Introduction to Higher-Order Categorical LogicIn this volume, Lambek and Scott reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher-order logic, and cartesian closed categories, are essentially the same. Part II demonstrates that another formulation of higher-order logic, (intuitionistic) type theories, is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained. Detailed historical references are provided throughout, and each section concludeds with a set of exercises. |
Contents
I | 3 |
II | 4 |
III | 8 |
IV | 12 |
V | 16 |
VI | 19 |
VII | 27 |
VIII | 35 |
XXXIII | 133 |
XXXIV | 139 |
XXXV | 145 |
XXXVI | 148 |
XXXVII | 153 |
XXXVIII | 160 |
XXXIX | 164 |
XL | 169 |
IX | 41 |
X | 42 |
XI | 47 |
XII | 50 |
XIII | 52 |
XIV | 55 |
XV | 57 |
XVI | 59 |
XVII | 62 |
XVIII | 65 |
XIX | 68 |
XX | 72 |
XXI | 77 |
XXII | 81 |
XXIII | 84 |
XXIV | 88 |
XXV | 93 |
XXVI | 98 |
XXVII | 101 |
XXVIII | 107 |
XXIX | 114 |
XXX | 123 |
XXXI | 124 |
XXXII | 128 |
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Common terms and phrases
A-calculus a₁ adjoint functors arrow f axioms Boolean C-monoid calculation canonical subobjects cartesian closed category category of sets closed term completes the proof computable construction Corollary deductive system defined Definition diagram easily elements epimorphism equations equivalence Example Exercise finite free topos free variables Freyd functional completeness functor categories functor F given graph hence Heyting algebra holds Hom(A indecomposable indeterminate arrow induction internal language intuitionistic type theory isomorphism Lambek left adjoint Lemma M-sets mapping monoid monomorphism Moreover morphism natural numbers object natural transformation numerical functions obtain pair polynomial preserves primitive recursive functions proof of Proposition Proposition 6.1 provable prove pure type theory reader representable rule of choice Section sheaf strict logical functor subset Suppose term forming operations term of type terminal object Theorem topos toposes triple unique arrow universal property variable of type weak natural numbers write