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Re: [aima-talk] Digest Number 303

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  • Paul Hsueh-Min Chang
    But my question is that being false does *not* equal being contradictory (i.e. necessarily false). If the sentence were merely false but not contradictory, he
    Message 1 of 10 , Sep 21, 2005
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      But my question is that being false does *not* equal being contradictory
      (i.e. necessarily false). If the sentence were merely false but not
      contradictory, he surely could consistently assert it.

      So lets review the argument:

      "The sentence cannot be false, because if it were then Lucas could not
      consistently assert it, so it would be true."

      Consider two conditions:
      1. If the sentence were meant to be merely false, then Lucas could
      consistently assert it, so the argument is invalid.
      2. If the sentence were meant to be contradictory, then nobody could
      consistently assert it, but then the argument would have a very strange
      form: "if p were contradictory then A could not assert it, so p would be
      true." Consider the following argument of the same form: "If 'Saddam
      Hussein is the US President' were contradictory, then I could not assert
      it, so he would be the US President". Clearly absurd.

      Again, please correct me if I am wrong.

      Paul

      The Geek wrote:

      > I believe if you look at the "setup" on the previous
      > page you'll see the authors intended the term
      > "consistent" to be the logical definition. (see pages
      > 137 and 353) That is, for something to be consistent,
      > it cannot be contradictory. Therefore, if the
      > sentance were false, he couldn't assert the sentance
      > and still be consistent, which therefore makes the
      > sentance true.
      >
      > But the point of the paragraph is that because of the
      > construction of the sentance, the agent "J.R. Lucas"
      > cannot assert something that other agents can.
      > However the authors are pointing out that this doesn't
      > make him inferior.
      >
      > Rob G.
      >
      > --- Paul Hsueh-Min Chang <avatar@...>
      > wrote:
      >
      > > Hi Bruce,
      > >
      > > I think "this statement is definitely false" is more
      > > a paradox than a
      > > contradiction, for when you decide it is false then
      > > it is true, and vice
      > > versa. A contradiction is always false. And
      > > "consistently" has two
      > > readings, one is the randomness you meant, and
      > > another reading common in
      > > philosophical literature is that it is possible for
      > > a set of
      > > propositions to all true. I'm just not sure which
      > > meaning the authors
      > > seem to imply.
      > >
      > > Paul
      > >
      > > Tommy Gun wrote:
      > >
      > > > Sounds like sort of a contradiction. The
      > > words "cannot
      > > > consistently" I think give it the flexability
      > > to sometimes be true
      > > > and sometimes not. If the sentence was
      > > definate all of the time,
      > > > then it would just be a contradiction. Take
      > > "this statement is
      > > > definately false" is a contradiction, but if
      > > it were, "this
      > > > statement is sometimes false" then there
      > > sometimes when it isn't a
      > > > contradiction. "cannot consistently"
      > > basically says that's it's
      > > > sorta random, so sometimes it could make
      > > sense.
      > > >
      > > > Not sure if that helps, but it's just my
      > > $.02...
      > > >
      > > > - Bruce
      > > >
      > > >
      > > > Message: 1
      > > > Date: Sun, 11 Sep 2005 23:08:35 +0800
      > > > From: Paul Hsueh-Min Chang
      > > > Subject: Problem about the J.R. Lucas sentence
      > > >
      > > > Hi,
      > > >
      > > > On page 950, the book argues that the sentence
      > > "J. R. Lucas cannot
      > > > consistently assert that this sentence is
      > > true." is necessarily true,
      > > > but Lucas cannot consistently assert it. There
      > > are two arguments
      > > > on that
      > > > page. I found no problem with the first
      > > argument, but could not
      > > > understand the second.
      > > >
      > > > Here is the second argument.
      > > >
      > > > "The sentence cannot be false, because if it
      > > were then Lucas could not
      > > > consistently assert it, so it would be true."
      > > >
      > > > But, why couldn't Lucas consistently assert it
      > > /if it were false/? One
      > > > can of course assert a false sentence and be
      > > consistent at the same
      > > > time, because one is inconsistent if and only
      > > if it is
      > > > /impossible/ that
      > > > all his beliefs are true. If Lucas happens to
      > > believe a false
      > > > sentence,
      > > > he is still consistent.
      > > >
      > > > Please help/correct me.
      > > >
      > > > Paul
      > > >
      >
    • The Geek
      I don t know if I m explaining this very well. Someone else should feel free to chime in if they ve got a better way of phrasing it... While they haven t
      Message 2 of 10 , Sep 23, 2005
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        I don't know if I'm explaining this very well.
        Someone else should feel free to chime in if they've
        got a better way of phrasing it...

        While they haven't spelled it out in detail, I think
        they're using a simple logic system and definition of
        "consistent". In other words, they're not trying to
        allow for an "agent belief" representation where we
        acknowledge that beliefs might be wrong. Instead, if
        you assert a sentance it must be true or you've
        introduced a contradiction.

        If we take 3 sentances:

        A = B.
        B = C.
        A != C.

        If we try to assert all three sentances, we cannot do
        it without creating a contradiction. Since this is
        the definition of "consistency", they would say that
        we cannot "consistently" assert all three sentances,
        since doing so would introduce a contradiction.

        The example is a little more complicated, since
        they're violating a couple of normal rules for logic
        systems - they have a sentance referring to the
        assertability of sentances. But if you reword the
        sentance to say "Agent A cannot assert this sentance
        without being wrong" I think it's a little more clear.

        In this case, as an agent **I** can consistently
        assert the sentance "Agent A cannot assert this
        sentance without being wrong" without introducing a
        contradiction. However if Agent A tries to assert the
        same sentance, he runs into a problem. If he asserts
        it as true and is right, then the sentance is false,
        so he's wrong.

        But the whole point of the illustration is simply to
        show that **sometimes** one agent is unable to
        assert/know/do things that another agent can, but that
        doesn't automatically imply that agent is inferior,
        it's inability to assert/know/do may be related to the
        specific situation.

        Rob G.

        --- Paul Hsueh-Min Chang <avatar@...>
        wrote:

        > But my question is that being false does *not* equal
        > being contradictory
        > (i.e. necessarily false). If the sentence were
        > merely false but not
        > contradictory, he surely could consistently assert
        > it.
        >
        > So lets review the argument:
        >
        > "The sentence cannot be false, because if it were
        > then Lucas could not
        > consistently assert it, so it would be true."
        >
        > Consider two conditions:
        > 1. If the sentence were meant to be merely false,
        > then Lucas could
        > consistently assert it, so the argument is invalid.
        > 2. If the sentence were meant to be contradictory,
        > then nobody could
        > consistently assert it, but then the argument would
        > have a very strange
        > form: "if p were contradictory then A could not
        > assert it, so p would be
        > true." Consider the following argument of the same
        > form: "If 'Saddam
        > Hussein is the US President' were contradictory,
        > then I could not assert
        > it, so he would be the US President". Clearly
        > absurd.
        >
        > Again, please correct me if I am wrong.
        >
        > Paul
        >
        > The Geek wrote:
        >
        > > I believe if you look at the "setup" on the
        > previous
        > > page you'll see the authors intended the term
        > > "consistent" to be the logical definition. (see
        > pages
        > > 137 and 353) That is, for something to be
        > consistent,
        > > it cannot be contradictory. Therefore, if the
        > > sentance were false, he couldn't assert the
        > sentance
        > > and still be consistent, which therefore makes the
        > > sentance true.
        > >
        > > But the point of the paragraph is that because of
        > the
        > > construction of the sentance, the agent "J.R.
        > Lucas"
        > > cannot assert something that other agents can.
        > > However the authors are pointing out that this
        > doesn't
        > > make him inferior.
        > >
        > > Rob G.
        > >
        > > --- Paul Hsueh-Min Chang
        > <avatar@...>
        > > wrote:
        > >
        > > > Hi Bruce,
        > > >
        > > > I think "this statement is definitely false" is
        > more
        > > > a paradox than a
        > > > contradiction, for when you decide it is false
        > then
        > > > it is true, and vice
        > > > versa. A contradiction is always false. And
        > > > "consistently" has two
        > > > readings, one is the randomness you meant, and
        > > > another reading common in
        > > > philosophical literature is that it is possible
        > for
        > > > a set of
        > > > propositions to all true. I'm just not sure
        > which
        > > > meaning the authors
        > > > seem to imply.
        > > >
        > > > Paul
        > > >
        > > > Tommy Gun wrote:
        > > >
        > > > > Sounds like sort of a contradiction. The
        > > > words "cannot
        > > > > consistently" I think give it the
        > flexability
        > > > to sometimes be true
        > > > > and sometimes not. If the sentence was
        > > > definate all of the time,
        > > > > then it would just be a contradiction.
        > Take
        > > > "this statement is
        > > > > definately false" is a contradiction, but
        > if
        > > > it were, "this
        > > > > statement is sometimes false" then there
        > > > sometimes when it isn't a
        > > > > contradiction. "cannot consistently"
        > > > basically says that's it's
        > > > > sorta random, so sometimes it could make
        > > > sense.
        > > > >
        > > > > Not sure if that helps, but it's just my
        > > > $.02...
        > > > >
        > > > > - Bruce
        > > > >
        > > > >
        > > > > Message: 1
        > > > > Date: Sun, 11 Sep 2005 23:08:35 +0800
        > > > > From: Paul Hsueh-Min Chang
        > > > > Subject: Problem about the J.R. Lucas
        > sentence
        > > > >
        > > > > Hi,
        > > > >
        > > > > On page 950, the book argues that the
        > sentence
        > > > "J. R. Lucas cannot
        > > > > consistently assert that this sentence is
        > > > true." is necessarily true,
        > > > > but Lucas cannot consistently assert it.
        > There
        > > > are two arguments
        > > > > on that
        > > > > page. I found no problem with the first
        > > > argument, but could not
        > > > > understand the second.
        > > > >
        > > > > Here is the second argument.
        > > > >
        > > > > "The sentence cannot be false, because if
        > it
        > > > were then Lucas could not
        > > > > consistently assert it, so it would be
        > true."
        > > > >
        > > > > But, why couldn't Lucas consistently
        > assert it
        > > > /if it were false/? One
        > > > > can of course assert a false sentence and
        > be
        > > > consistent at the same
        > > > > time, because one is inconsistent if and
        > only
        > > > if it is
        > > > > /impossible/ that
        > > > > all his beliefs are true. If Lucas happens
        > to
        > > > believe a false
        > > > > sentence,
        > > > > he is still consistent.
        > > > >
        > > > > Please help/correct me.
        > > > >
        > > > > Paul
        > > > >
        > >
        >
        >
        >
        >




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      • Paul Hsueh-Min Chang
        ... Maybe our sources are different. AFAIK a contradiction is the negation of a tautology (P. Tidman and H. Kahane, Logic and Philosophy: A Modern
        Message 3 of 10 , Sep 24, 2005
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          I will try to make some clarifications below:

          > While they haven't spelled it out in detail, I think
          > they're using a simple logic system and definition of
          > "consistent". In other words, they're not trying to
          > allow for an "agent belief" representation where we
          > acknowledge that beliefs might be wrong. Instead, if
          > you assert a sentance it must be true or you've
          > introduced a contradiction.

          Maybe our sources are different. AFAIK a contradiction is the negation
          of a tautology (P. Tidman and H. Kahane, Logic and Philosophy: A Modern
          Introduction, 8th edition, p48). For example, p^~p is a contradiction.
          So when I assert a contingently false sentence whose truth depends on
          the world, I introduce no contradiction. But maybe our definitions of
          contradiction differ.

          > If we take 3 sentances:
          >
          > A = B.
          > B = C.
          > A != C.
          >
          > If we try to assert all three sentances, we cannot do
          > it without creating a contradiction. Since this is
          > the definition of "consistency", they would say that
          > we cannot "consistently" assert all three sentances,
          > since doing so would introduce a contradiction.

          It is true that we would introduce inconsistency, but we would not
          introduce a contradiction. Of course, any set of sentences that contains
          any contradictory sentence is inconsistent, but not vice versa.

          > In this case, as an agent **I** can consistently
          > assert the sentance "Agent A cannot assert this
          > sentance without being wrong" without introducing a
          > contradiction. However if Agent A tries to assert the
          > same sentance, he runs into a problem. If he asserts
          > it as true and is right, then the sentance is false,
          > so he's wrong.

          I know what you are trying to say, but that I don't think we are talking
          about the same thing. The authors clearly intend to show two things:
          1. The sentence is a tautology (i.e. necessarily true).
          2. Yet, J. R. Lucas cannot assert it.
          They offer two arguments to show (1). What I do not understand is the
          their second argument to show (1). You are talking about (2), which I
          find no problem.

          > But the whole point of the illustration is simply to
          > show that **sometimes** one agent is unable to
          > assert/know/do things that another agent can, but that
          > doesn't automatically imply that agent is inferior,
          > it's inability to assert/know/do may be related to the
          > specific situation.

          I do understand the point; it is simply that particular argument (the
          one on p950 in parentheses) that confuses me. As a philosophical
          treatment, I think the chapter should reasonably be taken literally, but
          when I do so, I cannot make sense of that particular argument.

          Paul
        • The Geek
          I think the discussion all boils down to the definition of consistent and contradiction . I was kind of hoping one of the authors would chime in at some
          Message 4 of 10 , Sep 27, 2005
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            I think the discussion all boils down to the
            definition of "consistent" and "contradiction". I was
            kind of hoping one of the authors would chime in at
            some point to save me, but... :-) On page 137 they
            define consistency in reference to CSP problems as "an
            assignment that does not violate any constraints".
            That was the reason for my A,B,C example - looking at
            it as if it were a CSP.

            Their argument assumes the sentance must either be
            "true" or "false", (no quantuum physics or "unknown"
            values allowed) and hinges on the
            belief/assumption/fact one cannot "consistently"
            assert that a sentance is true if it can be shown to
            be false. If you buy into that, then their statement
            follows naturally - if the sentance was false, then he
            could not consistently assert it to be true, which
            would therefore make the sentance true - a
            contradiction since we assigned a value of "false" to
            the sentance at the beginning. The only way you can
            assign a value to the sentance and have everything
            hold together is if you assume it's true, which then
            implies that everyone else can assert it without a
            problem, but he cannot.

            But, at this point I don't think we're going to make
            any headway. Since they didn't define the logic
            system they're using, we can't really do anything
            formal.

            Thanks for the interesting discussion.

            Rob G.


            --- Paul Hsueh-Min Chang <avatar@...>
            wrote:

            > I will try to make some clarifications below:
            >
            > > While they haven't spelled it out in detail, I
            > think
            > > they're using a simple logic system and definition
            > of
            > > "consistent". In other words, they're not trying
            > to
            > > allow for an "agent belief" representation where
            > we
            > > acknowledge that beliefs might be wrong. Instead,
            > if
            > > you assert a sentance it must be true or you've
            > > introduced a contradiction.
            >
            > Maybe our sources are different. AFAIK a
            > contradiction is the negation
            > of a tautology (P. Tidman and H. Kahane, Logic and
            > Philosophy: A Modern
            > Introduction, 8th edition, p48). For example, p^~p
            > is a contradiction.
            > So when I assert a contingently false sentence whose
            > truth depends on
            > the world, I introduce no contradiction. But maybe
            > our definitions of
            > contradiction differ.
            >
            > > If we take 3 sentances:
            > >
            > > A = B.
            > > B = C.
            > > A != C.
            > >
            > > If we try to assert all three sentances, we cannot
            > do
            > > it without creating a contradiction. Since this
            > is
            > > the definition of "consistency", they would say
            > that
            > > we cannot "consistently" assert all three
            > sentances,
            > > since doing so would introduce a contradiction.
            >
            > It is true that we would introduce inconsistency,
            > but we would not
            > introduce a contradiction. Of course, any set of
            > sentences that contains
            > any contradictory sentence is inconsistent, but not
            > vice versa.
            >
            > > In this case, as an agent **I** can consistently
            > > assert the sentance "Agent A cannot assert this
            > > sentance without being wrong" without introducing
            > a
            > > contradiction. However if Agent A tries to assert
            > the
            > > same sentance, he runs into a problem. If he
            > asserts
            > > it as true and is right, then the sentance is
            > false,
            > > so he's wrong.
            >
            > I know what you are trying to say, but that I don't
            > think we are talking
            > about the same thing. The authors clearly intend to
            > show two things:
            > 1. The sentence is a tautology (i.e. necessarily
            > true).
            > 2. Yet, J. R. Lucas cannot assert it.
            > They offer two arguments to show (1). What I do not
            > understand is the
            > their second argument to show (1). You are talking
            > about (2), which I
            > find no problem.
            >
            > > But the whole point of the illustration is simply
            > to
            > > show that **sometimes** one agent is unable to
            > > assert/know/do things that another agent can, but
            > that
            > > doesn't automatically imply that agent is
            > inferior,
            > > it's inability to assert/know/do may be related to
            > the
            > > specific situation.
            >
            > I do understand the point; it is simply that
            > particular argument (the
            > one on p950 in parentheses) that confuses me. As a
            > philosophical
            > treatment, I think the chapter should reasonably be
            > taken literally, but
            > when I do so, I cannot make sense of that particular
            > argument.
            >
            > Paul
            >
            >
            >





            __________________________________
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            http://mail.yahoo.com
          • shivam shukla
            hey everybody am doing a project on image processing n am using the convolution theorem given in chapter 25.in tht am not able to get an image function. it
            Message 5 of 10 , Sep 27, 2005
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              hey everybody am doing a project on image processing n am using the convolution theorem given in chapter 25.in tht am not able to get an image function.

               it would be a great help for me if anyone could tell wat sort of a function is this image function.

                        thanx.


              Yahoo! India Matrimony: Find your partner now.
            • Robin
              The index in my edition (2nd) gives only two mentions of convolution: pg 869 and pg 899. These are in chapter 24, not ch. 25. So I m not sure what you re
              Message 6 of 10 , Sep 28, 2005
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                The index in my edition (2nd) gives only two mentions
                of convolution: pg 869 and pg 899. These are in chapter
                24, not ch. 25. So I'm not sure what you're looking at.
                But in general, when you apply a convolution to an image,
                the "image function" is just the pixel value at each
                (x,y) location. The value you use will depend on context
                - what you want to do. A commonly used value is pixel
                brightness. A white pixel has brightness 255, and a black
                one has brightness 0. If you're starting from a color
                image, you can calculate brightness by averaging the red,
                blue, and green values at pixel (x,y).

                Here's a good explanation of convolution in an image-
                processing context:
                http://www.cee.hw.ac.uk/hipr/html/convolve.html

                - Robin


                --- In aima-talk@yahoogroups.com, shivam shukla <shirohin@y...> wrote:
                >
                > hey everybody am doing a project on image processing n am using the
                convolution theorem given in chapter 25.in tht am not able to get an
                image function.
                >
                > it would be a great help for me if anyone could tell wat sort of a
                function is this image function.
                >
                > thanx.
                >
                >
                > ---------------------------------
                > Yahoo! India Matrimony: Find your partner now.
              • Paul Hsueh-Min Chang
                I doubt if I can make the issues more clear, but I ll try. I still think the main problem lies in the phrase if it were false . If a proposition in a set is
                Message 7 of 10 , Sep 29, 2005
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                  I doubt if I can make the issues more clear, but I'll try.

                  I still think the main problem lies in the phrase "if it were false". If
                  a proposition in a set is "shown" to be false from observation rather
                  than through logical proof, the proposition does not make the set
                  inconsistent, because it *could* be true in some other possible world.
                  If I recall correctly, that is how logical possibility is defined. Your
                  ABC example is indeed inconsistent because it can be proved without
                  observing the world state that the set contains at least one false sentence.

                  So, according to the defintion of consistency, the authors mostly likely
                  do not mean "if it were provably false". Otherwise, the argument would
                  be employing a tautologus sentence: "if the sentence were provably
                  false, then he (or in fact, anyone) could not consistently assert it to
                  be true". One derives nothing from a tautology

                  But, without the word "provably", I don't see how the contingent truth
                  or falsity of a sentence have anything to do with consistency, for
                  reasons above. Furthermore, it seems to me that J. R. Lucas just cannot
                  consistently assert the sentence anyway, whether it is true or false.

                  I do agree that we can do nothing formal without identifying the
                  underlying logic system. Perhaps somebody can enlighten us?

                  > Thanks for the interesting discussion.
                  Thank you.

                  Paul



                  The Geek wrote:

                  > I think the discussion all boils down to the
                  > definition of "consistent" and "contradiction". I was
                  > kind of hoping one of the authors would chime in at
                  > some point to save me, but... :-) On page 137 they
                  > define consistency in reference to CSP problems as "an
                  > assignment that does not violate any constraints".
                  > That was the reason for my A,B,C example - looking at
                  > it as if it were a CSP.
                  >
                  > Their argument assumes the sentance must either be
                  > "true" or "false", (no quantuum physics or "unknown"
                  > values allowed) and hinges on the
                  > belief/assumption/fact one cannot "consistently"
                  > assert that a sentance is true if it can be shown to
                  > be false. If you buy into that, then their statement
                  > follows naturally - if the sentance was false, then he
                  > could not consistently assert it to be true, which
                  > would therefore make the sentance true - a
                  > contradiction since we assigned a value of "false" to
                  > the sentance at the beginning. The only way you can
                  > assign a value to the sentance and have everything
                  > hold together is if you assume it's true, which then
                  > implies that everyone else can assert it without a
                  > problem, but he cannot.
                  >
                  > But, at this point I don't think we're going to make
                  > any headway. Since they didn't define the logic
                  > system they're using, we can't really do anything
                  > formal.
                  >
                  > Thanks for the interesting discussion.
                  >
                  > Rob G.
                  >
                  >
                  > --- Paul Hsueh-Min Chang <avatar@...>
                  > wrote:
                  >
                  > > I will try to make some clarifications below:
                  > >
                  > > > While they haven't spelled it out in detail, I
                  > > think
                  > > > they're using a simple logic system and definition
                  > > of
                  > > > "consistent". In other words, they're not trying
                  > > to
                  > > > allow for an "agent belief" representation where
                  > > we
                  > > > acknowledge that beliefs might be wrong. Instead,
                  > > if
                  > > > you assert a sentance it must be true or you've
                  > > > introduced a contradiction.
                  > >
                  > > Maybe our sources are different. AFAIK a
                  > > contradiction is the negation
                  > > of a tautology (P. Tidman and H. Kahane, Logic and
                  > > Philosophy: A Modern
                  > > Introduction, 8th edition, p48). For example, p^~p
                  > > is a contradiction.
                  > > So when I assert a contingently false sentence whose
                  > > truth depends on
                  > > the world, I introduce no contradiction. But maybe
                  > > our definitions of
                  > > contradiction differ.
                  > >
                  > > > If we take 3 sentances:
                  > > >
                  > > > A = B.
                  > > > B = C.
                  > > > A != C.
                  > > >
                  > > > If we try to assert all three sentances, we cannot
                  > > do
                  > > > it without creating a contradiction. Since this
                  > > is
                  > > > the definition of "consistency", they would say
                  > > that
                  > > > we cannot "consistently" assert all three
                  > > sentances,
                  > > > since doing so would introduce a contradiction.
                  > >
                  > > It is true that we would introduce inconsistency,
                  > > but we would not
                  > > introduce a contradiction. Of course, any set of
                  > > sentences that contains
                  > > any contradictory sentence is inconsistent, but not
                  > > vice versa.
                  > >
                  > > > In this case, as an agent **I** can consistently
                  > > > assert the sentance "Agent A cannot assert this
                  > > > sentance without being wrong" without introducing
                  > > a
                  > > > contradiction. However if Agent A tries to assert
                  > > the
                  > > > same sentance, he runs into a problem. If he
                  > > asserts
                  > > > it as true and is right, then the sentance is
                  > > false,
                  > > > so he's wrong.
                  > >
                  > > I know what you are trying to say, but that I don't
                  > > think we are talking
                  > > about the same thing. The authors clearly intend to
                  > > show two things:
                  > > 1. The sentence is a tautology (i.e. necessarily
                  > > true).
                  > > 2. Yet, J. R. Lucas cannot assert it.
                  > > They offer two arguments to show (1). What I do not
                  > > understand is the
                  > > their second argument to show (1). You are talking
                  > > about (2), which I
                  > > find no problem.
                  > >
                  > > > But the whole point of the illustration is simply
                  > > to
                  > > > show that **sometimes** one agent is unable to
                  > > > assert/know/do things that another agent can, but
                  > > that
                  > > > doesn't automatically imply that agent is
                  > > inferior,
                  > > > it's inability to assert/know/do may be related to
                  > > the
                  > > > specific situation.
                  > >
                  > > I do understand the point; it is simply that
                  > > particular argument (the
                  > > one on p950 in parentheses) that confuses me. As a
                  > > philosophical
                  > > treatment, I think the chapter should reasonably be
                  > > taken literally, but
                  > > when I do so, I cannot make sense of that particular
                  > > argument.
                  > >
                  > > Paul
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