Anthony rightly raised the problem of my ‘some men are not men’. He suggested (rightly I think) that this is a problem for ‘Brentano equivalence’, the thesis that ‘Some A is B’ is equivalent to ‘An A-B exists’, and that ‘some A is not B’ is equivalent to ‘An A-not-B exists’.
He is right. If Brentano is right, then ‘some men are not men’ is equivalent to ‘men that are not men exist’, which is clearly wrong. So what’s the problem? I suggest that Brentano is wrong. Clearly we can say that some of the men who landed on the moon have now died ( for example, Alan Shepard, the one who played golf on the moon). So, some men such as Shepard are no longer men. If Brentano is right, that implies that men who are non-men exist, which is false. Non valet consequentia, so Brentano is wrong.
The late thirteenth century philosophers of language, such as the early Scotus, were acutely aware of this problem. Many of them distinguished between so-called indefinite negation of the form ‘A is a non-B’, and pure negation ‘A is not B’. Indefinite negation is affirmative. It affirms the existence of an A that is non-B. In this sense ‘some man is a non-man’ is false. By contrast, pure negation denies everything, including the affirmation of existence. In that sense ‘some man is a not a man’ is true, pace Brentano.
The problem is to render this in predicate logic. The formal sentence ‘for some x, Ax and not Bx’ is affirmative in the traditional sense: it asserts that some x is both A and non-B. However, the pure negative for ‘not for some x, Ax and Bx’ is not equivalent to the medieval ‘some A is not B’. The predicate logic version simply denies the existence of anything that is both A and B, whereas the medievals understood it in the sense we understand ‘some men (such as Alan Shepard) are not men (i.e. are men no longer)’.
Brentano’s thesis was the first formulation of one of the key assumptions of the modern predicate calculus. It is wrong for the same reason the calculus is wrong. It does not translate the meaning of a standard English sentence in the way we want to translate it. So what is the meaning of the sentence, and into what formal language can we translate it?
Showing posts with label time. Show all posts
Showing posts with label time. Show all posts
Friday, January 13, 2012
Monday, December 12, 2011
Quantifying over
In an earlier post I asked whether some men are not men. Intuitively all men are men, and so ‘some men are not men’ is false. Yet if the term ‘man’, occurring in the subject position of a sentence, means something like ‘someone who is, was or will be a man’, and given that Caesar was once a man, but is no longer a man, i.e. not now a man, it seems to follow that some men (e.g. Caesar) are not men, i.e. were men but aren’t now. Thus, counterintuitively, some men are not men. From this we derived the even more puzzling ‘some present events are not present events’. Caesar’s crossing the Rubicon was once a present event. And if ‘present event’ means ‘event which is, or was once, or will be present’, and since Caesar’s crossing the Rubicon is not present, it apparently follows that at least one present event (crossing the Rubicon) is not a present event.
David Brightly, taking the approach of a contemporary logician, helpfully suggests that “we just stipulate in advance which men we propose to quantify over and this bounds what 'some man' may refer to. It could be men alive, dead or alive, living in Midsomer, ever been married, whatever is appropriate, as long as we make it explicit and make appropriate adjustments elsewhere.”
I object that it is a problem either way. If the English word ‘man’ in fact only means presently existing man, it immediately follows that we can’t quantify over men in the 13th century, for there are none to quantify over. Nor were there any. ‘Some man lived in the 13th century’ is false, unless there is a 700 year old man still alive. The question is what the word ‘man’ ranges over. If only presently existing entities, then there were no men alive 10 years except for those alive now, thus very few men who were alive 90 years ago. Census records are all false. There were not 7 million people living in London in the 1920s. More like a few thousand (namely all present Londoners who are old enough to have lived here 90 years ago).
If by contrast we accept that there were 7 million Londoners in the 1920s, we have to accept that most of these, namely the ones who have died, are no longer Londoners. Thus, some Londoners are not Londoners. You can’t escape the problem by specifying ‘domains of quantification’.
David Brightly, taking the approach of a contemporary logician, helpfully suggests that “we just stipulate in advance which men we propose to quantify over and this bounds what 'some man' may refer to. It could be men alive, dead or alive, living in Midsomer, ever been married, whatever is appropriate, as long as we make it explicit and make appropriate adjustments elsewhere.”
I object that it is a problem either way. If the English word ‘man’ in fact only means presently existing man, it immediately follows that we can’t quantify over men in the 13th century, for there are none to quantify over. Nor were there any. ‘Some man lived in the 13th century’ is false, unless there is a 700 year old man still alive. The question is what the word ‘man’ ranges over. If only presently existing entities, then there were no men alive 10 years except for those alive now, thus very few men who were alive 90 years ago. Census records are all false. There were not 7 million people living in London in the 1920s. More like a few thousand (namely all present Londoners who are old enough to have lived here 90 years ago).
If by contrast we accept that there were 7 million Londoners in the 1920s, we have to accept that most of these, namely the ones who have died, are no longer Londoners. Thus, some Londoners are not Londoners. You can’t escape the problem by specifying ‘domains of quantification’.
Thursday, December 30, 2010
Album fuit disputaturum
Vallicella continues here. What I find puzzling is how he continues with trifling, even frivolous objections to the extreme version of Ockhamism that I have proposed here and there, without picking up on any of the really serious objections to it.
So let’s talk about one possible objection here. Extreme or ‘London’ Ockhamism is the view that relations can only relate things. There are no ‘non-existent relata’. Where the Stanford Realist talks about ‘non-existent relata’, as apparently required by ‘Tom is thinking about Pegasus’, the Ockhamist sees only things. ‘Tom is thinking about Pegasus’ relates Tom to nothing (for Pegasus is not a thing). Thus ‘Tom is thinking about Pegasus’ is a linguistic relation only
Londonists allow present relations. Thus ‘Cameron is colleague of Clegg’ is true, and relates two things. Londonists also allow past relations. Thus ‘Churchill met Roosevelt’ is true, since it expresses a relation which was between two things, although now between no things.
The problem occurs when the relation is between something in the present, and something in the past, or when there is no time at which the relation could possibly have been between two things. Consider ‘Churchill (who died in 1965) died before Cameron (who was born in 1966) was born’. At no time when Churchill was something, was Cameron something. For when Churchill was something, Cameron was not born. And when Cameron was born, and so was something, Churchill was nothing. Thus at no time did ‘Churchill died before Cameron was born’ relate any two things.
The medieval philosophers discussed a similar problem in the sophisma ‘Album fuit disputaturum’ (a white person was going to argue). Assume Socrates was white, before he went into the Mediterranean sun. And assume that he only argued after acquiring a Mediterranean suntan. Thus ‘a white (i.e. untanned) person is arguing’ was never true (at least, in respect of Socrates). Thus ‘Album fuit disputaturum’ apparently relates the person of whom the subject is true (untanned and unarguing Socrates) with the person of whom the predicate is true (tanned and arguing Socrates). How is that possible?
So let’s talk about one possible objection here. Extreme or ‘London’ Ockhamism is the view that relations can only relate things. There are no ‘non-existent relata’. Where the Stanford Realist talks about ‘non-existent relata’, as apparently required by ‘Tom is thinking about Pegasus’, the Ockhamist sees only things. ‘Tom is thinking about Pegasus’ relates Tom to nothing (for Pegasus is not a thing). Thus ‘Tom is thinking about Pegasus’ is a linguistic relation only
Londonists allow present relations. Thus ‘Cameron is colleague of Clegg’ is true, and relates two things. Londonists also allow past relations. Thus ‘Churchill met Roosevelt’ is true, since it expresses a relation which was between two things, although now between no things.
The problem occurs when the relation is between something in the present, and something in the past, or when there is no time at which the relation could possibly have been between two things. Consider ‘Churchill (who died in 1965) died before Cameron (who was born in 1966) was born’. At no time when Churchill was something, was Cameron something. For when Churchill was something, Cameron was not born. And when Cameron was born, and so was something, Churchill was nothing. Thus at no time did ‘Churchill died before Cameron was born’ relate any two things.
The medieval philosophers discussed a similar problem in the sophisma ‘Album fuit disputaturum’ (a white person was going to argue). Assume Socrates was white, before he went into the Mediterranean sun. And assume that he only argued after acquiring a Mediterranean suntan. Thus ‘a white (i.e. untanned) person is arguing’ was never true (at least, in respect of Socrates). Thus ‘Album fuit disputaturum’ apparently relates the person of whom the subject is true (untanned and unarguing Socrates) with the person of whom the predicate is true (tanned and arguing Socrates). How is that possible?
Saturday, December 11, 2010
Internal and external time
I am still thinking about Brightly's objection to my arguments here and elsewhere. Is the impossibility of a Cantorean transfinite consciousness - a conscious moment preceded by an infinite number of conscious moments before that - simply down to the length of time such a consciousness would require? One simply cannot wait for so long. Or is it as I maintain, that the discontinuity between the transfinite consciousness and the finite one is such that the traversal of such a sequence is impossible?
Every conscious moment leads to the next, but that 'next moment' must be finitely far away. For conscious time and physical time are fundamentally different. I cannot 'cheat' infinity by sleeping for an infinite number of days then waking up on the infinitieth day. The physical time is irrelevant, for in my consciousness the passage of an infinite number of days goes unnoticed. The infinitieth day is indistinguishable from any other day in the finite series. Therefore (I would like to argue) in order to get to the infinitieth day, it is essential to be conscious of every one of the previous days. Every conscious moment (i.e. every moment of my consciousness) must be either (a) conscious of some previous moment or (b) conscious of nothing previous, in which case it is my first conscious moment, and nothing of mine preceded it.
Augustine (in the Confessions, e.g., see here) made a similar distinction between 'internal' (conscious) and 'external' (physical) time, holding that only internal time is real.
Every conscious moment leads to the next, but that 'next moment' must be finitely far away. For conscious time and physical time are fundamentally different. I cannot 'cheat' infinity by sleeping for an infinite number of days then waking up on the infinitieth day. The physical time is irrelevant, for in my consciousness the passage of an infinite number of days goes unnoticed. The infinitieth day is indistinguishable from any other day in the finite series. Therefore (I would like to argue) in order to get to the infinitieth day, it is essential to be conscious of every one of the previous days. Every conscious moment (i.e. every moment of my consciousness) must be either (a) conscious of some previous moment or (b) conscious of nothing previous, in which case it is my first conscious moment, and nothing of mine preceded it.
Augustine (in the Confessions, e.g., see here) made a similar distinction between 'internal' (conscious) and 'external' (physical) time, holding that only internal time is real.
On this view, however, it seems difficult to explain the passage of time.For if there are times past and future, I desire to know where they are. But if
as yet I do not succeed, I still know, wherever they are, that they are not
there as future or past, but as present. For if there also they be future, they
are not as yet there; if even there they be past, they are no longer there.
Wheresoever, therefore, they are, whatsoever they are, they are only so as
present. Although past things are related as true, they are drawn out from the
memory, -- not the things themselves, which have passed, but the words conceived
from the images of the things which they have formed in the mind as footprints
in their passage through the senses. My childhood, indeed, which no longer is,
is in time past, which now is not; but when I call to mind its image, and speak
of it, I behold it in the present, because it is as yet in my memory. Whether
there be a like cause of foretelling future things, that of things which as yet
are not the images may be perceived as already existing, I confess, my God, I
know not. This certainly I know, that we generally think before on our future
actions, and that this premeditation is present; but that the action whereon we
premeditate is not yet, because it is future; which when we shall have entered
upon, and have begun to do that which we were premeditating, then shall that
action be, because then it is not future, but present. (Confessions XI.18.23).
Monday, December 06, 2010
Cantor's Angel
The brief argument I gave here needs expanding. I wrote:
While I am waiting for the demon to replace my teeth, a Cantorean angel whispers to me. I must not despair. After this process has been repeated infinitely many times, my soul will enter a transfinite Cantorean paradise. I will still be conscious of every one of the infinite moments that has passed in hell. But those moments will now be behind me. They have all happened, infinitely many of them, an infinite number of teeth drilled out and replaced.
Now I ask. Does the pronouncement give me any hope? Surely not. I cannot hope ever to escape this infinite painful process; I have no hope. But if the consciousness in the Cantorean paradise were my consciousness, I would have such a hope. Therefore the consciousness in the Cantorean paradise cannot be my consciousness.
My consciousness is a set of conscious moments tied together by their belonging to a single consciousness. Any future moment must be such that I can hope or expect to experience it by the process of waiting. Thus no future moment of the same consciousness can be such that it is preceded by an infinite number of such moments belonging. For I cannot hope or expect the experience of such a moment. I would be waiting for and expecting something that will never happen to me. (I concede it is logically possible that such a moment could happen to someone else, who was remembering my conscious moments as if they were my own, but more on that later).
Every day I wake up from sleep, that 'little slice of death', and becomeImagine the following thought-experiment. My soul is in hell, and I am being tormented by a demon dentist who is removing my teeth by drilling through their nerves in an exquisitely painful way. My agonising screams are echoing through the halls of the inferno. After all my teeth are removed, they are supernaturally replaced, and the whole process begins again. I understand that this process is to continue infinitely. (There is a colourful depiction of the infinity of hell by James Joyce here).
conscious. Imagine the following thought-experiment. I wake up an infinite
number of times. Could I have a conscious moment after that infinite sequence?
Is it possible that there could be a waking moment belonging to my consciousness
such that there are an infinite number of waking moments before that? Surely
not. I can't think of an argument to prove it, rather, it seems an irreducible
part of my idea of consciousness that I cannot conceive of an actual or
'completed' infinity of conscious moments.
While I am waiting for the demon to replace my teeth, a Cantorean angel whispers to me. I must not despair. After this process has been repeated infinitely many times, my soul will enter a transfinite Cantorean paradise. I will still be conscious of every one of the infinite moments that has passed in hell. But those moments will now be behind me. They have all happened, infinitely many of them, an infinite number of teeth drilled out and replaced.
Now I ask. Does the pronouncement give me any hope? Surely not. I cannot hope ever to escape this infinite painful process; I have no hope. But if the consciousness in the Cantorean paradise were my consciousness, I would have such a hope. Therefore the consciousness in the Cantorean paradise cannot be my consciousness.
My consciousness is a set of conscious moments tied together by their belonging to a single consciousness. Any future moment must be such that I can hope or expect to experience it by the process of waiting. Thus no future moment of the same consciousness can be such that it is preceded by an infinite number of such moments belonging. For I cannot hope or expect the experience of such a moment. I would be waiting for and expecting something that will never happen to me. (I concede it is logically possible that such a moment could happen to someone else, who was remembering my conscious moments as if they were my own, but more on that later).
Friday, September 03, 2010
A perfect difficulty
In an earlier post I suggested that the semantics of a noun or referring phrase does not include time. The sense of time is what a verb brings in. Hence past, present and future tenses, and hence qualifying adverbs like 'now' and 'then'. This avoided the philosophical difficulty about treating the ship composed of the old planks ('the ship as it was then') being any different from the ship composed of the new planks ('the ship as it is now').
But perhaps there is a difficulty, indicated by the distinction between the simple past tense (the ship was composed of the old planks) and the perfect tense (the ship has been composed of the old planks). The distinction (though familiar and its ordinary usage well understood*) is not philosophically clear, and grammar books tend to fudge the explanation. But it seems to be: we use 'has been F' as a predicate to qualify the object 'as it as now', and 'was F' to qualify the object 'as it was then'. Otherwise the grammatical distinction makes no sense. It must distinguish something, and the distinction seems to be the philosophically dangerous one that I said we should avoid.
A similar distinction is evident in the difference between 'in 10 years time people will have a better standard of living' and 'in a 100 years' time, people will live for much longer'. In the first 'people' apparently ranges over people who exist now. In second, it clearly ranges over people who have not been born yet, but will exist in the future. Perhaps there is some notion of tense built into the subject of a proposition. But it's puzzling, and there is not much I can say about it, as things are now.
* The idea for this post came after reading a paper by Braakhuis, who wrote 'In 1940 Grabmann has drawn attention to this collection'. The use of the perfect, rather than the simple past, misleadingly suggests that Grabmann is still alive, which he isn't (he died in 1949).
But perhaps there is a difficulty, indicated by the distinction between the simple past tense (the ship was composed of the old planks) and the perfect tense (the ship has been composed of the old planks). The distinction (though familiar and its ordinary usage well understood*) is not philosophically clear, and grammar books tend to fudge the explanation. But it seems to be: we use 'has been F' as a predicate to qualify the object 'as it as now', and 'was F' to qualify the object 'as it was then'. Otherwise the grammatical distinction makes no sense. It must distinguish something, and the distinction seems to be the philosophically dangerous one that I said we should avoid.
A similar distinction is evident in the difference between 'in 10 years time people will have a better standard of living' and 'in a 100 years' time, people will live for much longer'. In the first 'people' apparently ranges over people who exist now. In second, it clearly ranges over people who have not been born yet, but will exist in the future. Perhaps there is some notion of tense built into the subject of a proposition. But it's puzzling, and there is not much I can say about it, as things are now.
* The idea for this post came after reading a paper by Braakhuis, who wrote 'In 1940 Grabmann has drawn attention to this collection'. The use of the perfect, rather than the simple past, misleadingly suggests that Grabmann is still alive, which he isn't (he died in 1949).
Thursday, August 26, 2010
Bikes, now and then
In discussing temporal identity, there is a strong temptation to use expressions like 'the bike then' or 'the bike as it was yesterday' or 'the bike as it was at that spatio-temporal moment' - see the comments to my previous post. As though 'the bike then' refers to some different object than 'the bike now' or 'the bike today'.
These expressions don't make any sense to me. Surely 'then' and 'now' or 'yesterday' and 'today' are just adverbs modifying the verb, rather than an adjective modifying a referring phrase or description. Consider
This bike, which was repaired yesterday, has a new mudguard today.
The sentence contains just one subject: 'this bike'. It is modified by two predicates. The first - 'was repaired yesterday' - tells us something that happened at some previous moment to the bike. The second - 'has a new mudguard' - tells us something that is true of it today, now. It makes no sense to divide the subject into something that is or was 'the bike yesterday', and another, different thing, that is 'the bike today'. Both 'was repaired' - which is a predicate in the past tense - and 'has a new mudguard' - which is in the present tense - qualify this single bike. Or 'this bike now', if you really want.
If there really is an object such as 'this bike then', it is an object to which the predicate 'is now being repaired' applies. For it was true to say of the bike, then, 'this bike is being repaired'. By contrast, the past tense predicate 'was being repaired' applies to 'the bike now'. But as I say, modifying a subject expression by means of an adverb doesn't make sense.
The medieval writers keenly appreciated this. A noun or referring phrase has no tense attached to it. Its semantics does not include time. The sense of time is what a verb brings in. Hence past, present and future tenses, and hence qualifying adverbs like 'now' and 'then'.
These expressions don't make any sense to me. Surely 'then' and 'now' or 'yesterday' and 'today' are just adverbs modifying the verb, rather than an adjective modifying a referring phrase or description. Consider
This bike, which was repaired yesterday, has a new mudguard today.
The sentence contains just one subject: 'this bike'. It is modified by two predicates. The first - 'was repaired yesterday' - tells us something that happened at some previous moment to the bike. The second - 'has a new mudguard' - tells us something that is true of it today, now. It makes no sense to divide the subject into something that is or was 'the bike yesterday', and another, different thing, that is 'the bike today'. Both 'was repaired' - which is a predicate in the past tense - and 'has a new mudguard' - which is in the present tense - qualify this single bike. Or 'this bike now', if you really want.
If there really is an object such as 'this bike then', it is an object to which the predicate 'is now being repaired' applies. For it was true to say of the bike, then, 'this bike is being repaired'. By contrast, the past tense predicate 'was being repaired' applies to 'the bike now'. But as I say, modifying a subject expression by means of an adverb doesn't make sense.
The medieval writers keenly appreciated this. A noun or referring phrase has no tense attached to it. Its semantics does not include time. The sense of time is what a verb brings in. Hence past, present and future tenses, and hence qualifying adverbs like 'now' and 'then'.
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