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    The formula (p → q) ∨ p is classically valid because if p... — Carmelics
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    Home/Philosophy of Language
    HistoryEditSee Inverse

    The formula (p → q) ∨ p is classically valid because if p is false, conditionals with false antecedents are vacuously true

    Modality & PossibilityPhilosophy of Language
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.If p fails, p is false
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    • 2.In classical logic, a conditional with a false antecedent is true
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    • 3.Therefore (p → q) holds whenever p is false, making (p → q) ∨ p a tautology
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Vacuous truth is a formal artifact of material implication, not a feature of genuine conditional reasoning (Anderson & Belnap, Entailment, 1975).
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    • 2.Relevant logic requires that antecedent and consequent share propositional variables for a conditional to be valid, which p→q need not satisfy.
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    • 3.Under relevant logic, (p→q)∨p fails when p and q are relevantly independent, undermining the claim's universality.
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    Reason against 2 of 2
    ?
    • 1.Intuitionistic logic rejects classical truth-value bivalence, so 'p is false' cannot be assumed whenever 'p fails' (Dummett, Elements of Intuitionism).
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    • 2.Without bivalence, a false antecedent cannot be established merely from the absence of proof of p, blocking the vacuous truth inference in constructive settings.
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    Topics

    Philosophy of LanguageModality & Possibility

    Related

    If p fails, p is falseIn classical logic, a conditional with a false antecedent is trueIntuitionistic logic rejects classical truth-value bivalence, so 'p is false' ca...Relevant logic requires that antecedent and consequent share propositional varia...
    +4 moreShow less
    Therefore (p → q) holds whenever p is false, making (p → q) ∨ p a tautologyUnder relevant logic, (p→q)∨p fails when p and q are relevantly independent, und...Vacuous truth is a formal artifact of material implication, not a feature of gen...Without bivalence, a false antecedent cannot be established merely from the abse...

    Similar

    Therefore (p → q) holds whenever p is false, making (p → q) ∨ p a taut...83%In classical logic, a conditional with a false antecedent is true82%Classical logic permits structural rules that validate (p → q) ∨ p via...81%K(p ∧ ¬Kp) is necessarily false (□¬K(p ∧ ¬Kp))80%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-substructural
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    shows that either \(p \rightarrow q\) or \(p\) must hold. This is classically valid (if \(p\) fails, \(p\) is false, and conditionals with false antecedents are true), but invalid in intuitionistic logic. The difference between classical and intuitionistic logic, then, can be understood formally as a difference between the kinds of structural rules permitted, and the kinds of structures appropriate to use in the analysis of logical consequence.
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit