## Abstraction and Application in (2001)

### BibTeX

@MISC{01abstractionand,

author = {},

title = {Abstraction and Application in},

year = {2001}

}

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### Abstract

The postulates of comprehension and extensionality in set theory are based on an inversion principle connecting set-theoretic abstraction and the property of having a member. An exactly analogous inversion principle connects functional abstraction and application to an argument in the postulates of the lambda calculus. Such an inversion principle arises also in two adjoint situations involving a cartesian closed category and its polynomial extension. Composing these two adjunctions, which stem from the deduction theorem of logic, produces the adjunction connecting product and exponentiation, i.e. conjunction and implication. Mathematics Subject Classification: 18A15, 18A40, 18D15 1

### Citations

986 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ...t be new, and a reader with some previous experience with cartesian closed categories (which he may have acquired by reading, for example, [10]), or with categories in general (for which many rely on =-=[12]-=-), should be able to perform them. 2 Set-Theoretical Postulates and Lambda Conversion The two grammatical categories of terms (i.e. individual terms) and of propositions are basic grammatical categori... |

144 |
Introduction to Higher-Order Categorical Logic. Number 7
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...eorem of logic, and which had been anticipated in combinatory logic, were first recognized by Lambek under the name functional completeness in his pioneering work in categorial proof theory (see [9], =-=[10]-=-, Part I, and references therein). Functional completeness is presented quite explicitly as adjunction in [7] and [3]. After a preliminary section on matters pertaining to the inversion principle of t... |

10 |
Fibrations with indeterminates: contextual and functional completeness for polymorphic lambda calculi
- Hermida, Jacobs
- 1995
(Show Context)
Citation Context ... name functional completeness in his pioneering work in categorial proof theory (see [9], [10], Part I, and references therein). Functional completeness is presented quite explicitly as adjunction in =-=[7]-=- and [3]. After a preliminary section on matters pertaining to the inversion principle of the postulates of set theory and of the lambda calculus, we shall turn to categorial proof theory and cartesia... |

8 |
Embedding of a free cartesian closed category into the category of sets
- Čubrić
- 1998
(Show Context)
Citation Context ...because of the arrow x and other arrows of K[x] involving x. It is also in general not one-one. Conditions that ensure that H is one-one on arrows are investigated in [9], [10] (I.5) and, especially, =-=[1]-=-. A necessary and sufficient condition, found in this last paper, is that the object D of x : T ⊢ D be “nonempty”, nonemptiness being expressed in a categorial manner by requiring that the arrow kD : ... |

8 |
Functional completeness of cartesian categories
- Lambek
- 1974
(Show Context)
Citation Context ...on theorem of logic, and which had been anticipated in combinatory logic, were first recognized by Lambek under the name functional completeness in his pioneering work in categorial proof theory (see =-=[9]-=-, [10], Part I, and references therein). Functional completeness is presented quite explicitly as adjunction in [7] and [3]. After a preliminary section on matters pertaining to the inversion principl... |

4 |
Deductive systems and categories III. Cartesian closed categories, intuitionist propositional calculus, and combinatory logic
- Lambek
- 1972
(Show Context)
Citation Context ...junctions, which are a refinement of the deduction theorem, were first considered by Lambek under the name functional completeness (see references above; in his first paper on functional completeness =-=[8]-=- Lambek actually envisaged rather unwieldy combinatorially inspired equalities, like those we mentioned in the previous paragraph). Through the categorial equivalence of the typed lambda calculus with... |

3 |
Modal Logic as Metalogic
- Doˇsen
- 1992
(Show Context)
Citation Context ...l. It will be the same for any kind of category. A step towards showing that conjunction and intuitionistic implication can be characterized by the adjunctions of functional completeness was taken in =-=[2]-=-, and, especially, [6]. What we need to show is that the assumptions made for cartesian, or cartesian closed categories, or categories that have only exponentiation and may lack product, are not only ... |

2 |
Modal functional completeness
- Dosen, Petric
- 1996
(Show Context)
Citation Context ... for any kind of category. A step towards showing that conjunction and intuitionistic implication can be characterized by the adjunctions of functional completeness was taken in [2], and, especially, =-=[6]-=-. What we need to show is that the assumptions made for cartesian, or cartesian closed categories, or categories that have only exponentiation and may lack product, are not only sufficient for demonst... |

1 |
Deductive completeness, Bull. Symbolic Logic 2
- Doˇsen
- 1996
(Show Context)
Citation Context ...nctional completeness in his pioneering work in categorial proof theory (see [9], [10], Part I, and references therein). Functional completeness is presented quite explicitly as adjunction in [7] and =-=[3]-=-. After a preliminary section on matters pertaining to the inversion principle of the postulates of set theory and of the lambda calculus, we shall turn to categorial proof theory and cartesian closed... |

1 |
Definitions of adjunction, in: W.A. Carnielli and I.M.L. D’Ottaviano eds
- Doˇsen
- 1999
(Show Context)
Citation Context ...has Φ in its name, while the function that corresponds to application has Γ. Before, it was the other way round. We make this switch to conform to the notation for adjoint situations of [3], [4], and =-=[5]-=-. Conforming to this same notation, in the next section matters will return to what we had in Sections 2 and 3. We can verify that Φ ′ x and Γ ′ x,A establish a bijection between the hom-sets K(D × A,... |