Today’s Topics
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Using CP and RAA
Things to watch for, things to avoid
Strategic hints for using CP and RAA
A Little Metalogic and some History of
Logic and Mathematics
Using CP and IP
• With the exception of A  B  A  (A 
B), CP and RAA are never required. You
can complete any other proof without these
methods.
• HOWEVER, CP and RAA are both
historically and culturally important and
they provide strategies for completing
proofs.
Using CP and IP
• CP and RAA give you both overall
strategies (e.g. if you need a conditional,
think CP) and sub-strategies (e.g., if you
have a conditional as goal, use CP to get it).
• CP and RAA are GREAT puzzle solving
strategies.
Things to Watch For:
• Carefully identify the dominant operator in a line
– Group rules by operators
• Be sure that the line on which you wish t use a
rule is a substitution instance of the statement
form in the argument form for the rule.
• Watch for the scope of a negation
• Think of the structure of the patterns on inference
and equivalence rules
• Make SURE the argument for which you are
attempting a proof is valid
Symbolization Hints
• Be careful when symbolizing conditionals
– “Only if” introduces the consequent of a
conditional
– “If and only if” introduces a biconditional
– Sufficient conditions are antecedents
– Necessary conditions are consequents
• Remember there are 2 ways to deal with
‘unless’
Still more strategy hints
• Think in terms of equivalences. Ask “To what is
the conclusion of this argument equivalent?”
• Break the problem down into smaller problems
and attack them independently. Think of
subroutines in a proof.
• Remember your strategy and don’t get lost. Make
notes to yourself.
• Be flexible—experiment!
Common Errors to AVOID:
• Trying to use an inference rule on a part of a
line
• Errors concerning the scope of a negation
• Confusing the role of tildes in WFF’s with
their role in argument forms
• Reluctance to use addition and distribution
• Reluctance to use CP and RAA
• Attempting the impossible
Hints for using CP:
• Remember, the line derived MUST be a
conditional whose antecedent was your
assumption.
• Use 2 CP subproofs followed by CONJ and
EQUI to derive a biconditional.
• You can do multiple steps of CP, including
nesting assumptions
Hints for using RAA:
• Scan the premises to identify a likely
contradiction.
• Remember that you are looking for a
contradiction—RAA provides you with an overall
strategy.
• Remember to discharge your assumption!
Sometimes you may derive what you are looking
for (your overall goal) within the scope of the
assumption, but you cannot use it.
RAA and Problem Solving
• Many standardized intelligence or aptitude
tests (e.g. the LSAT, the GRE, the MCAT)
include problems which can be solved
easily using indirect proof as a strategy.
• Use the strategy to discover when certain
claims can’t be right (namely, when they
lead directly to contradictions), and then use
that information to determine which claims
are correct.
Solving Puzzles Using IP
• Messrs. Fireman, Guard, and Driver are the
fireman, guard, and driver on a train. Each man
has only one job. When I tried to find out who
was what, I was given these four "facts":
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(1) Mr. Driver is not the guard.
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(2) Mr. Fireman is not the driver.
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(3) Mr. Driver is the driver.
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(4) Mr. Fireman is not the guard.
• It then transpired that, of the above four
statements, only one is true. Who is what?
• Solve this puzzle by applying IP. In order
to determine which of the 4 statements is
true, begin by assuming one to be true and
then look for a contradiction. Finding it lets
you know that statement is false. If you
assume (1) to be true, it leads to the
contradiction that Mr. Fireman is both the
driver and the guard, which is impossible.
Do you have it yet?
• Mr. Driver is the guard
• Mr. Fireman is the driver
• Mr. Guard is the fireman
Metalogic
• Logic is the study of what makes arguments
either good or bad.
• Metalogic is the critical examination of
systems of logic.
• Logic is the study and science of arguments.
Metalogic is the study and science of logic.
Tautologies and Theorems
• A truth-functional statement is a tautology
if and only if it is true on every line of its
truth table.
• A truth-functional statement is a theorem if
and only if it can be derived without using
any premises.
• The proof for a theorem must be either a CP
or an RAA.
Proving a Theorem
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Prove that ~(P • ~P) (the law of noncontradiction). There are no premises.
1.
2.
3.
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| ~~(P • ~P)
AP
|
P • ~P
1 DN
~(P • ~P)
1-2 IP
That’s all there is to it!
Semantics and Syntax (again)
• “Tautology” is a semantic concept, it turns
on the meaning of our logical operators.
• “Theorem” is a syntactic concept, it turns
on the logical structure of an argument.
• Core ideas in logic can be interpreted either
syntactically or semantically.
Validity
• Syntactic definition- An argument is valid
if, but only if, it is possible to construct a
proof of the conclusion from the premises.
• Semantic definition- An argument is valid
if, but only if, there is no row in the truth
table for the argument in which all the
premises are true and the conclusion false.
Consistency
• Syntactic definition- A set of statements is
consistent if, but only if, it is impossible to
derive a contradiction from them.
• Semantic definition- A set of statements is
consistent if, but only if, there is at least one
row in the truth table for the set in which
each member of the set is true.
A system of logic is COMPLETE
if every tautology in the system is
a theorem. A system of logic is
SOUND if every theorem is a
tautology. Our 18 rules plus CP
are both complete and sound.
A Little History of Logic and
Mathematics
• For a LONG time mathematicians and logicians
held a view called constructivism. According to
constructivism, we ought to be able to construct a
proof for any true claim in math or logic. Every
true claim in math or logic is a tautology (an
axiom)—it is necessarily true—so this is a
reasonable goal.
• Mathematics, constructivists believe, should be
complete and consistent, just like logic.
Constructivism & Growth of Logic
• Many of the most important advances in logic
in the late 19th century were the direct result of
the constructivist project.
• For examole, DeMorgan knew that the sentence
“If all horses are animals, then all heads of
horses are heads of animals” is a tautology, but
he couldn’t prove it.
• So he developed the DeMorgan rules as tools to
enable him to construct the proof.
Principia Mathematica
• B. Russell and A.N. Whitehead’s major book,
Principia Mathematica, was a constructivist
project. It sought to show that all of
arithmetic can be reduced to logic.
• Other major historical figures in the
constructivist camp include: Gotlob Frege,
Georg Cantor, David Hilbert, Paul Bernays
and Guiseppe Peano
Limits of completeness and
soundness
• In 1931 Kurt Gödel proved that it is
impossible to have a formal system that is
both complete and sound! This discovery
changed the nature of mathematics forever.
• Gödel’s result ended the constructivist
project and ended the quest for certainty in
mathematics.
• Gödel’s result was one of the major
conceptual revolutions of the 20th century.
A difficult choice
• Gödel showed that a formal system must be either
incomplete—there are true claims for which you
can’t construct a proof—or inconsistent—you can
prove both a claim and its negation!
• Since we know that anything follows from a
contradiction (if you don’t believe it, test the claim
with a truth table test), formal systems strive for
consistency at the expense of completeness.
• This means that there are true claims in formal
systems that we can’t prove to be true.