
Preparation file for Test 2:
Test 2, 77:55 p.m., Tuesday, 26 March,
IN TWO DIFFERENT ROOMS.
(Possibly, anyway  I have reserved Curtis J as the extra
room  we may need it or may not  check back here
later. People whose last names start with a letter toward
the end of the alphabet will go to Curtis J if we use it.)
What will appear in this file will surely not be a
complete list of what you should know for Tuesday's test.
Test preparation files never mention EVERYTHING.
You may NOT assume that if I don't mention something here
then it will not appear on our test.

READ instructions carefully and follow them, on the test.
At the test, you will have our two pages of lists of axiom schemes
and theorem schemes, etc., of Boolean and Predicate Logic. (I might
print only the things from predicate logic which we have talked
about already in class, from the predicate logic page.) Again: You
will have all the axioms from pp. 4243, numbered in the usual way, and
many theorems we will have discussed in class, with their usual numbers,
printed on a handout I will give out at the test (but don't bring
any such list yourself; I will have copies printed out for everyone
that day).
You will NOT have the basic inference rules printed out  you are
supposed to have used those, and other derived ones (such as "Trans"
and "other Equanimity" and M.P. and maybe others), and certain
metatheorems (like D.T., which you should be able to state fully
and precisely if asked to), so many times that you know them
in your sleep. You should basically know metatheorems and
derived inference rules discussed in class on your own 
they will not be given on a handout at the test.
Also: You are NOT allowed to use Completeness to prove
that wffs are theorems on our test Tuesday. If I ask you
to do this, I definitely expect a syntactic proof that
the wff is a theorem.
Where to go on test day
I will post more a little later this week about our test, in
particular, WHAT ROOM you should go to for the test.
Everyone should come to Curtis K after the tests are collected by
the invigilators; we will have class as usual, for about 95 minutes.
Know what splitting and merging are and how to use them in, say, a
Hilbert proof. (These are basically 2.5.1, parts (1) and (4).)
Regard them as derived inference rules or as metatheorems, whichever.
Know what the derived inference rule Modus Ponens (M.P., 2.5.3) is and
how to use it.
Know precisely what The Deduction Theorem (2.6.1) says and how to
use it to prove conditionals. I might ask you to state it precisely.
It works in predicate logic as well as boolean (as does M.P.)  as
do all of the inference rules we had before predicate logic. Just
replace, in all of them, "wff of boolean logic" by "wff of pred.
logic".
If you write out what M.P. and D.T. each say, you will see that
M.P. is basically the converse of D.T. (Reverse the "if" and "then"
parts of the English sentence which is the statement of the D.T.
to get a new "if, then" English sentence; convince yourself that it
is true iff M.P. is true. :) (I won't ask about any of this; I
just thought I'd mention it here.)
Be an expert at proofs of conditionals via the D.T., like the ones
done in class and the ones on Assignment 4.
Know what a pingpong proof is, and how to write them out. The
advantage of such proofs is that they allow one to bring to bear
the power of M.P. and D.T. on the proving of biconditionals.
Know the alphabet of predicate logic (PL) (not absolutely perfectly,
maybe, but well enough to know the difference between the two kinds
of variable and the letters used for them, well enough that you
know a term from a wff (terms and wffs are totally different things
 look at the early part of the chapter beginning predicate logic
if you want examples of terms). I am too lazy to type the x's below
as bold x's. Pretend they are bold (i.e., metaobject variables).
Know that if A is a wff then
((\exists x) A) is just an abbreviation for
&neg; ((\forall x) (&neg; A))
Know that, in the above, the wff A is the "scope" of (\exists x) A.
Understand thoroughly what free and bound occurrences of an object
variable in a wff are. I might well give you one or more fairly
complicated wffs and ask you which occurrences of this or that
variable in this or that subformula of the given formulas are free
and which are bound, and, in the case of the bound ones, which
(\forall x) binds each of them.
Have something of an understanding of the semantics of PL. Know in
particular that an interpretation always requires the choice of a
nonempty set called the "domain" or "universe of discourse".
One of you asked me after class why it is called the universe of
discourse (UD). "Discourse" is "discussion", "conversation". The
term may come from philosophy. Philosophers use formal logic,
and their nonempty set was the set D, or domain, "about which
they were talking".
Understand the examples done in class involving 2place predicate
letters φ (for instance) and their interpretation as directed
graphs.
By the way, a ONEplace predicate letter essentially interprets as
a subset of the universe of discourse. For instance, if &phi& is
a 1place predicate letter, then it interprets as ... a predicate!
I.e., as a statement about elements of the UD, which takes truth
value "true" or "false" depending on which element is fed into
it. It interprets, that is, as a function from the UD to {t, f}.
Example: Let the UD be D = {1, 2, 4, 5, 7, 11, 14}, and consider
the function (the predicate) (call it Joe) whose value at an
element of this set is t iff the integer is even. Then one
could, for instance, interpret a 1place predicate letter φ
as that function, or, more briefly, as the subset of the UD
where it takes the value t, i.e., as the subset {2, 4, 14}.
(The predicate letter interprets here as the statement
"n is even", which is true iff ... n is even!)
Joe: D > {t, f}, and such a function is completely determined
by giving its "truth set", i.e., the set of elements of its
domain D, at which it takes the value t.
We gave one or two examples involving "(universally) valid formulas"
of predicate logic, also known as "absolute truths". Tourlakis
uses both terms. x = x is a universally valid formula. So are,
indeed, all the theorems of PL. The word "tautology" is no longer
used in PL. It is replaced by "valid formula".
We spent quite a bit of time talking about substitution of terms
for object variables in wffs. Understand all of that. Know what
capture is and that a substitution is defined iff capture would
not result, from its being carried out.
Know 4.1.33 and 4.1.34 (not by number, of course).
Understand the proof in class that  (t = t) for any term t.
It involved a partial generalization and Ax 2 and M.P.
This last part finally added Sunday afternoon:
I said I would finish this up Thur. or Fri. That did not happen.
But I am just going through my notes and recording stuff here,
and that was almost done last Wednesday.
We did a proof or two or three using the very important metatheorem
Gen, also known as 6.1.1. Understand those proofs and be ready to
execute others like them.
6.1.1, for instance, gave us 6.1.3 in the text. (We remarked that
this does NOT come from "partial generalization" and that it does not
come from Ax 4 and M.P. either, because a theorem A might very well
have a free occurrence of x in it.)
We proved metatheorem 6.1.5 (Spec) easily from Ax 2, and 6.1.6 (Spec
spec) is the special case where the term t is x.
Look up all of these proofs in our text, by the way, if you like.
We gave a pingpong proof of 6.1.7. Understand that proof and
be able to carry out similar proofs. Both the ping and the pong
involve Spec spec and Gen (and D.T.).
An easy proof of 6.1.8 involves pingpong, with pong looking
just like ping, and ping requiring two Spec specs and two Gens and
D.T.
OK, that's about it. I don't know if Test 2 will feature any bullet
proofs. I sort of think that we moved on from bullet proofs after
Test 1; I only put some on Assignment 4 to give more practice (maybe
for the exam?) and a short one for some easy grading and assignment
marks. Centre stage has been sort of taken over by Hilbert and D.T.
and M.P. at this point. But we don't "forget" anything.
OK, here is the last thing I will post on this course page before
Tuesday's test:
Where you should go for Tuesday's test:
People whose LAST names start with any of the letters
V, W, X, Y, Z must write the test in Curtis J.
Everyone else must write in our usual room, Curtis K.
And we will have class starting within ten minutes of the
end of the test, in the usual room.
I wish everyone what she/he would want to be wished, for test
night, and I hope to see you Tuesday night. Ciao.
