I can’t find anything in Copi which disallows the following inference:
(x)(Ey)Bxy
therefore, (Ey)Byy, by Universal Instantiation
But unless I’m horribly confused, this inference is invalid.
So… what am I missing in the Copi text, if anything?
Thanks,
-FrL-
I don’t have copy of Copi (my class used Edgerton) but . . .
Are x and y both free variables? Is y dependent on x?
Here’s an example of why your inference may be wrong and I’m assuming B is a general proposition such as Bxy = x < y in the universe of the Naturals
Clearly the proposition x (Ey) Bxy is true by assigning y = x+1 but Byy is clearly not true.
Take a look at
http://www.cs.odu.edu/~toida/nerzic/content/logic/pred_logic/inference/univ_inst.html
It discusses exactly the error you’re making.
I also don’t know the specific text, but this would be simply disallowed by the form of predicate logic that I studied in school, along the following lines.
You cannot universally instantiate x to another variable, only an atom (?) (Not quite sure of the vocab any more.) So you could go to (Ey)Bay or something like that.
Now you can try to instantiate y, but the rules of existential instantiation keep you from using an atom that already exists. Therefore you’d get to Bac or something like that.
You can re-introduce the existentials at this point, but since the atoms are different you’d end up with something like (Ew)(Ez)Bzw , which is not what you were trying to go for.
And note that trying to instantiate y before x is also not allowed – you have to get rid of the variable quantifiers from the outside in, from left to right within a given term.
To prove (Ez)Bzz or something similar, you would probably need to introduce existential quantification from something like Bdd
I don’t know if this makes any sense at all in Copi, but I thought I’d try going over it.
That’s my problem: I know the inference is invalid precisely because of the kind of counterexample you give to it, yet I don’t see anything in Copi to disallow it.
-FrL-
In Copi you are explicitly allowed to instantiate into other variables. In fact, in Copi, the following is valid (in a technical sense of ‘valid’ which Copi defines in his text:
(x)Fx
therefore, Fx, by Universal Instantiation
Where the ‘x’ of the second line is not a constant/atom but rather is still a variable! Copi allows unbound variables to occur in lines in proofs.
I know. This threw me when I first encountered it as well.
Also note that for Copi, the following proof is valid. He explicitly endorses it in the text:
In other words, for Copi, universal quantifcation has existential import. He explains in the text that his system assumes a universe containing at least one object.
So… Yeah. Copi’s wierd.
For Copi, you must existintially instantiate into a variable (not a constant, though I don’t understand why he makes this restriction) which does not appear free in any line previous to the instantiation. You can see that an instantiation from (Ey)Byy in my inference would not break this restriction. That’s part of the problem: intuitively speaking, it seems like it should be thought of as breaking this restriction, since the first ‘y’ in ‘(Ey)Byy’ could have been absolutely any variable or constant/atom, and if it had been anything other than ‘y’ it would indeed have been free. Bur purely formally speaking, in the line as it stands, ‘y’ does not occur free.
In the inference in question, the outermost quantifier is instantiated, and the innermost quantifier is not, so I don’t think I’ve broken this restriction.
That’s how it feels like it should be, of course. But I’m trying to find out what in the Copi text tells me something like this. Right now I’m not seeing it.
His section on “Argument involving relations” is only like three pages, so I don’t see how I could have missed it. But surely I have.
Thanks,
-FrL-
Check and make sure that you’re not missing some restriction on what variable you can instantiate to. If there are none, that’s an error in the book.
I was just thinking, maybe Copi means it to have been understood that instantiations must be into free variables, never into bound variables. In the inference I am discussing, the instantiation is into a bound ‘y’ rather than a free ‘y’.
I think this makes perfect sense, (indeed it’s just part of a statement of what instantiation is) and it solves my problem. I haven’t seen anywhere where Copi explicitly says this, but still, it’s the kind of thing I can see him imagining is just “understood.” (Again, because it’s basically just part of what instantiation is–going from quantified to non-quantified…) I think it’s posisble for students to get tripped up over it, however. I wish he had explicitly said it.
Or maybe he did and I missed it…
-FrL-
Probably. When I was studying Edgerton, I was having trouble with a section. Rereading the chapter (a few times) I found the necessary info buried in a sentence a section beforehand. I know his book is highly regarded but it came across to me as not very explanitory.
Sounds like there might be a market for ‘Symbolic logic for dummies’ 
Which reminds me of my usual story about those ‘for dummies’ books. Their big selling point is supposed to be that they tell you all those really little things that make you feel foolish because they’ve left it out and you don’t know how to find it out, right??
“Java programming for dummies.”
Comes with a copy of the sun java development kit.
Chapter 1: "Type this code into your computer - it’ll be your first java program.
Now compile it!"
The only thing was… neither the book nor the JDK documentation gave a simple explanation of HOW, exactly, you compiled a java program. It would be easy enough to describe that the javac program is the compiler and you need to run it from a command prompt or batch file with the source code filename as a parameter… but they said nothing of the sort. (groan.)
They seem like good books though, except for the occasional bonehead omission like that.