You will be expected to know all the rules of SD, how to
write down properly the justification for any of the steps allowed in
an SD-derivation, etc. Whiteley and Ganong both feel that you
should have done enough derivations in SD by now (three dozen? five
dozen?) so that you know the rules.
You will, however, be given a
half a printed sheet or so, containing all the rules of SD+ that are
not rules of SD , so you need not memorize those for use on the
So what should you know for the test? Here is an indication (watch
also on this web page for samples of class tests given in the past in
- You should have a thorough mastery of truth tables. Know the
definitions of the connectives "in your sleep". (And of course use the
Symlog symbols for them, not the symbols used in some other course.)
Know what the "standard
ordering" for tva's is (Ganong also called it the lexicographic order),
and use ONLY that ordering if you are asked to produce any truth
tables, or if a truth table is necessary to justify something you
claim in one of your test answers.
- Be able to USE the definitions of any of the terms we have defined
in the course, and also able to WRITE DOWN clear and complete
definitions of any of those terms. You should be able to define,
and to use the definitions of, phrases such as:
"\alpha entails \beta", "SL set",
"\Gamma entails \beta", "\tau satisfies
\beta", "\alpha is satisfiable",
"\Gamma is satisfiable/consistent", "contradiction", "contingent wff",
"tautology", "\alpha is logically equivalent to \delta", "inconsistent
set", "premise/conclusion", "antecedent/consequent",
"valid/invalid argument", "SD (or SD+) -valid argument",
"SD-inconsistent set", "... is SD (or SD+) -derivable from ...",
"... is an SD (SD+) -theorem of ...", "... is SD (or SD+) -equivalent
- Be able to state and able to use both the "test for entailment"
(p. 111) and the "test for SD (or SD+) -derivability", each of which
involves checking whether a certain set is inconsistent in a certain
sense, and each of which is sometimes the easiest way to approach a
But do not go overboard and try to settle any question about
entailment or derivability by running to these "tests" -- after all,
before we had the tests, we had definitions of entailment etc., which
are often just as easy to apply as the tests, and sometimes easier.
(Of course, also understand why these tests "make sense" -- we proved
in class that the tests do in fact achieve what they claim to achieve.)
- Know all the synonyms we have encountered (e.g. "contingent" =
"truth-functionally indeterminate", "entails" = "logically implies",
etc.). You may be given test questions involving one of the synonyms,
and if you ask Ganong during the test whether a certain phrase means the
same thing as another phrase, his answer will probably be, "You are
supposed to know the answer to that." And he will not be happy that
you asked. Know that the same symbol may mean different things in
different contexts (e.g., \tau |= \alpha, \Gamma |= \alpha), and of
course know what all the symbols mean --
|---- , for example (with an "SD" or "SD+" written under it, hard for
me to typeset in html).
- Be comfortable with proof by cases and proof by contradiction
(this is what Ganong called these neato cool types of proof in SD,
SD+), and understand in your soul why they work, why it is
altogether fitting and proper that they are legitimate methods of
proof. Be comfortable with the notion that inconsistent sets entail
all wff's, and with the whole notion of the empty set, and able to
check with care, but with ease, statements made about the empty set.
(E.g., "Every element of the empty set is purple and has seven toes.")
Believe Ganong when he says that every time he has played
Bobby Fischer in a tournament chess game, he has checkmated BF in
under twenty moves; be skeptical when he claims to have beaten BF even
once in a tournament game.
- Know exactly what the Soundness and Completeness theorems say, and
be able to write them down if asked, and able to use them to solve
problems and to justify answers you give.
- Of course, the more precisely you can write test answers, the
more specific you can be with counterexamples, etc., the better will
be your test result. Do not give the markers any chance to
misunderstand you (if you do , they
will ). By the way, the people who
have been giving the tutorials and marking the assignments so far,
M. Rabus and J. Zhou, are both former instructors of Math
2090, and probably understand logic much better than Ganong does.
(Rabus even taught 2090 before Ganong did.) Your tests will be marked
in part by your instructor, and partly by Rabus and/or Zhou.
Ask again in public, DURING class, NOT in private ,
on Friday, 14 February, for updates to this test
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