 WVOQ 55 Willard Van Orman Quine - cleancut
Descrição do Vídeo:
http://www-groups.dcs.st-and.ac.uk/~history/Printonly
/Quine.html - such as the constant function of one
argument that returns true, so that our beliefs have
not been fully formalised. To go further would require
some explicit notion of termination, and some way of
handling general recursive definitions formally in HOL.
We discuss this further in Sect. 5. To execute the
prover in Isabelle, we can rewrite the prove function
to prove, then use the lemma prove to rewrite prove.
Further rewriting can occur. Ex- amining the definitions,
one sees that all functions involved are com
Data: November 22, 2008
Palavras Relacionados: Philosophy Willard Van Orman Quine
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WVOQ 54 Willard Van Orman Quine - cleancut
Descrição do Vídeo:
http://www-groups.dcs.st-and.ac.uk/~history/Printonly
/Quine.html - OCaML definition. lemma prove : prove s =
(if s = [] then True else prove (step s)) At this point,
we believe that if the OCaML function terminates with
true, then the derivation is finite. Conversely, we
believe that if the derivation is finite, there is some
n such that the nth iteration is empty, and our algorithm
will discover this at stage n and terminate with true.
On the other hand, we also believe that if the
derivation is not finite, then the algorithm will fail
to terminate. We have shown formally in HO
Data: November 22, 2008
Palavras Relacionados: Philosophy Willard Van Orman Quine
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WVOQ 53 Willard Van Orman Quine - cleancut
Descrição do Vídeo:
http://www-groups.dcs.st-and.ac.uk/~history/Printonly
/Quine.html - this in HOL we run into a problem. HOL is
a logic of total functions, whilst the general recursive
OCaML definitions just given are clearly partial.
However, we can improve our confidence in the OCaML
definitions as follows. Firstly, we define our function
prove in a non-constructive way. Page 12 12 constdefs
prove :: nseq list ? bool prove s ? ? m. iter step s m =
[] constdefs prove :: seq ? bool prove s ? prove [ns-of-s
s] From lemma finite-deriv, prove s corresponds to the
finiteness of the deriva- tion from s
Data: November 22, 2008
Palavras Relacionados: Philosophy Willard Van Orman Quine
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WVOQ 51 Willard Van Orman Quine - cleancut
Descrição do Vídeo:
http://www-groups.dcs.st-and.ac.uk/~history/Printonly
/Quine.html - a ? fold for nats primrec iter g a 0 = a
iter g a (Suc n) = g (iter g a n) So iterating the step
function n times on an initial sequent s gives the
sequentsat depth n in the derivation. lemma ? x. ((n,x) ?
deriv s) = (x ? set (iter step [s] n)) Now we know that
the derivation from s is finite iff there is some n
such that there are no sequents at depth n in the
derivation, in which case the nth iteration of the step
function on s will be empty. lemma finite-deriv: finite
(deriv s) = (? m. iter step [s] m = []) No
Data: November 22, 2008
Palavras Relacionados: Philosophy Willard Van Orman Quine
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WVOQ 50 Willard Van Orman Quine - cleancut
Descrição do Vídeo:
http://www-groups.dcs.st-and.ac.uk/~history/Printonly
/Quine.html - all sequents at depth n, to obtain the
list of sequents at depth n + 1. If there are no
sequents at a given depth, then all branches have been
closed and we have found our finite derivation. We define
a global version of the algorithm as a function that
takes an initial sequent s, and a number n, and gives
back the list of sequents at depth n in the derivation
rooted at s. We need a step function that takes a list
of sequents at depth n in the derivation, and gives back
a list of sequents at depth n + 1. constd
Data: November 22, 2008
Palavras Relacionados: Philosophy Willard Van Orman Quine
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WVOQ 46 Willard Van Orman Quine - cleancut
Descrição do Vídeo:
http://www-groups.dcs.st-and.ac.uk/~history/Printonly
/Quine.html - the notion of when a formula is
considered at a point n in a path, which is when the
formula is at the head of the sequent at position n.
constdefs contains :: (nat ? (nat × nseq)) ? nat ?
nform ? bool contains f n nf ? nf ? set (snd (f n))
constdefs considers :: (nat ? (nat × nseq)) ? nat ?
nform ? bool considers f n nf ? case snd (f n) of [] ?
False | (x#xs) ? x = nf 3.8 Models 2 A falsifying model
will in general consist of at least countably many
elements. So far, we have said nothing about the size
of
Data: November 22, 2008
Palavras Relacionados: Philosophy Willard Van Orman Quine
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WVOQ 44 Willard Van Orman Quine - cleancut
Descrição do Vídeo:
http://www-groups.dcs.st-and.ac.uk/~history/Printonly
/Quine.html - provable. Alternatively, if s is not
provable, we must exhibit a model where s is false. If
s is not provable, then when we attempt to prove it
using the rules of our system, we will not end up with a
finite derivation. If the derivation tree is infinite
then, since it is finitely branching, we can use Konigs
lemma to find an infinite path in the tree. We call this
infinite path a failing path. We define a failing path
as a function from a derivation tree to a path through
the derivation tree. Paths are simply
Data: November 22, 2008
Palavras Relacionados: Philosophy Willard Van Orman Quine
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WVOQ 37 Willard Van Orman Quine - cleancut
Descrição do Vídeo:
http://www-groups.dcs.st-and.ac.uk/~history/Printonly
/Quine.html - that the proofs in the rest of the
mechanisation are uniform. ? Nil ? Then to represent
the rules, we use a function mapping the conclusion of
arule to a list of premises. consts subs :: nseq ? nseq
list primrec subs [] = [[]] subs (x#xs) = Page 7 7
(let (m,f ) = x in case f of PAtom p vs ? if NAtom p vs
? set (map snd xs) then [] else [xs@[(0,PAtom p vs)]] |
NAtom p vs ? if (PAtom p vs) ? set (map snd xs) then []
else [xs@[(0,NAtom p vs)]] | FConj f g ? [xs@[(0,f
)],xs@[(0,g)]] | FDisj f g ? [xs@[(0,f ),(0,
Data: November 22, 2008
Palavras Relacionados: Philosophy Willard Van Orman Quine
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WVOQ 32 Willard Van Orman Quine - cleancut
Descrição do Vídeo:
http://www-groups.dcs.st-and.ac.uk/~history/Printonly
/Quine.html - x and produces a result f x is denoted by
?x. f x. The function which is exactly the same as f,
except that x is mapped to y, is written f (x := y). The
type of a function with domain a and codomain b is a ?
b. ML style datatypes are present, as is definition by
primitive and wellfounded recursion. Lists are a
particularly important datatype. The type of lists over
a base type a is formed by applying the list type
constructor, viz. a list. The empty list is written [],
whilst the list xs with an additional x o
Data: November 22, 2008
Palavras Relacionados: Philosophy Willard Van Orman Quine
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WVOQ 11 Willard Van Orman Quine - cleancut
Descrição do Vídeo:
http://www-groups.dcs.st-and.ac.uk/~history/Printonly
/Quine.html - Quine developed a new type of philosophy,
which he called naturalized epistemology. He claimed
that epistemology's only legitimate role is to describe
the way knowledge is actually obtained so, according to
Quine, its function is to describe how present science
arrives at the beliefs accepted by the scientific
community. Among Quine's publications are works on
logic, metaphysics, the philosophy of language, and the
philosophy of logic. His 22 books include A System of
Logistic (1934), Mathematical Logic
Data: November 22, 2008
Palavras Relacionados: Philosophy Willard Van Orman Quine
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Data: November 22, 2008
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