Logic and Proofs

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Created by
Raphael Elorah

Cards (27)

  • Proposition

    Sentence that is either True or False, but not both
  • Propositions

    • Washington D.C. is the capital of the USA
    • Quezon City is the capital of the Philippines
    • 1 + 1 = 3
    • 5 + 5 = 10
  • Not a proposition
  • Compound propositions

    Formed from existing propositions using logical operators
  • Logical operators

    • ~ for "not/negation"
    • ^ for "and"
    • V for "or"
    • for "implication/if-then"
    • for "if and only if"
  • Compound propositions

    • If it is cloudy, then its going to rain.
    • If it is going to rain, I should take my umbrella.
    • Therefore, if it is cloudy, I should take my umbrella.
  • Translating compound propositions

    1. p q
    2. q r
    3. ∴ p r
  • Compound propositions

    • If x is 3, then x squared is 9
    • The x cubed is 27, it is necessary that x is 3
  • Truth tables

    Displays the relationships between the truth values of propositions
  • Negation (¬)

    • The negation of a proposition A (written as ¬A) is false when A is true and is true when A is false
  • Conjunction (AND, ∧)

    • The AND operation of two propositions A and B (written as A∧B) is true if both the propositional variable A and B is true
  • Disjunction (OR, ∨)

    • The OR operation of two propositions A and B (written as A∨B) is true if at least any of the propositional variable A or B is true
  • Implication (if-then, →)

    • An implication A→B is the proposition "if A, then B". It is false if A is true and B is false. The rest cases are true
  • Material Equivalence (If and only if, ⇔)

    • Also known as biconditional, A⇔B is bi-conditional logical connective which is true when p and q are same, i.e. both are false or both are true
  • Compound propositions

    • (p v q) ^ ~(p q)
    (p ^ ~q) [(p v ~ q) (~p ^ q)]
  • If all premises are true but the conclusion is false, the argument is invalid
  • Arguments and argument forms
    • If it rains, then the ground is wet.
    It rains,
    Therefore, the ground is wet.
  • Tautologies

    A formula which is always true for every value of its propositional variables
  • Tautologies

    • Prove [(A→B)∧A]→B is a tautology
  • Contradiction

    A formula which is always false for every value of its propositional variables
  • Contradiction

    • Prove (A∨B)∧[(¬A)∧(¬B)] is a contradiction
  • Contingency

    A formula which has both some true and some false values for every value of its propositional variables
  • Contingency

    • Prove (A∨B)∧(¬A) a contingency
  • Logical Equivalence

    Denoted by ⇔, if they have the same truth values
  • Logical Equivalence rules

    • Double Negation
    2. Commutation
    3. De Morgan's theorem
    4. Association
    5. Distribution
    6. Transposition
    7. Identity laws
    8. Tautology
    9. Material Implication
    10. Material Equivalence
    11. Exportation
  • Rules of inference

    • Modus Ponens
    2. Modus tolens
    3. Hypothetical syllogism
    4. Disjunctive syllogism
    5. Constructive Dilemma
    6. Destructive Dilemma
    7. Simplification
    8. Conjunction
    9. Addition
  • Rules of inference

    • 1.(r ⇔ ~s) ⊃ (t ⊃ u)
    r ⇔ ~s M.P.
    t ⊃ u
    2.(a ⊃ ~b) . [c ⊃ (d.e)]
    ~~b v ~(d.e) D.D
    ~a v ~c
    3.~[g ⊃ (h v x )] . ~[(j . k) ⊃ l]
    ∴~[g ⊃ (h v x )] Simplification