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Propositional Logic & Truth Tables

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Logical connectives, truth tables and rules of inference.

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Propositional logic

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  1. 1Propositional logic
    A branch of classical logic that deals with propositions (which can be true or false) and relations between propositions
  2. 2Another name for propositional logic
    Sentential logic, propositional calculus, statement logic, or zeroth-order logic
  3. 3Proposition
    A declarative sentence that has a truth value (can be either true or false)
  4. 4Principle of bivalence
    The principle that every proposition has exactly one of two possible truth values: true or false
  5. 5Five main logical connectives
    Conjunction (AND), disjunction (OR), negation (NOT), implication (IF-THEN), and biconditional (IF AND ONLY IF)
  6. 6Conjunction
    A logical connective (∧ or &) that is true only when both operands are true (p AND q)
  7. 7Disjunction
    A logical connective (∨) that is true when at least one operand is true (p OR q)
  8. 8Negation
    A logical connective (¬ or ~) that reverses the truth value (NOT p)
  9. 9Material implication
    A logical connective (→ or ⊃) that is false only when the antecedent is true and consequent is false (IF p THEN q)
  10. 10Biconditional
    A logical connective (↔ or ⇔) that is true when both operands have the same truth value (p IF AND ONLY IF q)
  11. 11Truth table
    A table showing the truth value of a logical expression for every possible combination of truth values of its variables
  12. 12How many rows does a truth table have for n variables?
    2^n rows (for example, 2 variables = 4 rows, 3 variables = 8 rows)
  13. 13Tautology
    A formula that is true under every possible interpretation (true in all rows of its truth table)
  14. 14Contradiction
    A formula that is false under every possible interpretation (false in all rows of its truth table)
  15. 15Contingency
    A formula that is neither a tautology nor a contradiction (true in some cases, false in others)
  16. 16Valid argument
    An argument where it is impossible for all premises to be true while the conclusion is false
  17. 17Sound argument
    An argument that is both valid and has all true premises
  18. 18Modus ponens
    If p then q; p; therefore q. A fundamental valid argument form
  19. 19Modus tollens
    If p then q; not q; therefore not p. Another fundamental valid argument form
  20. 20Hypothetical syllogism
    If p then q; if q then r; therefore if p then r
  21. 21Disjunctive syllogism
    Either p or q; not p; therefore q
  22. 22Atomic formula
    A propositional variable (letter) that cannot be broken down further by logical connectives
  23. 23Molecular formula
    A compound formula built from atomic formulas using logical connectives
  24. 24Law of excluded middle
    Every proposition is either true or false; there is no third option
  25. 25Semantic consequence
    φ ⊨ ψ means there is no interpretation where φ is true and ψ is not true
  26. 26Syntactic consequence
    φ ⊢ ψ means ψ can be derived from φ using the rules of the formal system
  27. 27Sheffer stroke (NAND)
    A logical connective (| or ↑) that is false only when both operands are true (NOT (p AND q))
  28. 28Logical NOR
    A logical connective (↓) that is true only when both operands are false (NOT (p OR q))
  29. 29XOR (exclusive or)
    A logical connective (⊕) that is true when exactly one operand is true (p XOR q)
  30. 30Can all logical connectives be defined using only NAND?
    Yes, NAND is functionally complete (all other connectives can be derived from it)

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