Logic Test 2
Constructive Dilemma
(p > q) • (r > s) p v r / q v s
What steps are you supposed to follow in order to construct a truth table for an argument?
1. Symbolize the arguments using letters to represent the simple propositions. 2. Write out the symbolized argument, placing a coma between the premises and slash between the last premise and the conclusion. 3. Draw a truth table for the symbolized argument as if it were a proposition broken into parts, outlining the columns representing the premises and conclusion. 4. Look for a line in which all of the premises are true and the conclusion is false. If such a line exists, the argument is..invalid; if not, it is valid.
Natural Deduction
A proof procedure by which the conclusion of an argument is validly derived from the premises through the use of rules of inference.
Noncontingent Statement
A statement such that the truth values in the main operator column do not depend on the truth values of the component parts. -Tautology -Self-contradiction
Self-contradiction
A statement that is necessarily false.
Tautology
A statement that is necessarily true
Statement Variable
A statement variable p, q, r, s .... can stand for any statement, simple or complex
Truth Value
A statement's truth value is either true or false
Definition of Ponens
Affirming
Argument Form
An arrangement of logical operators and statement variables such that a substitution instance of statement variables results in an argument
Implication rules 2
Constructive Dilemma Simplification Conjunction Addition
Definition of Tollens
Denying
What is L=2n used for?
Determining the number of lines in a truth table
Fallacy of Denying the Antecedent
If P then Q Not P Therefore, not Q
Fallacy of Affirming the Consequent
If P then Q Q Therefore P
Hypothetical Syllogism
If P, then Q If Q, then R If P then R
Modus Tollens
If P, then Q Not Q Therefore, not P
Modus Ponens
If P, then Q P Therefore, Q
Propositional logic
Its basic elements are statements rather than classes. Provides precise definitions for sentences containing and, or, if, and only if. Captures much more of ordinary language than is possible with categorical logic.-Its basic elements are statements rather than classes. Unlike A, E, I, O, contains an unlimited number of complex statements.
Rules of inference
Justify the steps of the proof
Strategy 4
Look at the conclusion (Try working backward from the conclusion by imagining what the next to last line of the proof might be. Use this to help determine a short-term strategy to derive that line.)
Strategy 3
Look for conditionals (MP, MT, HS)
Strategy 2
Look for negation (MT, DS)
Definition of Modus
Method
Implication rules 1
Modus Ponens Modus Tollens Hypothetically Syllogism Disjunctive Syllogism
What does L stand for?
Number of lines?
When will a conditional be false?
Only when the antecedent is true and the consequent is false
Disjunction indicators
Or Unless Otherwise Either... or
Addition
P / P v Q
Conjunction
P Q / P• Q
Disjunctive Syllogism
P or Q Not P Therefore, Q
Rule 4 WFF
Parentheses, brackets, and braces are required in order to eliminate ambiguity in a complex statement.
What are the two types of rules of inference
Rules of implication and rules of replacements
Proof
Sequence of steps in which each step is either a premise or follows from earlier steps in the sequence according to the rules of inference.
Truth Tables
Show every possible truth value for compound propositions. Provide definitions of logical operators. Truth tables can determine the validity of arguments.
Strategy 1
Simplify and isolate (MP, MT, DS)
Contingent Statement
Statements that are neither necessarily true nor necessarily false (they are sometimes true, sometimes false)
Inconsistent Statements
Statements that do not have even one line on their respective truth tables where the main operators are both true.
Consistent Statements
Statements that have at least one line on their respective truth tables where the main operators are both true.
Logical operators
Symbols used in translations of ordinary language statements
Rule 1 of WFF
The dot, wedge, horseshoe, and triple bar symbols must go between two statements (either simple or compound)
What does n stand for?
The number of different simple propositions in the statement
What does the 2 represent in L=2n?
The number of truth values (true and false)
Rule 3 WFF
The tilde (~) cannot, by itself, go between two statements
Rule 2 WFF
The tilde (~)....goes in front of the statement it is meant to negate.
Truth Function
The truth value of a compound proposition is a function of the truth values of its component statements and the logical operators
Tatics
The use of small-scale maneuvers or devices
Contradictory Statements
Two statements that have opposite truth values on every line of their respective truth tables.
Strategy
Typically understood as referring to a greater overall goal
WFF
Well formed formula. Compound statement forms that are grammatically correct.
When will a conjunction be true?
When both conjucts are true
When will a disjunction be false?
When both disjuncts are false
When is a biconditional true?
When p and q have the same truth value
Logically Equivalent
When two truth-functional statements appear different but have identical truth tables,
Statement Form
a pattern of statement variables and logical operators~ ( p v q )
Conjunction indicators
and but also Moreover Still While However Although Yet Nevertheless Whereas
When is a proof valid?
each step is either a premise OR is validly derived using the rules of inference
Negation indicators
not; it is not the case that It is false that It is not true that
Simplification
p • q / p
Justification
refers to the rule of inference that is applied to every validly derived step in a proof.
When does a proof end?
when the conclusion of the argument has been derived