- •Sense, Denotation and Semantics
- •Sense and denotation in logic
- •The algebraic tradition
- •The syntactic tradition
- •The two semantic traditions
- •Tarski
- •Heyting
- •Natural Deduction
- •The calculus
- •The rules
- •Interpretation of the rules
- •The Curry-Howard Isomorphism
- •Lambda Calculus
- •Types
- •Terms
- •Operational significance
- •Conversion
- •Description of the isomorphism
- •Relevance of the isomorphism
- •The Normalisation Theorem
- •The weak normalisation theorem
- •Proof of the weak normalisation theorem
- •Degree and substitution
- •Degree and conversion
- •Conversion of maximal degree
- •Proof of the theorem
- •The strong normalisation theorem
- •Sequent Calculus
- •The calculus
- •Sequents
- •Structural rules
- •The intuitionistic case
- •Logical rules
- •Some properties of the system without cut
- •The last rule
- •Subformula property
- •Asymmetrical interpretation
- •Sequent Calculus and Natural Deduction
- •Properties of the translation
- •Strong Normalisation Theorem
- •Reducibility
- •Properties of reducibility
- •Atomic types
- •Product type
- •Arrow type
- •Reducibility theorem
- •Pairing
- •Abstraction
- •The theorem
- •The calculus
- •Types
- •Terms
- •Intended meaning
- •Conversions
- •Normalisation theorem
- •Expressive power: examples
- •Booleans
- •Integers
- •Expressive power: results
- •Canonical forms
- •Representable functions
- •Coherence Spaces
- •General ideas
- •Coherence Spaces
- •The web of a coherence space
- •Interpretation
- •Stable functions
- •Parallel Or
- •Direct product of two coherence spaces
- •The Function-Space
- •The trace of a stable function
- •Representation of the function space
- •The Berry order
- •Partial functions
- •Denotational Semantics of T
- •Simple typed calculus
- •Types
- •Terms
- •Properties of the interpretation
- •Booleans
- •Integers
- •Sums in Natural Deduction
- •Defects of the system
- •Standard conversions
- •The need for extra conversions
- •Subformula Property
- •Extension to the full fragment
- •Commuting conversions
- •Properties of conversion
- •The associated functional calculus
- •Empty type
- •Sum type
- •Additional conversions
- •System F
- •The calculus
- •Comments
- •Representation of simple types
- •Booleans
- •Product of types
- •Empty type
- •Sum type
- •Existential type
- •Representation of a free structure
- •Free structure
- •Representation of the constructors
- •Induction
- •Representation of inductive types
- •Integers
- •Lists
- •Binary trees
- •Trees of branching type U
- •The Curry-Howard Isomorphism
- •Coherence Semantics of the Sum
- •Direct sum
- •Lifted sum
- •dI-domains
- •Linearity
- •Characterisation in terms of preservation
- •Linear implication
- •Linearisation
- •Linearised sum
- •Tensor product and units
- •Cut Elimination (Hauptsatz)
- •The key cases
- •The principal lemma
- •The Hauptsatz
- •Resolution
- •Strong Normalisation for F
- •Idea of the proof
- •Reducibility candidates
- •Remarks
- •Reducibility with parameters
- •Substitution
- •Universal abstraction
- •Universal application
- •Reducibility theorem
- •Representation Theorem
- •Representable functions
- •Numerals
- •Total recursive functions
- •Provably total functions
- •Proofs into programs
- •Formulation of HA2
- •Translation of HA2 into F
- •Representation of provably total functions
- •Semantics of System F
- •What is Linear Logic?
10.6. THE ASSOCIATED FUNCTIONAL CALCULUS |
79 |
Having said this, we shall give no proof, because the theoretical interest is limited. One tends to think that natural deduction should be modi ed to correct such atrocities: if a connector has such bad rules, one ignores it (a very common attitude) or one tries to change the very spirit of natural deduction in order to be able to integrate it harmoniously with the others. It does not seem that the (?; _; 9) fragment of the calculus is etched on tablets of stone.
Moreover, the extensions are long and di cult, and for all that you will not learn anything new apart from technical variations on reducibility. So it will su ce to know that the strong normalisation theorem also holds in this case. In the unlikely event that you want to see the proof, you may consult the references above.
10.6The associated functional calculus
Returning to the idea of Heyting, it is possible to understand the Curry-Howard isomorphism in the case of ? and _ (the case of 9 will receive no more consideration than did that of 8).
10.6.1Empty type
Emp is considered to be the empty type. For this reason, there will be a canonical function "U from Emp to any type U: if t is of type Emp, them "U t is of type U. The commutation for "U is set out in ve cases:
1("U V t) |
"U t |
2("U V t) |
"V t |
("U!V t) u |
"V t |
"U ("Emp t) |
"U t |
x: u y: v ("R+S t) |
"U t |
In the last case ( x: u y: v t is introduced |
below) U is the common type of |
u and v. It is easy to see that "U corresponds exactly to ?E and the ve conversions above to the ve commutations of ?.
80 |
CHAPTER 10. SUMS IN NATURAL DEDUCTION |
10.6.2Sum type
For U + V , we have the following schemes:
1. |
If u is of type U, then 1u is of type U + V . |
2. |
If v is of type V , then 2v is of type U + V . |
3.If x, y are variables of respective types R, S, and u, v, t are of respective types U, U, R + S, then
x: u y: v t
is a term of type U. Furthermore, the occurrences of x in u are bound by this construction, as are those of y in v. This corresponds to the pattern matching
match t with inl x ! u j inr y ! v
in a functional programming language like CAML.
Obviously the 1, 2 and schemes interpret _1I, _2I and _E. The standard conversions are:
x: u y: v ( 1r) u[r=x]
The commuting conversions are
1( x: u y: v t)
2( x: u y: v t)
( x: u y: v t) w
"W ( x: u y: v t)x0: u0 y0: v0 ( x: u y: v t)
x: u y: v ( 2s) |
v[s=y] |
1 |
1 |
U = V W |
x: ( 2u) y: ( 2v) t |
||
x: ( u) y: ( v) t |
U = V W |
|
x: (u w) y: (v w) t |
U = V !W |
|
( x: ("W u) y: ("W v) t) |
U = Emp |
|
x: ( x0: u0 y0: v0 u) y: ( x0: u0 y0: v0 v) t
U = V + W
which correspond exactly to the rules of natural deduction.
10.6.3Additional conversions
Let us note for the record |
the analogues of h 1t; 2ti t and x: t x t: |
"Emp t t |
x: ( 1x) y: ( 2y) t t |
Clearly the terms on both sides of the \ " are denotationally equal. However the direction in which the conversion should work is not very clear: the opposite one is in fact much more natural.