Chinese University of Hong Kong

1
Revising Our Logic

• Chinese University of Hong Kong
• March 5, 2012

2
Outline

• 1. Paradoxes of truth, definability, (e.g.
  Liar)
• Why they arent idle puzzles
• How they make a case for revising logic (
  fundamental rules of reasoning)
• 2. But can we rationally revise logic
  (fundamental rules of reasoning)?
• 3. The nature of logic. How the right account of
  this helps with (but doesnt by itself fully
  answer) the puzzles about rational revision of
  logic.

3
The Liar paradox

• What Im saying is false.
• Seems that
• (1) what he says (S) is true if and only if
  its false.
• (2) its false if and only if its not false.
• (1) is surprising it seems to imply that S is
  either both true and false, or else neither true
  nor false.
• (2) is more than surprising, it seems
  contradictory
• If S is false, its not false, so both false and
  not false.
• If S is not false, its false, so again both
  false and not false.
• So either way, S is both false and not false!
  

4
Not take it seriously?

• This is the Liar paradox. Its been around for
  2000 years, with no agreement as to how best to
  deal with it.
• One might not be inclined to take it seriously,
  on one of two grounds
• that it only arises for artificial examples that
  could never arise in practice
• that it can never affect anything were really
  interested in.
• Both grounds are mistaken.

5
(a) Non-artificial examples

• Two candidates for the same office, on election
  night.
• Polls have closed. Initial returns strongly
  support Candidate A.
• Candidate B is on TV, denouncing Candidate A.
• Candidate A, watching the TV, says What the
  idiot who lost the election is now saying is
  false.
• The initial returns were wrong Candidate B won.
• And of course both candidates are idiots.
• So Candidate As remark is equivalent (given the
  facts) to the claim that it itself is false.
• So again, its false if and only if its not
  false.

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Why its a paradox

• This is a paradox elementary assumptions about
  truth and falsity, together with very simple
  principles of standard logic, have led to
  inconsistency.
• The point isnt that the politician was being
  inconsistent. That wouldnt be paradoxical at
  all then what hed be saying is false.
• Rather, the point is that whatever we say to
  describe the politicians utterance inevitably
  leads us into inconsistency. If we call it
  false, we seem committed to also calling it not
  false, and conversely.
• The politician story is an illustration of how
  paradox can arise out of normal conversation in
  which people dont intend to be speaking
  paradoxically.

7
(b) Do such paradoxes matter?

• Dismiss the paradoxes on the ground that they
  never affect anything were really interested in?
  
• That would be mistaken. There are infinitely
  many other paradoxes that turn on the same
  principles that the Liar paradox turns on.
• Even very good mathematicians and logicians are
  sometimes led to serious errors by using
  reasoning which on closer inspection turns on
  some paradox or other.
• We need a notion of truth, and will be led to
  error if we dont get its principles consistent,
  and indeed correct.

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Dangers of inadvertent paradoxical reasoning

• People sometimes are led to serious errors by
  using reasoning which on closer inspection turns
  on paradoxes.
• Examples
• Königs purported disproof of the Axiom of
  Choice.
• Soundness arguments in logic arguments that
  certain systems of derivation can never lead from
  truth to non-truth.
• More on (1)

9
König

• Axiom of Choice a once-controversial
  mathematical axiom, now a cornerstone of
  mathematics.
• König (prominent early 20th century set theorist)
  thought hed disproved it, by arguing that it
  implies that there are real numbers that both are
  and are not definable in a given language.
• The argument is extremely persuasive indeed,
  there is no agreement as to where it goes wrong!
  
• People came to agree that its wrong only when
  Berry came up with a similar argument for
  contradiction not relying on controversial
  assumptions like AC.

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Other examples where were fooled

• Königs paradox, like the Liar, is a paradox of
  truth. (Reason definability is explained in
  terms of an expression being true of an object.)
• It is just one of many examples where its easy
  to be taken in by proofs that, on closer
  analysis, turn on principles about truth that
  jointly lead to paradox.
• Soundness arguments in logic---arguments that
  certain systems of derivation can never lead from
  truth to non-truth---fool many philosophers today.

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REPRISE Do such paradoxes matter?

• Dismiss the paradoxes on the ground that they
  never affect anything were really interested in?
  
• Mistaken.
• Even very good mathematicians and logicians are
  sometimes led to serious errors by using
  reasoning which on closer inspection turns on
  paradoxes
• We need a notion of truth, and will be led to
  error if we dont get its principles consistent
  and indeed correct.

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Need of truth (1)

• We need a notion of truth, in daily life and in
  mathematics.
• Within the standard framework of mathematics
  (Zermelo-Fraenkel set theory) we can define
  restricted notions of truth, but not the full
  thing.
• This limits natural reasoning within ZF set
  theory in many ways many informal mathematical
  arguments that everyone wants to make simply
  cant be formalized within the official ZF.

13
Need of truth (2)

• We get a much more flexible system if we add a
  notion of truth to the set theoretic framework.
• But different assumptions about truth yield
  different expansions of the framework.
• Some mathematical sentences not settled in ZF set
  theory are settled in such expansions.
• But we want the right expansion, to settle the
  questions in the right way. For this, we need to
  know the right morals of the Liar and similar
  paradoxes. ///

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REPRISE Outline

• 1. Paradoxes of truth, definability,
• Why they arent idle puzzles
• How they make a case for revising logic (
  fundamental rules of reasoning)
• 2. But can we rationally revise logic
  (fundamental rules of reasoning)?
• 3. Understanding the nature of logic, and how it
  helps with (though doesnt by itself fully
  answer) the puzzles about rational revision of
  logic.

15
Back to Liar paradox (1)

• Simpler version a sentence L that says of itself
  that it is not true (rather than false).
• (Such sentences can arise naturally in
  conversation.)
• Its puzzling what to say about L
• If we declare L not true, then that declaration
  is equivalent to L itself so were asserting L
  while in the same breath declaring it not true.
  Very odd!
• And if we declare L true, then our declaration is
  equivalent to a denial of L itself so were
  denying L while in the same breath declaring it
  true. Also very odd!

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Back to Liar paradox (2)

• What if we neither declare it not true nor
  declare it true?
• If we accept standard logic, we must at least
  declare that its one or the other.
• But if its absurd to declare it true, and absurd
  to declare it not true, isnt it equally absurd
  to say that its either true or not true, even if
  I dont say which?
• (If I say Either there are square circles or
  triangular squares,
• Ive said something absurd, even if I dont
  say which one.)

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Yogi Berra

• There are people who think its OK to assert that
  either the Liar sentence is true or its not
  true, even while holding that each is absurd.
• Yogi Berra once offered the following advice
• If you come to a fork in the road, take it.
• These people reject that advice.

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Incoherence Principles

• Upshot If we dont restrict classical logic, we
  must violate at least one of the following
  Incoherence Principles
• First Incoherence Principle it is incoherent to
  accept A while accepting A isnt true.
• Second Incoherence Principle it is incoherent to
  accept A is true while accepting not-A.
• Third Incoherence Principle If accepting either
  of two claims (B, C) would be incoherent, then it
  is incoherent to accept that either B or C.
• Each of these principles seems compelling.
• For this and other reasons, my preferred approach
  is to weaken classical logic slightly, in such a
  way that the problem doesnt arise.

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My approach

• The law of excluded middle is the principle
• p or not-p (for any p you like).
• The idea is to reject that it holds generally.
  In particular, we reject
• () Either L is true or it isnt true.
• Without (), we can accept that its absurd to
  call the Liar sentence true and that its absurd
  to call it not true, as the first two Incoherence
  Principles require, without violating the Third
  Incoherence Principle.
• We can restrict excluded middle here in a way
  that allows for unrestricted classical logic
  within mathematics etc.

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Contrast to intuitionism

• LEM (Law of excluded middle) p or not-p (for
  any p you like).
• Famously criticized by the Dutch intuitionists
  (Brouwer, Heyting), who proposed doing math
  without it.
• Hilbert Depriving the mathematician of LEM would
  be like depriving the boxer the use of his fists.
• I have a way of restricting logic thats immune
  to the critique it leaves logic unaffected in
  math, physics, etc.
• Also intuitionist logic doesnt ultimately help
  with the paradoxes.

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Results of my approach

• An adequate treatment of the Liar must deal also
  with far more complicated paradoxes (infinitely
  many).
• One of my main areas of research over the last
  few years was to work out such an account.
• I showed how one can get a logic that
• is in some sense close to classical logic,
• reduces to classical logic in contexts where
  paradoxes dont threaten, e.g. math and physics,
• where even in paradoxical contexts, the claim
  that a given sentence is true is equivalent to
  that sentence.
• allows for all three incoherence principles.

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Costs and benefits

• Is restricting LEM to handle the paradoxes worth
  the cost? Requires detailed cost/benefit
  analysis of this and competing approaches.
  (Given in my recent book Saving Truth From
  Paradox.)
• Some people think a cost/benefit analysis
  inappropriate they think it intrinsically wrong
  to suggest an alteration of the laws of classical
  logic.
• Similar arguments have been given in the case of
  other cases of radical change in view
• Gauss/Riemann on weakening Euclidean views of
  geometric structure
• Einstein on weakening Newtonian views of
  simultaneity.

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Physics as precedent (1)

• Before Einstein, it was hard to see how Newtonian
  views of simultaneity could fail. (Similarly for
  Gauss and Riemann re Euclidean geometry.)
• It seemed part of our conceptual scheme that
  there is an objective simultaneity relation
  satisfying Newtonian assumptions.
• No one even made these assumptions explicit until
  Einstein questioned them. (Similarly
  Gauss/Riemann, with variable-curvature geometry.)

24
Physics as precedent (2)

• Gauss/Riemann and Einstein developed generalized
  theories that
• accommodated the successes of the earlier
  theories,
• while allowing exceptions to handle contrary
  observations.
• Even after their work, some conservatives argued
  that theres something intrinsically wrong with
  trying to change such basic conceptual apparatus.
• Generally agreed to be disreputable there. But
  if there, why not equally so in the case of logic?

25
Summary so far

• Logical change shouldnt be made lightly.
• Perhaps it shouldnt be made at all.
• But if a way is proposed to
• avoid serious anomalies by an alteration of logic
  
• that will
• accommodate ordinary reasoning in most
  circumstances, and
• allow for something close to ordinary reasoning
  everywhere,
• then it should not be dismissed out of hand.
  ///

26
REPRISE Outline

• 1. Paradoxes of truth, definability,
• Why they arent idle puzzles
• How they make a case for revising logic (
  fundamental rules of reasoning)
• 2. But can we rationally revise logic
  (fundamental rules of reasoning)?
• 3. Understanding the nature of logic, and how it
  helps with (though doesnt by itself fully
  answer) the puzzles about rational revision of
  logic.

27
Can we change logic rationally?

• Revision of logic does raise some special
  epistemological problems. We need an account of
  how rational change in logic is possible.
• Ive said its rational to change our logic when
  we have an alternative that
• is enough like the old logic to serve ordinary
  purposes just as well,
• but is just different enough to avoid certain
  anomalies that arise only in special
  circumstances.
• But this is too vague to answer all worries.

28
An apparent obstacle

• A precise account would be something like a
  computer (or probabilistic automaton) model of an
  agent rationally changing her logic. Or anyway,
  a sketch of one.
• An apparent obstacle to providing one you need
  logic in arguing for the change in logic. This
  may seem to raise a threat of some kind of
  vicious circularity.
• To elaborate

29
Fundamental rules

• It is natural (inevitable?) to think of rational
  thought in terms of the employment of fundamental
  rules
• To ascertain whats true we rely on a set of
  epistemic rules, or norms, that tell us in some
  general way what it would be most rational to
  believe under different epistemic circumstances.
  (Boghossian 2008)
• The idea isnt just of rules, but of fundamental
  ones general rules under which all our rational
  practices can be subsumed, and which in
  particular dictate all rational revision.
  

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Rational change of fundamental rules?

• Fundamental rules that dictate all rational
  revision?
• If so, its hard to see how these fundamental
  rules can rationally change.
• Do the rules say dont follow me, follow some
  rules that conflict with me? Then following them
  would require not following them!
• But if the (alleged) fundamental rules include
  rules of logic (as apparently they must), then
  its hard to see how the logic could rationally
  change.

31
Representative quotes

• Not everything can be revised, because something
  must be used to determine whether a revision is
  warranted. Tom Nagel 1997, p. 65
• There is no intellectual position we can occupy
  from which it is possible to scrutinize our
  logical beliefs without presupposing them.
  That is why they are exempt from skepticism. They
  cannot be put into question by an imaginative
  process that essentially relies on them. Ibid.
  p 64
• The only way to revise ones logic is by brain
  surgery (Louise Anthony)

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Core vs. peripheral?

• Solution? Distinguish between
• core principles of logic, built into the
  fundamental epistemic rules, and
• peripheral principles not part of the rules.
• Allow for rational change only in the peripheral
  principles.
• But I doubt that this is adequate.

33
Not just after-the-fact account

• I concede that given any serious proposal for a
  change of logic,
• one could find, after the proposal is made,
• a common core between that and the old logic
  sufficient to reconstruct some epistemic
  principles that would have licensed the change.
• But the fundamental rule picture requires more a
  single set of rules that could be used to
  evaluate any proposal for an altered logic (and
  handle ordinary reasoning tasks too).
• It should be possible to give the rules for
  handling all possible reasonable proposals for
  altering our logic, in advance of any particular
  proposed alteration.

34
Alternative to rules? Mental models (1)

• A standard alternative to rules in the psychology
  of logic Johnson-Laird on mental models in place
  of rules of deduction.
• This doesnt help much one seems to need rules
  for the construction and interpretation of the
  models.
• If so, those rules need to be revised to
  accommodate new logics.
• More explicitly

35
Mental models (2)

• The generation of models for classical reasoning
  seems to go by rules, according to which
• in any describable circumstances, a given
  sentence is either true or not true.
• Only by giving this up and constructing a more
  general conception of model could the paradoxes
  be adequately accommodated.
• The rules of model-generation arent rules of
  deduction, but present the same difficulty it
  isnt clear how to model their change.
• Ive put this in terms of rules for generating
  models. Maybe theres an alternative to the use
  of rules. But talk of models doesnt provide one.
  

36
Default program

• A better approach is to take logic to be part of
  a default program, rewritable under exceptional
  circumstances.
• The procedure for rewriting it is assumed
  unrevisable (a non-rewritable program or part of
  basic architecture).
• Its unrevisability doesnt raise the same
  problems, because it doesnt need to carry out
  the burden of ordinary reasoning---thats in the
  default logic.

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A Direction

• But we need a version of this that is detailed
  enough to make substantive and correct
  predictions about the circumstances under which
  one might rationally change ones logic.
• I think that recent work by others on how we
  handle inconsistent information provides a clue.
  
• Our frequent need to handle inconsistent
  information and our occasional need to revise
  logic are both aspects of our logical
  imperfection, something that philosophers
  unfortunately tend to idealize away.
• But to see how to implement this, we need to
  become clearer about the nature of logic itself.
  ///

38
REPRISE Outline

• 1. Paradoxes of truth, definability,
• Why they arent idle puzzles
• How they make a case for revising logic (
  fundamental rules of reasoning)
• 2. But can we rationally revise logic
  (fundamental rules of reasoning)?
• 3. Understanding the nature of logic, and how it
  helps with (though doesnt by itself fully
  answer) the puzzles about rational revision of
  logic.

39
The nature of logic (1)

• A common view is that logic is the science of
  which forms of inference necessarily preserve
  truth.
• The usual alternative explains logical validity
  in normative terms in terms of what we ought to
  believe and not believe.
• Both views seriously misconceive the nature of
  logic.
• The paradoxes can be shown to decisively refute
  the first.
• The second would sully the precision of logic by
  bringing in imprecise normative notions.

40
The nature of logic (2)

• Theres a more subtle normative view of logic.
• On it, a logic is a description of a component of
  a possible set of norms for believing.
• A good logic is a description of a component of a
  good set of norms. (Purpose-relative, but
  truth-oriented purposes are especially
  important.)
• We can theorize about which possible norms are
  good, relative to the purposes we have. For
  this, we use our current logic.
• If we come to think an alternative better, we can
  try to train ourselves to employ it.

41
Connection to rational change (1)

• This doesnt by itself give the sort of account
  of rational change of logic Id like a (sketch
  of) a computer or probabilistic automaton model
  of an agent rationally changing her logic.
• But I think it gives the proper framework for
  that. It divides the problem into
• a theoretical investigation (using currently
  accepted logic) into (i) what possible systems of
  logic there are and (ii) what life would be like
  if we employed them
• a practical problem of training ourselves to
  alter our inferential practices.

42
Connection to rational change (2)

• This split lessens the worries about using logic
  to revise logic
• the use of logic is mostly confined to the
  theoretical task at stage 1, of figuring out
  which logic is best
• The revision comes at the practical level of
  retraining ourselves, in stage 2.

43
Disallow Question Begging. (How?)

• This doesnt completely remove all circularity
  worries, since there are arguments available at
  stage 1, using the old logic, that would
  undermine any change.
• But such arguments would seem question-begging.
• When change of logic and other change of central
  practices (e.g. observational practices) is at
  issue, we need a way to block the use of
  question-begging reasoning.
• I think we can adapt models of our use of
  inconsistent information to do this.

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Summary of this part

• So my claim is
• Logic is not the science of what preserves truth.
  (I think the paradoxes decisively refute that
  view, though I didnt argue that today.)
• It is not the science of what one ought to
  believe. (One shouldnt bring normativity into
  the subject matter of logic.)
• The normativity is external a logic is a partial
  description of norms for believing, and the
  normativity comes into the question of what makes
  a logic good.
• This helps clear the ground for a proper approach
  to the subject of the rational revision of logic,
  and gives some clues as to the direction of a
  solution.

45
FINAL REPRISE Outline

• 1. Paradoxes of truth, definability,
• Why they arent idle puzzles
• How they make a case for revising logic (
  fundamental rules of reasoning)
• 2. But can we rationally revise logic
  (fundamental rules of reasoning)?
• 3. Understanding the nature of logic, and how it
  helps with (though doesnt by itself fully
  answer) the puzzles about rational revision of
  logic.