Logic & Reasoning Flashcards
Types of Argument in mathematical Reasoning
Deductive and Inductive Reasoning
Structure: moving from general premise to specific conclusion
Deductive reasoning
All dogs are animals, so my dog is an animal.
Deductive Reasoning
Structure: specific premise to general conclusion
Inductive reasoning
Strength: If the premise is true, conclusion is more likely to be true (but could be false)
Inductive Reasoning
May be used to visually evaluate the validity of a deductive argument
Euler Diagram
Applies when a conditional and its antecedent are given as premises, and the consequent is the conclusion.
Law of Detachment (Modus Ponens- mode that affirms)
The general form for Law of Detachment is ___
Premise: p implies q
Premise p
Conclusion: q
When a conditional and the negation of its consequent are given as premises, and the negation of its antecedent is the conclusion.
Law of Contraposition (Modus Tollens- mode that denies)
Arises when a conditional and its consequent are given as premises, and the antecedent is the conclusion.
Fallacy of the Converse
General form of Law of Contraposition
Premise: p implies q
Premise: ~q
Conclusion: ~p
General form of Fallacy of a Converse
Premise: p implies q
Premise: q
Conclusion: p
Occurs when a conditional and the negation of its antecedent are given as premises, and the negation of the consequent is the conclusion.
Fallacy of the Inverse
General from of Fallacy of the Inverse
Premise: p implies q
Premise: ~p
Conclusion: ~q
A complete declarative sentence P(x) involving variable x.
Propositional Function (Predicate)
In a Propositional Function or Predicate, the variable x is called
Argument
Another method of changing propositional function into proposition. may be universal or existential
Quantification
Universal quantification define__
“P(x) is true for all values of x in the domain of discourse”
What does the domain of Discourse specify?
specifies the possible value of X
” There exist an element in x in the domain of discourse such that P(x) is true”
Existential Quantification
What are the Methods of Proof?
Direct Proof, Indirect proof, Proof by contradiction, and Existence Proof.
A proof that p implies q is true by showing that q is true, if p is true
Direct proof
A proof that p implies q is true by showing that p must be false, if q is false.
Indirect proof
A proof that the proposition p is true based on the truth of ~p implies q, where q is a contradiction.
Proof by Contradiction
A proof of the proposition of the form ∃x: P(x)
Existence Proof
Classification of Existence Proof____
Constructive Existence Proof and Non-Constructive Existence Proof
Establishes the assertion by exhibiting a value c such that P(c) is true.
Constructive Existence Proof
The ¬ operator has higher precedence than
∧/ conjunction/ and
∧ has higher precedence than
∨/ disjunction/ or
∨ has higher precedence than
⇒/ implication/ conditional/ if and then
⇒ has higher precedence than
⇔/ biconditional/ if and only if
(p ∧ q), is true if both p and q are true, and is false if either p is false or q is false or both are false.
Conjunction
(p ∨ q) is true if either p is true or q is true, or both p and q are true, and is false only if both p and q are false.
Disjunction
(p ⇒ q) is false if p is true and q is false, and is true if either p is false or q is true (or both).
Implication or Conditional
(p ⇔ q) is regarded as true if p and q are either both true or both false, and is regarded as false if they have different truth-values.
Biconditional
called logical truths or truths of logic
Tautologies
statements that come out as false regardless of the truth-values of the simple statements
Contradiction
true for some truth-value assignments to its statement letters and false for others.
Contingent
neither contradictory nor tautological
Contingent
statements are said to be __________ if and only if all possible truth-value assignments to the statement letters making them up result in the same resulting truth-values for the whole statements.
logically equivalent
An argument is ____________ if and only if its conclusion is a logical consequence of its premises.
logically valid
