By Giovanni Sommaruga
A accomplished survey of Martin-Löf's optimistic style thought, huge components of that have in basic terms been awarded by means of Martin-Löf in lecture shape or as a part of convention talks. Sommaruga surveys the prehistory of kind idea and its hugely advanced improvement via 8 various levels from 1970 to 1995. He additionally offers a scientific presentation of the newest model of the speculation, as provided through Martin-Löf at Leiden collage in Fall 1993. This presentation supplies a fuller and up-to-date account of the procedure. prior, short displays took no account of the problems on the topic of the type-theoretical method of common sense and the principles of arithmetic, whereas the following they're accorded a whole a part of the publication.
Readership: entire money owed of the heritage and philosophy of optimistic variety idea and a large amount of similar fabric. Readers want a sturdy historical past in average good judgment and a primary, easy acquaintance with variety theory.
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Extra info for History and Philosophy of Constructive Type Theory
X r -fresh' means that x is a variable not occurring in r. 2) What do judgements of the form 'y : r' and 'y = 8 : r' (where r : context) mean? or What does it mean for y to be an environment for the context r? And what does it mean for the environments y and 8 for r to be identical? 34 CHAPTER 1 'y : r' means that for every variable declaration x : a in r (i) ay : type, and (ii) xy : ay. 'y = 8 : r' means that for every variable declaration x : a in r (i) ay = a8 : type, and (ii) xy = x8 : ay.
Removal 1) I : (X : a)f3 x : a I-- I(x) = g(x) : 13 ---'-----=-------'-- I =g : (X : a)f3 if f does not depend on x (arg. removal 2) same proviso This is the ~ - or extensionality rule in combinatory logic. Justification (arg. removal I) Thejudgement' I : (x : a)f3' was semantically explained as meaning I(x) : 13 provided x : a (cf. 3)). (arg. removal 2) And the judgement' I = g : (x : a)f3' was semantically explained as meaning I(x) = g(x) : 13 provided x : a (cf. 4)). D The argument move rules are a:a x:al--b:f3 - - - - - - - - - ' - (arg.
Ay is read as the closure of a with y. C,n&o This presentation of the semantical explanations of T : context', 'y : r' and 'a : type [rr may hide the fact that these three forms of judgement are actually defined simultaneously. 2) What does a judgement of the form 'a = f3 : type [rr mean? or What does it mean for two relative types to be identical? Presuppositions in the order of conceptual priority: a : type [r], f3 : type [r] 'a = f3 : type [rr means that for any y : r, ay = f3y : type. 3) What does a judgement of the form 'a : a [r], mean?
History and Philosophy of Constructive Type Theory by Giovanni Sommaruga