Wednesday, December 14, 2011


David Brightly has rightly questioned whether my argument is valid:
Caesar was a man
Caesar is not (any longer) a man
Some man is not a man
It's not, unless our domain of quantification consists only of men.  But I can easily restate the argument
No y is identical with Caesar
Some x was identical with Caesar
Some x is not identical with any y
I don't see any way round that.  Of course, some x was identical with some y (for y = Caesar, where the verb "=" has past tense).  And perhaps we could read the existential quantifer as tensed - there was an x such that x is not identical with any y.  But there's no way we could make any sense of it in standard predicate logic.  Moreover, the standard way of understanding quantification as a kind of relation between variables or open sentences or predicates on the one side, and objects on the other, makes no sense either.  For example, logicians say that the predicate "- was an emperor" is satisfied.  Is satisfied?  Or was satisfied?


Anthony said...

>> No y is identical with Caesar
>> [....]
>> I don't see any way round that.

The easy way around it is to say that Caesar exists, but not as a [currently living] man.

And if you don't say that, you get into trouble way before then. How can you say that Caesar *is* dead, if there *is* no y that *is* identical to Caesar?

J said...
This comment has been removed by the author.
Edward Ockham said...

>>How can you say that Caesar *is* dead, if there *is* no y that *is* identical to Caesar?

By recasting it as a negative: "Caesar is dead" = "It is not the case that Caesar is alive". Since 'Caesar is alive' is false, it follows that the corresponding denial is true.

David Brightly said...

Morning Ed. I'm not sure how you intend us to read your syllogism and I'm afraid you rather lose me in the ensuing commentary. Are we perhaps to find concept terms that can substitute for x and y to make the sentences true? If so, then 'Roman' for x and 'Briton' for y would work. Also 'Ancient Man' for x and 'Modern Man' for y. And surely this suggests a resolution of the puzzle. Let me reword slightly:

1. No y is Caesar = None of the ys is Caesar
2. Some x was Caesar = one of the xs was Caesar
3. Some x is no y = one of the xs is none of the ys

Trouble comes when we put x=y=Man. But 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. (3) follows from (1) and (2) only if the quantified expressions in (3) have the same extensions as they do in (1) and (2), Caesar being the 'middle term'. And indeed one of the men who ever were is not one of the presently existing men. Fine. But the present tense 'is' in (3) restricts the 'some x' quantifier to the presently existing men and under this interpretation (3) is self-contradictory. In summary, the argument is invalid because the 'some man' in (3) has a narrower range than the 'some man' in (2). Or in other words, the xs in (3) are a subset of the xs in (2). I'm seeing this very clearly now but may not have explained myself perfectly. Does it make sense to you?

David Brightly said...

Further to my last comment, I'm not sure I share your pessimism wrt standard predicate logic. My feeling is that MPL, plus the language of sets perhaps, resolves the paradox. One point about tense: we might think that the tense variations in our example make a difference. I think they do, but by way of modifying the quantifier ranges. Not by affecting the identities or set membership assertions. 'Caesar was an ancient man' and 'Caesar is one of the ancient men' say the same thing, I believe. We could semantically ascend to 'Caesar refers to one of the individuals that the ancient men refers to.' Likewise talk of the satisfaction of '_ was an emperor' is talk about the sentences currently before us, for which the present tense is all we need. One last point: 'some man was Caesar' could be seen as a surface grammatical transformation that occurs to the true assertion 'some man is Caesar' said during Caesar's lifetime when it is translated to the present temporal point of view. Thus we might want to say that in (2) 'some man' more narrowly ranges over the contemporaries of Caesar. My analysis goes through under this interpretation too.

Anthony said...

>> "Caesar is dead" = "It is not the case that Caesar is alive".

So my great-great-great-grandson is dead?

Edward Ockham said...

>>So my great-great-great-grandson is dead?

"It is no longer the case that Caesar is alive"

Anthony said...

>> "It is no longer the case that Caesar is alive"

So "(Existed X : X equaled Caesar and X was alive) and (not-exists Y : Y=Caesar and Y is alive)"?

I guess that works for that one particular case.

J said...

Mortality is part of the extension of living humans (along with say, mammalian-ness). Being bone, fertilizer, ash, etc that of humans who died.

Solved! Term logic doesn't deal with individuals anyway a point lost of most scholastic wannabes