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2 Be Logical

INTERMEDIATE Logic Lessons 22-24 plus some review

QuestionAnswer
consistent when two or more propositions can be true at the same time
inconsistent when two or more propositions cannot be true at the same time
dilemma a valid argument which presents a choice between two conditionals
formal proofs of validity step-by-step deduction of a conclusion from a set of premises, each step being justified by an appropriate basic rule
rules of inference a valid argument form which can be used to justify steps in a proof
QED Quod erat demonstrandum (what was to be demonstrated)
Start a proof by . . . . comparing the conclusion with the premises.
Try saying the premises out loud or in your head to . . . . help you recognize which rule of inference to use.
conditional proof a special rule in a formal proof which allows us to assume the antecedent of a conditional and, once we deduce the consequent, to conclude the entire conditional
CPA conditional proof assumption
Reductio ad Absurdum a special rule which allows us to assume the negation of a proposition, deduce a self-contradiction, then conclude the proposition
truth trees a diagram that shows a set of propositions being decomposed into their literals
decomposed when a compound proposition is broken down into simple propositions (or the negation of simple propositions) which are called the literals
literals simple propositions or negated simple propositions which together compose a compound proposition
recover the truth values to determine the truth values of the simple propositions for which the propositions in the set would all be true
open branch a path on a truth tree which includes no contradictions
closed branch a path on a truth tree for which a contradiction has been found
A conjunction decomposes by separating into its conjuncts.
SM set member
3x5 mean line 3 contradicts line 5
No truth values can be recovered for inconsistent sets.
A disjunctions decomposes by branching into its disjuncts.
Truth trees may need to "branch" so that you can consider several possible truth values for the literals.
When using the truth tree method for consistency, you know that the set is consistent IF at LEAST one branch is an OPEN branch.
Conjunctions do NOT branch, but negated conjunctions DO branch.
Disjunctions BRANCH, but negated disjunctions do NOT branch.
Do NOT continue to decompose a branch on a proposition that has already ended in a contradiction or a CLOSED branch.
Do NOT skip the explicit decomposition of DOUBLE negations.
Conditionals BRANCH, but negated conditional statements do NOT branch.
Bi-conditionals AND negated bi-conditionals BOTH branch.
Whenever possible, the propositions which do NOT branch should be decomposed before the propositions that DO branch.
If you can tell which branches will end in a contradiction, decompose those branches FIRST. This will simplify the truth tree.
Stop creating the truth tree when you have answered the question which has been asked. For example, if you are ONLY asked if a set is consistent, you MAY stop once you have ONE open branch.
If none of the other decomposition suggestions for simplification apply, then try decomposing the most complex propositions first. This will save you from rewriting them under multiple branches later.
You do NOT need to decompose set member in the ____________ which they are given. Use the order which will generate the simplest truth tree. ORDER
Created by: MrsHough
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