Today’s Topics

• 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":

• (1) Mr. Driver is not the guard.• (2) Mr. Fireman is not the driver.• (3) Mr. Driver is the driver.• (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

• Prove that ~(P • ~P) (the law of non-contradiction). There are no premises.

1. | ~~(P • ~P) AP

2. | P • ~P 1 DN

3. ~(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.