[extropy-chat] Bioethics Essay- Revised

[kevinfreels.com]

What a wonderful piece. Just what I needed to lighten up my day. Thank you!
:-)

I have a long held belief that the line between being a non-person and
developing into a person is whan a baby makes their first true smile. It is
this moment when they truly have crossed the line into human emotion. I'm
sure this concept is upsetting to some, but it is a more distinct moment
than any of the other arbitrary lines that are used to determine personhood.
Of course there is still vagueness when you start to consider the definition
of a smile. Can an action such as a smile be reduced to a logical
description represented by numbers? This would be more difficult than
defining baldness, yet it is even easier for us to readily identify a true
smile. Or is there maybe a different part of the brain which processes
information in a manner totally different from what we are capable of
representing with numbers and language?


----- Original Message ----- 
From: "scerir" <scerir at libero.it>
To: "ExI chat list" <extropy-chat at lists.extropy.org>
Sent: Tuesday, May 24, 2005 11:43 AM
Subject: Re: [extropy-chat] Bioethics Essay- Revised


> From: "The Avantguardian"
> > If one believes that people have souls
> > one can make the argument that a fetus
> > has a soul and is therefore entitled to rights.
> > One cannot however, based on the evidence,
> > make such an argument for a cloned embryo
> > in the blastula stage.
>
> It seems that questions like that above,
> or questions like "when does a fertilized
> egg develop into a person?", have much to
> do with the concept of logical 'vagueness'
> http://plato.stanford.edu/entries/vagueness/
> (Perhaps also concepts like Heisenberg's
> 'cut' or von Neumann's 'cut' share the same
> vagueness).
>
> s.
>
> There is a long, interesting post, by Stewart
> Shapiro, on that topic, in [FOM]. --->
>
> There probably is no way to say what it is for an expression to be vague
> without begging the question against some substantial view.  Intuitively,
a
> predicate is vague if it has, or can have borderline cases, instances in
> which it is not determinate that the predicate applies, nor is it
> determinate that it fails to apply.  A borderline case of "red" would be
an
> object whose color lies about halfway between red and pink.
>
> Typical vague predicates give rise to the ancient *sorites paradox*,
> sometimes called the *paradox of the heap*.  Here is an example:
>
> Premise:  A man with no hair at all is bald.
>
> Premise:  If a man has only n hairs on his head is bald, then so is a man
> with n+1 hairs.
>
> Conclusion:  A man with 50,000 hairs on his head is bald.
>
> Clearly, the argument is valid (once supplemented with the obvious
premises
> about the natural numbers).  Its first premise is true and its conclusion
> is false.  But its second premise *seems* to be true.  How can a single
> hair make the difference between someone who is bald and someone is not?
>
> Another version of the sorites uses tons of premises, instead of the
> induction-like premise that invokes natural numbers.  If a man with 0
hairs
> is bald, then so is a man with 1; if a man with 1 hair is bald, then so is
> a man with 2; ...  This argument has 50,002 premises.
>
> The traditional example uses the word "heap", and turns on how many grains
> make up a heap.
>
> Intuitively, vague predicates are what Crispin Wright calls
> "tolerant".  Many of the predicates are applied on the basis of casual
> observation, and small differences should not matter.
>
>
> The problem is to give a semantics of natural-language vague predicates
> which reveals what goes wrong (if something does) with sorites reasoning.
>
> Just about every account agrees that something does go wrong with sorites
> reasoning.  The only serious candidate is the inductive premise.  Most of
> the different accounts have that premise coming out untrue (if not
> false).  The burden that remains is to show why we find the inductive
> premise so appealing, when it leads to paradox.
>
>
> Here are the main theses (or slogans) of some of the main theories of
> vagueness:
>
> I. Nihilism (Dummett)
>
> Advocates of this view hold that the semantic rules for vague predicates
of
> natural languages are inconsistent.  The rules sanction the premises of
the
> sorites arguments, and yet they reject the conclusion.  Natural language
is
> incoherent.  Logic does not apply to it.
>
>
> II. Epistemicism (Williamson, Sorenson)
>
> This is the result of applying classical semantics to natural
> languages.  Each predicate of natural language has a precise
> extension.  There is a an exact nanosecond at which a person stops being a
> child;  there is a real number n such that a person who is n feet tall is
> short, but a person who is n+.00000001 feet tall is not short;  there is a
> real number m such that m minutes after noon is noonish, but m+.00000001
> minutes is not.
>
> The view is called "epistemicism" since the sharp borderlines of vague
> predicates cannot be known.  That is, it is impossible to know that values
> of n and m, nor how many hairs make for non-baldness, how many grains of
> sand make for a heap, etc.
>
>
> Most theorists find this view fantastic.  How can natural language
> predicates get extensions that are that precise?  The main advantage of
the
> view is that it maintains classical model-theoretic semantics, and
> classical, two-valued logic.  In sorites arguments, the inductive premise
> is simply false.
>
>
> III. Supervaluationism (Fine, Keefe)
>
> A given vague predicates does not have a single extension.  Instead, there
> are a number of different ways of making it precise, which are consistent
> with its meaning.  A sentence P that contains vague predicates is
> super-true if it comes out true under all acceptable ways of sharpening
the
> predicate.  The sentence P comes out false if it comes out false under all
> acceptable ways of sharpening the predicate.  P is indeterminate
otherwise.
>
> One advantage of supervaluationism is that it maintains much of classical
> logic.  All tautologies are super-true.  But the semantics has some
strange
> properties.  Let t be a rose whose color is midway between red and
> pink.  Then the sentence "t is red or t is pink" is super-true.  But is
not
> super-true that t is red, nor is it super-true that t is pink.
>
> Something similar happens with the inductive premise of a sorites
> argument.  Recall the instance above, formulated more carefully:
>
> For every number x, if a man has only x hairs on his head is bald, then so
> is a man with x+1 hairs.
>
> This is super-false, since it comes out false on every acceptable way to
> sharpen the predicate "bald".  But there is no number n such that it is
> super-false that "if a man has only n hairs on his head is bald, then so
is
> a man with n+1 hairs".
>
> It is generally agreed that this theory pushes the problem up to the
> meta-language.  What makes a sharpening acceptable is itself vague.
>
>
>
> III.  Fuzzy logic.  (Machina, Edgington)
>
> This one involves truth values intermediate between truth and
> falsehood.  The most common proposal is to use real numbers between 0 and
> 1.  That is, 0 is complete falsehood, 1 is complete truth.  .4 would be a
> partial truth.  If s is borderline bald, leaning toward baldness, then the
> statement "s is bald" might get truth value .623.
>
> Another, less developed, proposal, is to use the elements of a Boolean
> algebra as truth values.  This gives the theorist a shot a preserving
> classical logic.
>
>
> Below, I'll insert the handout for some lectures I gave on
> supervaluationist semantics and fuzzy logic.
>
> IV.  Contextualism.  (Raffman, Shapiro, Graff)
>
> The extensions of vague predicates shift with the context of
> utterance.  Everyone agrees that the extensions of vague predicates vary
> with factors like the comparison class and paradigm cases.   A family can
> be wealthy with respect to professors in mathematics departments, but not
> wealthy with respect to corporate executives.  A person can be short with
> respect to NBA players but not short with respect to philosophers.  The
> contextualist argues that extensions also vary in the course of a given
> conversation (or psychological state), even if the comparison class is
held
> fixed.
>
> There are a few proposals in this direction.  Some preserve classical
logic
> and some don't.
>
>
>
> If you want to read a bit more, without leaving your computer, here are a
> couple of links to the Stanford Internet Encyclopedia of Philosophy.  The
> third is to a closely related matter.
>
>
> http://plato.stanford.edu/entries/vagueness/
>
> http://plato.stanford.edu/entries/sorites-paradox/
>
> http://plato.stanford.edu/entries/problem-of-many/
>
>
> I'd be glad to supply more specific references, if anyone wants them.
Just
> send me a (private) email.  I just finished a book that develops and
> defends a contextualist account.  The model theory is much like that of
> Kripke structures for intuitionistic languages.
>
> Here are a couple of handouts, one on fuzzy logic and one on
> supervaluationist model theory.
>
>
> A crash course in many-valued model theory
>
> If Harry is borderline bald, then perhaps it is natural to say that "Harry
> is bald" is not fully true, but not quite false either.  Perhaps the
> sentence has some intermediate truth value.
>
> The idea behind the many-valued or degree-theoretic approach to vagueness
> is that sentences can take any of a number of truth values.  The most
> common collection of truth-values is the set of real numbers from 0 to 1
> (inclusive).  The truth value 1 indicates complete truth and the truth
> value 0 indicates complete and utter falsity.  So if "Harry is bald" is
> true to degree .87, then he is almost bald.  If it is true to degree .5
> then he is exactly halfway between determinately bald and determinately
> non-bald, etc.  Other sets of truth values may be considered.
>
> A fuzzy interpretation M of a formal language is a pair <d,I> where d is a
> non-empty set, the domain of discourse (as usual), and I assigns
> denotations and extensions to the non-logical terminology.  Let us assume
> that we have no function letters in the language.  If a is a constant,
then
> Ia is a member of d (as usual).  If R is an n-place relation symbol and m1
>  . . mn are members of the domain d, then IRm1 . . . mn is a real number.
> between 0 and 1.
>
> A fuzzy interpretation is classical if the only truth-values it assigns
are
> 0 and 1.
>
>
> Let M = <d,I> be a fuzzy interpretation.  The semantics assigns a "truth
> value" (i.e., a real number) to every sentence of the language.  If P is a
> sentence, then let VP be its truth value in M.
>
> For convenience, assume that the members of d have been added to the
> language, as constants denoting themselves.  (So we can deal with
sentences
> without introducing variable assignments).
>
> Only the atomic case and perhaps the negations are straightforward:
>
> If R is an n-place relation letter and t1 . . . tn are terms, then VRt1 .
.
>  tn  = IRm1 . . . mn, which each mi is the denotation of ti..
>
> V(not-P)  = 1-VP.
>
> So if "Harry is bald" is half true, then so is "Harry is not bald".
>
> One main issue concerns whether the other connectives should be
> truth-functional, or what is sometimes called "degree-functional".  The
> question is whether the truth value of a compound is determined completely
> by the truth values of its parts.  I am convinced by Dorothy Edgington's
> arguments that the semantics should not be truth-functional.  But I will
> present the most common semantics, which is truth-functional:
>
> V(P&Q)   is the minimum of VP    and VQ.
>
> V(PvQ)   is the maximum of VP    and VQ.
>
> if VP<=VQ, then V(if P then Q)        = 1,
> if VP>VQ, then  V(if P then Q)        = 1-(VP-VQ).
>
> The existential quantifier gives the least upper bound of its instances,
> and the universal quantifier gives the greatest lower bound of its
> instances.
>
>
> Notice that if M is classical (i.e., if the only truth values it assigns
to
> the atomic sentences are 0 and 1), then the semantics agrees with
classical
> semantics (exercise).
>
>
> There are a number of different definitions of validity.  You can decide
> which of them accords with your intuitions.
>
> One slogan is that validity is the necessary preservation of truth.  Since
> (complete) truth is the truth value 1, then we might define strict
validity
> as the necessary preservation of truth value.  So an argument is strictly
> valid iff there is no fuzzy interpretation that makes its premises 1
> (completely true) and its conclusion less than 1.
>
> Every strictly valid argument is classically valid, but the converse
> fails.  There are no strictly logical truths.  However, classical
> tautologies can never get a truth value less than .5.
>
> Why should validity be limited to the preservation of complete truth?  We
> often reason with sentences that (on this view) are less then completely
> true, and need a notion of validity for those.  We can say that an
argument
> is valid if it preserves whatever truth value the premises may have:
>
> Say that an argument is fuzzy-valid if there is no fuzzy interpretation M
> such that the truth value of the conclusion is less than the greatest
lower
> bound of the truth values of the premises.
>
> Edgington has an interesting definition of validity.  Define the unverity
> of a sentence P in a fuzzy interpretation M to be 1-VP in  M.  Then an
> argument is E-valid if the unverity of the conclusion is no smaller than
> the sum of the unverities of the premises.  (But as noted above, she uses
a
> different, non-truth-functional semantics.)
>
>
> Now, what of sorites?
>
> Suppose we have a series of 1000 men lined up.  The first m1 has no hair
> whatsoever, and so is clearly bald.  The last m1000 has a fine head of
> hair.  Moreover, each man in the list has only slightly more hair than the
> one before (and his hair is arranged in roughly the same way).  Let B be
> the predicate for bald.  So a reasonable fuzzy interpretation would be one
> in which, say,  Bm1  =   Bm2  = . . . =  Bm60  = 1, and  Bm61  = .999.
The
> truth values fall slowly from there, until  Bm925 =0, and everything after
> that is completely false.  Suppose that the difference in truth value
> between adjacent formulas is never more than .002.
>
> Recall the two versions of the sorites argument:
>
> Bm1
> for all i<1000(Bmi ->  Bmi+1)
> therefore Bm1000
>
> Bm1
> Bm1 ->  Bm2
> Bm2 ->  Bm3
>       . . .
> Bm999 ->  Bm1000
> therefore Bm1000
>
> In both cases, the first premise has truth value 1, and the conclusion has
> truth value 0.  The first few conditionals and the last few conditionals
> each have truth value 1.  Given the assumption about the differences
> between adjacent formulas, each of the conditionals has truth value of at
> least .998, and some are not fully true.  So the second premise of the
> first argument has truth value somewhere between .998 and .999.
>
> On the first definition of validity (as the necessary preservation of
> complete truth), both arguments are valid.  The only inferences used are
> modus ponens and (for the first argument) universal elimination, and both
> are valid on that conception of validity.  But neither argument is sound,
> since it has premises that are less than fully true.
>
> One puzzling feature is that we can have a valid argument whose premises
> are almost completely true (at least .998), but which has a completely
> false conclusion.
>
> On the second definition of validity (as the preservation of minimal truth
> value), neither argument is valid.  Modus ponens fails.
>
> I think that the Edgington definition of validity is the most satisfying
> here.  She shows that (on her semantics), all classically valid arguments
> are valid in her sense.  In the second argument, the many small
> "unverities" in the premises add up to 1.  But on her semantics, the
> conditional in the first argument ends up completely false.
>
>
> Here is the other handout:
>
> A crash course in supervaluation model theory
>
> The first modification to ordinary model theory introduced here is the
> notion of a *partial interpretation*.  This is a pair <d,I>, where d is a
> non-empty set (the domain of discourse) and I is a function.
>
> Let R be an n-place relation letter.  In the partial interpretation
> M=<d,I>, IR is a  pair <p,q> such that p and q are disjoint sets of
> n-tuples of members of the domain.  The idea is that p is the extension of
> R in the partial interpretation, the n-tuples of objects that the relation
> (determinately) applies to; q is the anti-extension of R, the n-tuples of
> objects that the relation (determinately) fails to apply to.  If IR=<m,n>,
> then define IR+= m and IR-=n.  From now on, we write "interpretation" for
> "partial interpretation".
>
> Any n-tuples from the domain of discourse d that are in neither IR+ nor
IR-
> are borderline cases of R in the partial interpretation M.  If there are
no
> such borderline cases, then we say that R is *sharp* in M.  A partial
> interpretation M is completely sharp or classical if every relation in the
> language is sharp in M.  A completely sharp interpretation corresponds to
a
> classical interpretation since, in that case, the anti-extension of each
> predicate is just the complement of the extension.  So we sometimes call a
> completely sharp partial interpretation "classical".
>
> We introduce a "three-valued" semantics on partial interpretations.  The
> values are T (truth), F (falsehood), and I (indeterminate).  It does not
> make much difference formally, but it is more convenient if we do not
think
> of I as a truth-value on a par with truth and falsehood.  Rather, we take
> "I" to indicate the lack of a (standard) truth value in the given partial
> interpretation.
>
> Assume that we have expanded the language to include the members of the
> domain as new constants (so we do not have to futz with variable
> assignments).
>
>
> If a and b are terms, then a=b is true in M if and only if the denotation
> of a in M is identical to the denotation of b in M; a=b is false in M
> otherwise.  Notice that a=b is never indeterminate.
>
> Let R be an n-place predicate letter, and t1, ...tn terms.  For each i,
let
> mi be the denotation of ti under M.  Then Rt1...tn is true in M if
> <m1...mn> is in IR+; Rt1...tn is false in M if <m1...mn> is in IR-; and
> Rt1...tn is indeterminate in M otherwise.
>
> We use the strong, internal negation.  Its truth table is the following:
>
> P not-P
> T F
> F T
> I I
>
> For the other connectives, we employ the strong Kleene truth-tables:
>
> P&Q T F I PvQ T F I
> T T F I T T T T
> F F F F F T F I
> I I F I I T I I
>
> P->Q T F I
> T T F I
> F T T T
> I T I I
>
> A universally quantified sentence is true if all of its instances are
true,
> it is false if it has a false instance, and it is indeterminate
> otherwise.  Existentially quantified sentences are analogous.
>
> So far, we do not have a plausible model for a semantics of
> vagueness.  Suppose, for example, that we have a patch whose color is on
> the borderline between red and orange.  To capture this, we would make
> statement Rt that the patch is red indeterminate; and we would make the
> statement Ot that the patch is orange indeterminate.  In such a partial
> interpretation, both RsvOs and not-(Rs&Os) would both be
> indeterminate.  But this seems implausible.  Even if it is indeterminate
> that the patch is red and it is indeterminate that it is orange, it surely
> true that it is either red or orange.  After all, what other color could
it
> have?  And it is surely true that it is not both red and orange--nothing
> can be red (all over) and orange (all over) at the same time.  Moreover,
> Rs->Rs is also indeterminate in the given interpretation.  Even if Rs can
> go either way, it surely is determinately true that if s is red, then s is
> red.
>
> For a second example, consider a pair of men, r, s.  Both are borderline
> bald, but r has a bit more hair than s (arranged the same way).  A partial
> interpretation that reflects this situation would have Br->Bs
> indeterminate.  But surely, if we sharpen in order to make Br true (as we
> can, presumably), then we would thereby made Bs true, since s has even
less
> hair than r.  So, intuitively, Br->Bs should come out determinately true.
>
> Let M1=<d1,I1> and M2=<d2,I2> be partial interpretations.  Say that M2
> sharpens M1 if:
>
> (1) d1=d2,
>
> (2) the interpretation functions I1 and I2 agree on each constant and
> function letter.
>
> (3) for each relation letter R, I1+Ris a subset ofI2+R and I1-Ris a subset
> of I2-R.
>
> The idea is that M2 sharpens M1 if and only if the two interpretations
have
> the same domain and agree on the constants and function letters, and M2
> extends both the extension and anti-extension of each relation.  The two
> interpretations agree on the clear or determinate cases of M1, but M2 may
> decide some borderline cases of M1.
>
> Kit Fine [1975] calls M2 a "precisification" of M1.  We have that each M1
> sharpens itself (although that does sound awkward).
>
> Theorem 1.  Suppose that M2 sharpens M1.  If a sentence P is true (resp.
> false) in M1, then P is true (resp. false) in M2.
>
> The proof of Theorem 1 is a straightforward induction on the complexity of
P
> .
>
> Recall our decision to not think of "indeterminate" as a truth value, but
> rather as lack of a (standard) truth value.  In these terms, it follows
> from this result that the semantics is monotonic.  As we sharpen, we do
not
> change the truth values of formulas that have truth values; we only give
> truth values to formulas that previously lacked them.
>
> Notice that since this nice result is an induction on the complexity of
> formulas, it depends on the particular expressive resources in the object
> language.
>
> Recall that a partial interpretation is completely sharp if every relation
> in the language is sharp in the interpretation.  A straightforward
> induction shows that a completely sharp interpretation corresponds to a
> classical interpretation.
>
> So far, things are pretty rigorous.  Now we wax intuitive for a bit.  Not
> every sharpening is legitimate.  Suppose we have a predicate for "rich"
and
> there are two members of the domain, Jon and Joe, who are borderline rich
> (so that neither fall in the extension nor the anti-extension of
> "rich").  Suppose that Joe's total worth is a bit greater than the total
> worth of Jon.  A sharpening that declares Jon rich and fails to declare
Joe
> rich (or declares that he is not rich) would clearly be unacceptable-it is
> incompatible with the meaning of "rich".  Similarly, a sharpening that
> declares a man bald and declares a man with less hair (arranged similarly)
> to be not bald is likewise unacceptable.
>
> Fine [1975] uses the term "penumbral connection" for analytic (or all but
> analytic) truths for the terms in question.  An example of a penumbral
> connection is that if someone is rich then (other things equal) anyone
with
> more money is also rich.  Another penumbral connection is that if someone
> is not bald then someone with more hair (arranged the same way) is also
not
> bald.  Another is that nothing is completely red and completely orange.
>
> Of course, formal languages do not have such analytic (or almost analytic)
> truths.  The formulas are just sequences of characters, and have no
meaning
> beyond the stated truth conditions for the logical terminology and perhaps
> any premises and axioms that are explicitly added.  So, as a first
> approximation, we assume a collection of conditions on partial
> interpretations.  Suppose, for example, that the object language has a
> predicate B, to represent the English word "bald", and a binary relation
R,
> such that Rab represents the statement that a is less hairy than b.  Then
a
> partial interpretation M is not acceptable if there are objects m,n in the
> domain of discourse such that m is in the extension of B, M satisfies Rnm,
> but n is not in the extension of B.  Similarly, M is not acceptable if m
is
> in the anti-extension of B, M satisfies Rmn and n is not in the
> anti-extension of B.
>
> Say that a partial interpretation M is acceptable if it satisfies all of
> its penumbral connections, and we say that a partial interpretation M2 is
> an acceptable sharpening of a partial interpretation M1 if they are both
> acceptable on the same collection of penumbral connections and M1~M2.
>
> Let P be a sentence and let M be a partial interpretation.  Say that P is
> super-true in M if (1) there is no acceptable sharpening M' of M such that
> P is false in M' and (2) there is at least one acceptable sharpening M' of
> M such that P is true in M'.
>
> While we are at it, define P to be super-false in M similarly.
>
> The idea, of course, is that P is super-true if it comes out true under
> every acceptable way of making the predicates sharp.  That is, if we
> sharpen, P comes out true if it gets a truth value at all (and we make
sure
> that P does get a truth value in some acceptable sharpenings).
>
> Let P be a sentence, and assume, for now, that each acceptable partial
> interpretation has at least one acceptable, completely sharp
> sharpening.  In light of monotonicity (Theorem 1), P is super-true if and
> only if P is true in every acceptable, completely sharp sharpening of
> M.  Thus, in a sense, the definition of super-truth (and super-falsity)
> depends only on the semantics of completely sharp interpretations, and, in
> effect, those are ordinary, classical interpretations.  So the notion of
> super-truth does not invoke the three-valued semantics above.
>
> Most defenders of the supervaluation approach insist that the primary (or
> perhaps the only) notion of truth is super-truth. The mantra is that truth
> is super-truth.  Accordingly, they define validity as necessary
> preservation of super-truth:
>
> An argument is externally valid if every partial interpretation that makes
> its premises super-true also makes its conclusion super-true.
>
> Call this external validity.  Under the prevailing assumption about
> acceptable, completely sharp interpretations, this definition does not
> depend on the three-valued semantics introduced above.  All of the
relevant
> action takes place in completely sharp interpretations.
>
> Theorem 2.  External validity coincides with classical (two-valued)
> consequence.
>
> So the supervaluationist accepts classical logic for vague expressions.
>
> What of sorites?  Suppose we have a series of 1000 men lined up.  The
first
> m1 has no hair whatsoever, and so is clearly bald.  The last m1000 has a
> fine head of hair.  Moreover, each man in the list has only slightly more
> hair than the one before (and his hair is arranged in roughly the same
> way).  Let B be the predicate for bald.  So a reasonable partial
> interpretation would be one in which, say, m1,..., m247 are all in the
> extension of B, and m621, ..., m1000 are in the anti-extension.  The
> penumbral connection would be that mi is in the extension of B and if j<i,
> then mj is in the extension of B; and if mi is in the anti-extension of B
> and if i<j, then mj is in the anti-extension.  The various acceptable
> (complete) sharpenings would put the border somewhere between m247 and
m621.
>
> Recall the two versions of the sorites argument:
>
> Bm1
> for every i<1000(Bmi->Bmi+1)
> Therefore, Bm1000
>
> Bm1
> Bm1->Bm2
> Bm2->Bm3
>       ...
> Bm999->Bm1000
> Therefore, Bm1000
>
> Both of these are classically valid, and so they are valid on the
> supervaluational approach.  But the conclusions are super-false (in the
> envisioned partial interpretation), and the first premise is
> super-true.  So at least one other premise fails to be super-true.
>
> In the first argument, the only other premise is the inductive
> one.  Actually, it is super-false.  In each complete sharpening of the
> partial interpretation, there is a sharp boundary, an i such that Bmi but
> not-Bmi+1.
>
> In the second argument, many of the conditionals fail to be super-true
> (although none are super-false).  If mi is in the borderline region (in
the
> original partial interpretation), then there is a sharpening in which
> Bmi->Bmi+1 is false.
>
>
>
>
>
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