(1) No y is identical with Caesarto which David Brightly objected that
(2) Some x was identical with Caesar
(3) Some x is not identical with any y
For (1) to be true it's clear that the range of the quantified expression 'no y' cannot be all men who ever were. Rather the present tense 'is' modifies the quantifier 'no man' restricting the ys in (1) to the presently existing men. Similarly in (2) the past tensed 'was' modifies the quantifier 'some x', restricting the xs to the men who ever were, ie, no restriction at all.Well that’s true, and that’s one solution proposed by some of the scholastics, who thought that the present tense of the main verb of the sentence restricts (Latin: restringit) a common term like ‘man’ to suppositing (i.e. ranging over) presently existing men (praesentibus).
What’s wrong with that solution is the present tense that we have to use when we say what things are in the domain or range of quantification. David says that using a verb in the past tense allows the quantifier it to range over all the man that ever were. The problem is the implied present tense of the ‘ranging’. How is it that the quantifier ‘ranges’ – present tense – over past men, men who longer exist? Surely it can’t. Nor can the domain now ‘contain’ all such men. It used to contain them, but now it doesn’t. So the second premiss (2) cannot be true. There cannot be any x that was identical with Caesar, because however wide the domain or range of quantification, the domain exists in the present. It has to exist in the present because we say that it is the range of our quantification, and to say that we must use the present tense.
Anthony is closer when he says that the real problem is presentism, but there are problems with presentism also, which I will talk about later.
24 comments:
>> Anthony is closer when he says that the real problem is presentism
Well, that's the first problem. The next is that you're equivocating between "does not exist" as in "is not part of existence" and "does not exist" as in "cannot be interacted with".
>> How is it that the quantifier ‘ranges’ – present tense – over past men, men who longer exist? Surely it can’t.
Can we not say then that Caesar was one of the emperors Caesar, Augustus, and Tiberius?
>>Well, that's the first problem. The next is that you're equivocating between "does not exist" as in "is not part of existence" and "does not exist" as in "cannot be interacted with".
<<
In no way would I ever mean the second. I don't understand the first, unless 'is not part of existence' is a more complicated way of saying 'does not exist'.
>>Can we not say then that Caesar was one of the emperors Caesar, Augustus, and Tiberius?
<<
Are you saying these emperors are (present tense) part of the domain?
>> In no way would I ever mean the second.
You might say that something does not exist presently, right? What do you mean by "does not exist presently", if you don't mean "cannot be interacted with"?
>> I don't understand the first, unless 'is not part of existence' is a more complicated way of saying 'does not exist'.
It's another way to say "does not exist", not a more complicated way.
>> Are you saying these emperors are (present tense) part of the domain?
Not at all. I'm saying that the tense of the verb and the quantified expression combine to tell us of which individuals the predicate is asserted, and of these how many, some or all.
>> I'm saying that the tense of the verb and the quantified expression combine to tell us of which individuals the predicate is asserted, and of these how many, some or all.
It's not that simple, though:
"A ten-year-old was a 9-year-old" (current ten-year-olds)
"My wife was a marathon runner" (my current wife)
"An 11-year-old graduated from college." (s/he might not be 11 any more)
"A U.S. President was confined to a wheelchair" (he's not even alive any more, let alone US President)
"A student at this high school was born in Mexico" (a current student)
"A student at this high school was shot and killed" (not a current student)
"A student at this high school was President of the United States" (not a current student)
"A student at this high school was caught smoking pot" (may or may not be a current student, but was a student while caught smoking pot, wasn't merely a former student)
>> It's not that simple, though:
Sure. There are plenty of sources of ambiguity in ordinary language. You have alerted Ed to an example where the perhaps natural interpretation of an ambiguous quantified expression leads to a paradoxical conclusion. My claim is that when the quantified expression is correctly understood, taking tense into account, the conclusion no longer follows from the premises and the paradox evaporates. Which is as it should be.
>>My claim is that when the quantified expression is correctly understood, taking tense into account, the conclusion no longer follows from the premises and the paradox evaporates.
I'm still not seeing it. You ask whether we can say e.g. "Caesar was one of the emperors Caesar, Augustus, and Tiberius?"
But for this to work, the predicate "--- was one of the emperors Caesar, Augustus, and Tiberius" must apply to something that is within the domain. My point is that Caesar cannot be in the domain, on the assumption that he no longer exists. Perhaps he was in the domain, but he isn't now. So how can any predicate apply to him – even a past tense predicate? To repeat: the fact that the predicate "--- was one of the emperors Caesar, Augustus, and Tiberius" contains a past tense verb, does not alter the fact that it must 'apply to' or 'be satisfied by' something. The verbs apply and 'satisfied by' are in the present, not the past tense.
I've said this so many times I'm wondering if I am missing something.
I've been trying to avoid talking about the domain in my latest comments, and also to avoid talking about sets of individuals. Since set talk comes very naturally I find the latter is harder. It's all rather metalinguistic though possibly unavoidable. Perhaps 'Caesar is in the domain' is a shorthand for 'Caesar is the name of one of the individuals we are talking about'. Likewise, talk of verbs applying and predicates being satisfied is primarily talk about bits of sentences, surely?
'is an animal is truly predicated of man' means that the sentence 'man is an animal' is true.
'was an emperor is truly predicated of Caesar' means that 'Caesar was an emperor' is true.
>>Perhaps 'Caesar is in the domain' is a shorthand for 'Caesar is the name of one of the individuals we are talking about'.
Same problem. If N is the name of X, then it's a present tense 'is'. But it will be objected that it was the name of X. Similarly for 'we are talking about'. '-- is being talked about' is predicated now of Caesar. But predicated of what? There is no such person to be talked about.
To be clear, this is not an issue abouts sets or domains per se. It's about the problem of 'ranging over' being necessarily a relation that is expressed in the present tense. How can one term of a present tense relation (ranging over, satisfying, referring to etc) be a no longer existing object?
>> How can one term of a present tense relation (ranging over, satisfying, referring to etc) be a no longer existing object?
Aha! We have touched on this before at the maverick's I think. We have different ideas about relations, I suspect, because I have absolutely no problem with this. For me a relation is a logical gadget describing the 'shapes' of certain sentences. You and Bill sound rather more realist about relations. Would you agree?
Here is how I see _is/was the grandmother of_:
I'd say that we use the form of words 'x is/was the grandmother of z' when we recognise an instance of a certain pattern in time. That pattern can be described as follows: for some y and some times t1 and t2, x gave birth to y at t1 and y gave birth to z at t2. If x is still extant we tend to say 'x is...' else we say 'x was...'.
But this may not be germane to the present discussion if we are really talking about relations between bits of sentences. Note the second pair of quotes in
'was an emperor is truly predicated of Caesar' means that 'Caesar was an emperor' is true.
>>But this may not be germane to the present discussion if we are really talking about relations between bits of sentences.
<<
Agree absolutely. This intra-linguistic approach to resolving the difficulty is entirely consistent with what I have argued elsewhere. You remember my claim that the 'refers to' in 'X refers to Y' is a logically intransitive verb?
That was sort of where I was trying to get to.
yes indeed I do remember. In addition to singular referring terms construed this way I think we need plural terms like 'the first triumvirate' or 'the ancient Romans' which behave exactly like mathematical sets (seen as I have suggested as plural referring devices rather than collections) and we can then explain quantified terms in the language of sets.
>> we can then explain quantified terms in the language of sets.
On a nominalistic interpretation of 'sets' i hope.
naturellement.
I make a start at this here. Is this sufficiently nominalistic?
>> is this sufficiently nominalistic?
So this is yours, yes? I didn't realise.
I'll take a look.
I remember this. In answer to your question, then, yes.
Much work needs to be done on elementary first order semantics, though. How would elementary quantification theory be expressed in this new way?
And we never resolved the problem of whether Ockham sets are consistent with standard set theory.
>> How would elementary quantification theory be expressed in this new way?
That sounds a rather technical question. I guess the idea would be to show how to translate quantified sentences into a first order language of sets over certain ur-elements, as I almost do in the comments above. But I'm getting out of my depth here...
>> Is this sufficiently nominalistic?
It treats numbers as "ordinary objects".
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