The tasks of logic
Why we need more versatile tools
Philosophy and logic 2013
Kyiv 25 May 2013
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The task of logic – the classical view
• To analyse reasoning or arguments.
Requires two things:
• that logic can correctly represent the
components of reasoning (thoughts) and
arguments (sentences), and
• that it can also correctly represent correct
dynamic flows of such components
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The task of logic – in common practice
• A tool for the analysis of concepts and
definition of terms
Requires two things:
• that logic can correctly represent entities at
the level of concepts and words, and
• that it can also correctly represent how these
combine to form entities at the next level
(thoughts and sentences)
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A useful table
sphere
language
world
mind
content
element
word
object
idea
concept
statics
sentence
fact
judgem.
propos.
dynamics
text
process
thinking
reasoning
General problem: what sort of morphisms exist
between the spheres?
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Inessential limitations of logic
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Cannot represent modalities (add operator)
Cannot express finitude (generalise quantifier)
Cannot represent finitude (go second order)
Cannot represent dependencies (branch
quantifiers)
• Cannot represent X (add new symbol)
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Essential limitation of logic
There is no formal language where a categorical set
of arithmetical sentences can be formulated and the
syntactic and semantic concepts of consequence
coincide.
In other words: either content cannot be
represented in its entirety, or reasoning cannot.
This is a also conflict between the two most
fundamental concepts of epistemology, knowing
and thinking, the static and the dynamic aspect of
knowledge.
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Lack of isomorphism between
language and the mind for ”belief”
• If a belief is acquired by seeing or otherwise
experiencing something directly, then it has a
fullness that cannot be exhaustively described
by any number of sentences because there
are no words for most of the concepts
involved.
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Lack of isomorphism between
language and the mind for ”belief”
• If a belief is acquired by reading or being told
something, it is already given in a linguistic
form and therefore presumably propositional
in character. It has been depleted in relation to
the original experience (if there was one).
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Is a logic of belief possible?
• Beliefs (like thoughts) no doubt have content
• Some (like Isaac Levi) think that these
contents form a Boolean structure
• If so, they can be represented in
”propositional logic”, but it doesn’t follow that
they are propositional
• More likely, though, negation is problematic
and we need some other representation
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Is a logic of belief possible?
• The simplest solution would be to use a
Heyting algebra instead, but it would not
really be informative
• Preferably, salient aspects of beliefs should be
taken into account
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Beliefs have properties along many
dimensions
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Certainty (of content)
Justification (logical relation to other beliefs)
History of coming into being (pedigree)
Robustness (sensitivity to further inquiry)
Satisfactoriness (relation to holder,
determining willingness to further inquiry)
• Etc.
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Fundamental question of the general
theory of knowledge
Is knowledge is just a species of belief (whether
those beliefs that happen to be true and justified,
or singled out in any other way); can proper
knowledge be studied without regard to beliefs in
general?
Suspected answer:
Knowledge is a limiting case, only the surface of a
deep sea of beliefs (or a hyperplane in the space of
beliefs, for those with a more formal mind), and
one cannot well understand the ripples on the
surface unless one studies what goes on beneath.
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Problem to be studied
Which is the nature and structure of belief states? (We
know fairly well that they cannot be sets of propositions
as modelled in a formal language.) And how can we
model them if we want to study them more precisely?
Suspicions
Many problems in belief revision are spurious and selfgenerated, resulting from inappropriate modelling.
Probability is too one-dimensional for modelling the
partiality of beliefs; it cannot, for example, render the
disposition to react to new information, i.e. robustness.
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Tactical question
Shall one concentrate on states of full belief, as
Isaac Levi does, rather than on belief states in
general? Pro: full beliefs are a special case and it is
therefore easier to start with them. Contra: one
may be tempted to seek ad hoc solutions to
problems about full belief, whose drawbacks do not
become obvious unless one studies the general
case.
Solomonian answer
Do both, and see how they interact.
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The measurement of power
• Power consists in exerting influence over
other people
• But it counts more if you exert influence over
powerful people, in proportion to their power
• This is a circular definition, but that is not an
argument for its inadmissability
• Use a vector/matrix equation!
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Powers as a vector
• pi = power of individual i
• cij = influence of i over j
• P = n x 1 matrix of powers
• C = n x n matrix of influences
Then solve the equation: CP = P
The same idea applicable to other concepts with
apparently circular definition, like coherence.
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Voting, or preference aggregation
• Traditional: function from sequence of
individual preferences to group preference
• Alternative: look at decisive groups, i.e. groups
that can force their will if united
• They form an ultrafilter
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Ultrafilters (”large subsets”)
Set of subsets of a given set satisfying:
• Empty set does not belong
• Closed under intersection
• Closed under supersets
• Either a set or its complement belongs
Ultrafilters are either principal (= all subsets
containing a given element) or free.
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Ultrafilters
The ultrafilter technique is applicable to all sorts
of aggregation problems, and also for the
construction of infinitesimals in mathematics.
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The philosopher’s toolbox
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Logics
Set theory and transfinite numbers
Lattices
Algebraic structures
Vectors and linear spaces
Topology, metric spaces
Graph theory
Category theory??
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