Read e-book online A System of Logic Ratiocinative and Inductive, Part II (The PDF

By John Stuart Mill

ISBN-10: 0710075065

ISBN-13: 9780710075062

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Extra resources for A System of Logic Ratiocinative and Inductive, Part II (The Collected Works of John Stuart Mill - Volume 08)

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A) ¬???? ∨ ????, ¬???? ⊨ ¬???? (b) ¬(???? ∧ ????), ???? ⊨ ¬???? (c) ???? → ????, ???? ⊨ ???? ∨ ???? (d) ???? → ????, ???? → ????, ???? ⊨ ???? (e) ???? ∨ ???? ∧ ????, ¬???? ⊨ ???? 2. Show the following using truth tables. (a) ¬(???? ∧ ????) ̸⊨ ¬???? (b) ???? → ???? ∨ ????, ???? ̸⊨ ???? (c) ???? ∧ ???? → ???? ̸⊨ ???? → ???? (d) (???? → ????) ∨ (???? → ????), ???? ∨ ???? ̸⊨ ???? ∨ ???? (e) ¬(???? ∧ ????) ∨ ????, ???? ∧ ???? ∨ ???? ̸⊨ ???? ∧ ???? (f) ???? ∨ ????, ???? ∨ ????, ???? ↔ ???? ̸⊨ ???? ∧ ???? 3. 10. (a) ???? → ???? → ???? , ???? → ???? ⇒ ???? (b) ???? , ???? ∨ ???? ⇒ ???? ∧ (???? ∨ ????) (c) ???? ⇒ ???? ∨ (???? ↔ ¬???? ∧ ¬[???? → ????]) (d) ???? , ???? → (???? ↔ ????) ⇒ ???? ↔ ???? (e) ???? ∨ ???? ∨ ????, (???? ∨ ???? → ????) ∧ (???? → ???? ∧ ???? ) ⇒ ???? ∨ ???? ∧ ???? (f) ???? ∨ (???? ∨ ????), ¬???? ⇒ ???? ∨ ???? (g) ???? → ¬????, ¬¬???? ⇒ ¬???? (h) (???? → ????) ∧ (???? → ????), ¬???? ∨ ¬???? ⇒ ¬???? ∨ ¬???? (i) (???? → ????) ∧ (???? → ????) ⇒ ???? → ???? 29 30 Chapter 1 PROPOSITIONAL LOGIC 4.

Show using truth tables. (a) ¬???? ∨ ????, ¬???? ⊨ ¬???? (b) ¬(???? ∧ ????), ???? ⊨ ¬???? (c) ???? → ????, ???? ⊨ ???? ∨ ???? (d) ???? → ????, ???? → ????, ???? ⊨ ???? (e) ???? ∨ ???? ∧ ????, ¬???? ⊨ ???? 2. Show the following using truth tables. (a) ¬(???? ∧ ????) ̸⊨ ¬???? (b) ???? → ???? ∨ ????, ???? ̸⊨ ???? (c) ???? ∧ ???? → ???? ̸⊨ ???? → ???? (d) (???? → ????) ∨ (???? → ????), ???? ∨ ???? ̸⊨ ???? ∨ ???? (e) ¬(???? ∧ ????) ∨ ????, ???? ∧ ???? ∨ ???? ̸⊨ ???? ∧ ???? (f) ???? ∨ ????, ???? ∨ ????, ???? ↔ ???? ̸⊨ ???? ∧ ???? 3. 10. (a) ???? → ???? → ???? , ???? → ???? ⇒ ???? (b) ???? , ???? ∨ ???? ⇒ ???? ∧ (???? ∨ ????) (c) ???? ⇒ ???? ∨ (???? ↔ ¬???? ∧ ¬[???? → ????]) (d) ???? , ???? → (???? ↔ ????) ⇒ ???? ↔ ???? (e) ???? ∨ ???? ∨ ????, (???? ∨ ???? → ????) ∧ (???? → ???? ∧ ???? ) ⇒ ???? ∨ ???? ∧ ???? (f) ???? ∨ (???? ∨ ????), ¬???? ⇒ ???? ∨ ???? (g) ???? → ¬????, ¬¬???? ⇒ ¬???? (h) (???? → ????) ∧ (???? → ????), ¬???? ∨ ¬???? ⇒ ¬???? ∨ ¬???? (i) (???? → ????) ∧ (???? → ????) ⇒ ???? → ???? 29 30 Chapter 1 PROPOSITIONAL LOGIC 4.

To make rigorous which propositional forms can be inferred from given forms, we establish some rules. These are chosen because they model basic reasoning. They are also not proved, so they serve as postulates for our logic. 10 Let ????, ????, ????, and ???? be propositional forms. ∙ Modus Ponens [MP] ???? → ????, ???? ⇒ ???? ∙ Modus Tolens [MT] ???? → ????, ¬???? ⇒ ¬???? ∙ Constructive Dilemma [CD] (???? → ????) ∧ (???? → ????), ???? ∨ ???? ⇒ ???? ∨ ???? ∙ Destructive Dilemma [DD] (???? → ????) ∧ (???? → ????), ¬???? ∨ ¬???? ⇒ ¬???? ∨ ¬???? ∙ Disjunctive Syllogism [DS] ???? ∨ ????, ¬???? ⇒ ???? ∙ Hypothetical Syllogism [HS] ???? → ????, ???? → ???? ⇒ ???? → ???? ∙ Conjunction [Conj] ????, ???? ⇒ ???? ∧ ???? ∙ Simplification [Simp] ????∧???? ⇒???? ∙ Addition [Add] ???? ⇒ ???? ∨ ????.

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A System of Logic Ratiocinative and Inductive, Part II (The Collected Works of John Stuart Mill - Volume 08) by John Stuart Mill


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