An explanation of the difference between Hurley’s use of PL Rules and Mine

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Information on the format and content of the final exam (test 3)

 

Use of PL Rules:

 

When you look on page 420 of the text where Hurley explains the use of the rules UI, UG, EI, and UG, you see that he says that (Ex)Fx therefore Fy is an illicit move.  It is, but with a qualification.  Note that in the paragraph immediately above the summary of the rules, he says that Fy is reserved to represent any statement function – that is, an arrangement containing individual variables.  To put this in ordinary language, think of it like this:

 

If you know that “Something is fair” (Ex)Fx, then you know that there is some individual thing that has the property F.  What Hurley tells you is that you can’t instantiate existentially to “x” or “y” (that is, you can’t instantiate to an individual variable).  THAT IS RIGHT.  But here’s where the problem appears to arise.  He says that you can’t use “y” or “x,” and I say you can.  But when I say you can, I say you can because when you use existential instantiation, you can’t instantiate to a known individual, or to an arbitrarily selected individual, but you CAN instantiate to an UNKNOWN individual (which Hurley calls a “hypothetical individual”).  The terms “unknown” and “hypothetical individual” are the same thing, it is just that he says you can use a “d” or a “b” or whatever, but not a variable, where a variable is “x,” “y,” or “z.”  The difference is purely a practical one.  If you are using my way of understanding the rules, when you use “x” or “y” from an existentially quantified statement, you are essentially saying that the letter you use is NOT A VARIABLE ANYMORE, BUT THAT IT IS AN UNKNOWN INDIVIDUAL.

 

His reason for saying that you would choose a hypothetical individual is so that you never make the mistake of universally generalizing from a statement that has been existentially instantiated.  But even if you use “x” or “y” and YOU REMEMBER THAT YOU ARE USING ONE OF THEM TO INDICATE AN UNKNOWN (HYPOTHETICAL INDIVIDUAL), then you wouldn’t make that mistake, anyway.  But it does make sense to use an individual constant LETTER when you are thinking of it as an unknown (hypothetical individual) just so long as you don’t use that same constant again in another existential instantiation.

 

Another slight distinction is this.  When he explains UG, he says that you can go from Fy to (x)Fx but that you can’t go from Fa to (x)Fx.  THIS IS ALSO RIGHT.  Since “a” is either a constant (a known individual) or an unknown individual, it is still a very particular individual, and just because something applies to some individual does not imply that the characteristic or property that belongs to a particular individual belongs to every individual.

 

In essence, then, and to get right to the point,  Hurley’s rules aren’t wrong, and neither are mine.  They represent different ways of doing the same thing.  So to make things simple, remember the rules like this:

 

  1. When you use UNIVERSAL INSTANTIATION, you can instantiate to anything you want or need.
  2. When you use EXISTENTIAL INSTANTIATION, you can instantiate only to an unknown (or hypothetical) individual.  Hurley says you should use “a” or “b” or “c” or “d” or whatever, JUST SO LONG AS THOSE LETTERS NEVER APPEAR AT ANY OTHER TIME PRIOR TO YOUR USE OF IT IN THE ARGUMENT.  My point is that you can use ANY LETTER YOU WANT (an x, y, z, a, b or c), JUST SO LONG AS THOSE LETTERS NEVER APPEAR AT ANY OTHER TIME PRIOR TO YOUR USE OF IT IN THE ARGUMENT.  In short, there is no difference between the way Hurley explains the rules and uses them, and the way I do.  It is just a matter of which letters to use.  And just so long as you care consistent about it, I don’t care which letters you use in existential instantiation.  Just make sure that you always existentially instantiate to something “new” – i.e., to something previously unused in the proof, and that you never universally generalize from a known individual.
  3. When you use EXISTENTIAL GENERALIZATION, you can existentially generalize from anything, whether it is unknown (a hypothetical individual), an arbitrarily selected individual, or from a known individual.  If  anything has a particular property, then clearly, SOMETHING has that property.
  4. When you use UNIVERSAL GENERALIZATION, you have to be very careful.  You can only universally generalize from a statement containing an arbitrarily selected individual, AND ARBITRARILY SELECTED INDIVIDUALS IN A PROOF EXIST ONLY WHEN YOU CREATE THEM BY UNIVERSAL INSTANTIATION.  So when you universally generalize, it must ALWAYS be from a variable, and that variable must never be the result of an existential instantiation.

 

OK, so what does all this mean?  Hurley is right, and so am I.  It’s a matter of which letters you use.  And you can use the system Hurley has in place.  What this means is that you can use individual constants in an existential instantiation JUST SO LONG AS YOU NEVER UNIVERSALLY GENERALIZE FROM THEM, AND JUST SO LONG AS YOU NEVER INSTANTIATE TO THAT SAME LETTER AGAIN, OR USE THAT LETTER WHEN IT ALREADY APPEARS IN THE ARGUMENT.

 

Think of it like this.

 

Ann is a logician.

Bob is not a mathematician

All mathematicians are brilliant.

Some logicians are not men.

 

If these were premises in an argument, and you wanted to existentially instantiate statement 4, you COULD NOT USE a or b SINCE THEY ARE BOTH RESERVED FOR KNOWN INDIVIDUALS – THAT IS, THEY STAND FOR ANN AND BOB, and a and b as individual constants (known individuals) already appear in the argument.  But you could use “c” if you wanted to, because it never comes up in the argument again.  Or, if you want to instantiate line 4 to an “x” or “y,” you would simply have to remember that when you did so, the letter you use stands for an UNKNOWN and not to universally generalize from it.

 

IT DOESN’T MATTER TO ME WHICH OF THESE YOU CHOOSE TO DO (THAT IS, WHICH APPROACH YOU TAKE – HURLEY’S OR MINE ). WHAT MATTERS IS THAT YOU UNDERSTAND WHAT YOU’RE DOING.

 

There are some typos in the answers in the back.  Based on my system, for example, the answer on p. 630 to number 13 looks wrong, but it is not.  You CAN use “m” as an unknown (hypothetical individual) if you want to.  But line 8 DOES have a typographical error in it.  Line 8 should be 7, Simp., not 7, Com.  The difference between Hurley’s view and mine in this case is simply the use of “m” as a hypothetical (unknown) individual, and my claim that you ought to use a variable (x or y) to indicate such an individual.  But it ultimately doesn’t make a difference (especially since this is the book you have for the course).  In all, however, I think Hurley’s use of “m” or “d” or other letters set aside as individual constants is MORE confusing than my way of doing it.  And in case you’re looking for the original of my way of doing it, it is not actually my way at all.  It is the way it is done by Irving Cop in his Introduction to Logic and in Howard Kahane’s Logic and Philosophy, among other logic texts.  It is essentially a pedagogical difference, not a difference in the way logic works.

 

Format of the Final Exam (Test 3):

 

Much of your final exam will be in multiple choice and true/false format.  There will be some translation problems, a couple of short proofs, and questions about the use and meaning of the rules QN, UI, EI, UG, and EG.  Here are some examples of the types of questions.

 

  1. If you see this statement ---  (x)(Fx à Gx), why can’t you instantiate to Fa à Gb?  In a multiple choice format, you would have options, and one of them might be the right answer, which is:  UI is an inference rule, and inference rules apply only to entire lines of a proof.  So you could instantiate to Fa à Ga, but not to multiple variables or constants when the properties and the individual variables are all bound by the same quantifier.

 

  1. If you see this statement --- (x)Fx à (Ex)Bx, why can’t you instantiate to this:  Fx à (Ex)Bx.  The reason is that, again, the instantiation rules are inference rules, and they apply only to entire lines.  In fact, you couldn’t even do this:  Fx à Bx because what has the property F might not be the same thing that has property b.

 

  1. If you know that “Something is not plastic,” you also know that ~(x)Px.  This statement is true, and it is true because the statement (Ex)~Px (Something is not plastic) means the same thing as “it is not the case that everything is plastic” ~(x)Px  by QN.

 

I don’t intend to torture you on the final exam.  The idea is to learn something, not to see whether you can jam everything into your head and do it mechanically.  Instead, the idea is to understand how PL works, and to be able to do some simple to medium difficulty proofs and to understand the implications of the use of the rules.

 

Here’s the general plan for the final.

 

Translation problems as multiple choice or true/false, or both.

Use of the rules questions in multiple choice or true/false format, or both.

Proofs (as justifications – i.e., the kind where the proof is worked out, and you either provide the justification showing which rule was used, or where you are given the rule and you write in the line that belongs in the empty space)

Proofs (some done as direct and some as indirect)

 

So concentrate on understanding what is going on with PL.  Don’t give yourself any more of a headache than this type of logic can give you on its own!