Brouwer–Heyting–Kolmogorov interpretation explained

In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the realizability interpretation, because of the connection with the realizability theory of Stephen Kleene. It is the standard explanation of intuitionistic logic.

The interpretation

The interpretation states 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

P\wedgeQ

is a pair

\langlea,b\rangle

where

is a proof of

and

is a proof of

A proof of

P\veeQ

is either

\langle0,a\rangle

where

is a proof of

\langle1,b\rangle

where

is a proof of

A proof of

P\toQ

is a

that converts a proof of

into a proof of

A proof of

(\existsx{\in}S)(Px)

is a pair

\langlex,a\rangle

where

is an element of

and

is a proof of

A proof of

(\forallx{\in}S)(Px)

is a function

that converts an element

into a proof of

The formula

\negP

is defined as

P\to\bot

, so a proof of it is a function

that converts a proof of

into a proof of

\bot

There is no proof of

\bot

, the absurdity or bottom type (nontermination in some programming languages).

The interpretation of a primitive proposition is supposed to be known from context. In the context of arithmetic, a proof of the formula

x=y

is a computation reducing the two terms to the same numeral.

Kolmogorov followed the same lines but phrased his interpretation in terms of problems and solutions. To assert a formula is to claim to know a solution to the problem represented by that formula. For instance

P\toQ

is the problem of reducing

; to solve it requires a method to solve problem

given a solution to problem

Examples

The identity function is a proof of the formula

P\toP

, no matter what P is.

\neg(P\wedge\negP)

expands to

(P\wedge(P\to\bot))\to\bot

A proof of

(P\wedge(P\to\bot))\to\bot

is a function

that converts a proof of

(P\wedge(P\to\bot))

into a proof of

\bot

A proof of

(P\wedge(P\to\bot))

is a pair of proofs <a, b>, where

is a proof of P, and

is a proof of

P\to\bot

A proof of

P\to\bot

is a function that converts a proof of P into a proof of

\bot

.Putting it all together, a proof of

(P\wedge(P\to\bot))\to\bot

is a function

that converts a pair <a, b> – where

is a proof of

, and

is a function that converts a proof of

into a proof of

\bot

– into a proof of

\bot

.There is a function

that does this, where

f(\langlea,b\rangle)=b(a)

, proving the law of non-contradiction, no matter what P is.

Indeed, the same line of thought provides a proof for the modus ponens rule

(P\wedge(P\toQ))\toQ

as well, where

is any proposition.

P\vee(\negP)

expands to

P\vee(P\to\bot)

, and in general has no proof. According to the interpretation, a proof of

P\vee(\negP)

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

P\to\bot

. Thus if neither P nor

P\to\bot

is provable then neither is

P\vee(\negP)

Definition of absurdity

It is not, in general, possible for a logical system to have a formal negation operator such that there is a proof of "not"

exactly when there isn't a proof of

; see Gödel's incompleteness theorems. The BHK interpretation instead takes "not"

to mean that

leads to absurdity, designated

\bot

, so that a proof of

lnotP

is a function converting a proof of

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 that states that 0 is "not" the successor of any natural number. This makes 0 = 1 suitable as

\bot

in Heyting arithmetic (and the Peano axiom is rewritten 0 = Sn → 0 = S0). This use of 0 = 1 validates the principle of explosion.

Definition of function

The BHK interpretation will depend on the view taken about what constitutes a function that converts one proof to another, or that 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.

If one takes lambda calculus as defining the notion of a function, then the BHK interpretation describes the correspondence between natural deduction and functions.

References

Web site: Troelstra. A.. Anne Sjerp Troelstra. 1991. History of Constructivism in the Twentieth Century.
Troelstra. A.. Anne Sjerp Troelstra. 2003. Constructivism and Proof Theory (draft). 10.1.1.10.6972.

External link

Van Atten. Mark. 4 May 2022. The Development of Intuitionistic Logic. intuitionistic-logic-development.

Brouwer–Heyting–Kolmogorov interpretation explained

The interpretation

Examples

Definition of absurdity

Definition of function

See also

References

External link