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Logical Implication (Conditional Statement)


Definition and Notation
Truth Table for Logical Implication
Properties of Implication




Definition and Notation

Logical implication is a fundamental concept in logic and mathematics. It describes a conditional relationship between two statements.

Definition:

Logical implication (π‘β†’π‘žπ‘β†’π‘ž) is a conditional statement meaning "if pp, then π‘žπ‘ž." It asserts that whenever 𝑝𝑝 (the antecedent or hypothesis) is true, π‘žπ‘ž (the consequent or conclusion) must also be true.
An implication expresses a dependency between two propositions, where the truth of the antecedent guarantees the truth of the the consequent.
However, if 𝑝𝑝 is false, the implication is still considered to be true regardless of π‘žπ‘ž. This is very important point that follows directly from the definition. We will see the meaning of it while dealing with implication truth table.

Notation:

  • β€’
    p→qp→q (standard notation)
  • β€’
    p⇒qp⇒q (sometimes used in formal logic)
  • β€’
    "If 𝑝𝑝, then π‘žπ‘ž" (verbal expression)
  • β€’
    "𝑝𝑝 implies π‘žπ‘ž" (another verbal expression)

Truth Table for Logical Implication

Logical implication (P→Q)(P → Q) is a fundamental operation in propositional logic, represented as "if P, then Q." This relationship is only false in one specific case: when the antecedent P is true but the consequent Q is false.
The truth table below shows all possible combinations of truth values for 2 variables:
PQP β†’ QExplanation
TTTWhen P is true and Q is true, the implication is true.
TFFWhen P is true but Q is false, the implication is false.
FTTWhen P is false and Q is true, the implication is true.
FFTWhen P is false and Q is false, the implication is true.
The counterintuitive aspect is that P β†’ Q is true whenever P is false, regardless of Q's value. This is called "vacuous truth". A helpful way to understand this is with a promise analogy: "If it rains, I'll bring an umbrella." If it doesn't rain, the promise remains unbroken whether I bring an umbrella or not.
Promise may be broken only if it has been given. In case it had been never given (last 2 rows, when PP is equal to false)- it can not be broken and the whole statement evaluates to true.
And to conclude this discussion, the rule of thumb with logical implication is that whenever the hypotesis is false-the overall expression evaluates to true.
See more truth tables for expressions involving implications here.
Or use this interactive truth table generator to evaluate your own expressions.

Properties of Implication

Logical implications have several important properties that define how it behaves in reasoning and formal logic. These properties help establish relationships between statements, simplify logical expressions, and form the basis for proofs. Understanding these propertiesβ€”such as transitivity, contraposition, and material implicationβ€”is essential for working with logical arguments and mathematical reasoning.
Here are some important properties of logical implication:
  • β€’
    Reflexivity:
    pβ†’ppβ†’p is always true for any proposition 𝑝𝑝.
    This follows from the truth table since whenever
    𝑝𝑝 and π‘žπ‘ž are the same-π‘β†’π‘žπ‘β†’π‘ž is always true.
  • β€’
    Transitivity:
    If π‘β†’π‘žπ‘β†’π‘ž and π‘žβ†’π‘Ÿπ‘žβ†’π‘Ÿ, then π‘β†’π‘Ÿπ‘β†’π‘Ÿ.
    Example:
    "If it rains, the ground gets wet." (π‘β†’π‘žπ‘β†’π‘ž)
    "If the ground gets wet, the grass grows." (π‘žβ†’π‘Ÿπ‘žβ†’π‘Ÿ)
    Conclusion: "If it rains, the grass grows." (π‘β†’π‘Ÿπ‘β†’π‘Ÿ)
  • β€’
    Contraposition:
    π‘β†’π‘žπ‘β†’π‘ž is logically equivalent to Β¬π‘žβ†’Β¬π‘Β¬π‘žβ†’Β¬π‘.
    This means: If "If it rains, then the ground is wet" is true, then "If the ground is not wet, then it did not rain" must also be true.
    This equivalence is useful in proof techniques, especially proof by contrapositive.
  • β€’
    Material Implication (Alternative Form):
    π‘β†’π‘žπ‘β†’π‘ž is equivalent to Β¬p∨qΒ¬p∨q.
    This means that "If 𝑝𝑝 then π‘žπ‘ž" can be rewritten as "Either 𝑝𝑝 is false or π‘žπ‘ž is true."
    Example:
    "If it's a dog, then it's an animal."
    This is logically the same as saying: "It's not a dog, or it's an animal."
    This equivalence is a key rule in propositional logic and is used in proofs and simplifications.
  • β€’
    Asymmetry:
    (π‘β†’π‘žπ‘β†’π‘ž) is equivalent to (Β¬π‘βˆ¨π‘žΒ¬π‘βˆ¨π‘ž), but not equivalent to (π‘žβ†’π‘π‘žβ†’π‘).
    This means that implication is not symmetric. Just because π‘β†’π‘žπ‘β†’π‘ž is true does not mean π‘žβ†’π‘π‘žβ†’π‘ is true.
    Example:
    "If you are a mother, then you are a woman" (π‘β†’π‘žπ‘β†’π‘ž) is true.
    But "If you are a woman, then you are a mother" (π‘žβ†’π‘π‘žβ†’π‘) is not necessarily true.