluded middle 
expands to 
, and in general has no proof.
[edit] What is absurdity?
It is not possible for a logical system to have a formal negation operator such that there is a proof of "not" P exactly when there isn't a proof of P (in general -- see Gödel's incompleteness theorems). The BHK interpretation instead takes "not" P to mean that P leads to absurdity, designated , so that a proof of ¬P is a function converting a proof of P into a proof of absurdity.
A standard example of absurdity is found in dealing with arithmetic. Assume that 0=1, and proceed by mathematical induction: 0=0 by the axiom of equality. Now (induction hypothesis), if 0 were equal to a certain natural number n, then 1 would be equal to n+1, (Peano axiom: Sm = Sn if and only if m = n), but since 0=1, therefore 0 would also be equal to n+1. By induction, 0 is equal to all numbers, and therefore any two natural numbers become equal.
Therefore, there is a way to go from a proof of 0=1 to a proof of any basic arithmetic equality, and thus to a proof of any complex arithmetic proposition. Furthermore, to get this result it was not necessary to invoke the Peano axiom which states that 0 is "not" the successor of any natural number. This makes 0=1 suitable as in Heyting arithmetic (and the Peano axiom is rewritten 0=Sn → 0=S0). This use of 0=1 validates the principle of explosion.
[edit] What is a function?
The BHK interpretation will depend on the view taken about what constitutes a function which converts one proof to another, or which converts an element of a domain to a proof. Different versions of constructivism will diverge on this point.
Kleene's realizability theory identifies the functions with the computable functions. It deals with Heyting arithmetic, where the domain of quantification is the natural numbers and the primitive propositions are of the form x=y. A proof of x=y is simply the trivial algorithm if x evaluates to the same number that y does (which is always decidable for natural numbers), otherwise there is no proof. These are then built up by induction into more complex algorithms.
