Primitive Rules of Proof From Logic Primer E. Barrett Turnstile Primitive Rules of Proof Sequent Proof From Logic Primer Annotation Assumption Set Line Number Colin Allen, Michael Hand Line of Proof Beamer by Eric Barrett Proof for a Given Argument Primitive Rules Chapman University Final Comment January 27, 2013 Primitive Rules of Primitive Rules of Proof - Turnstile Proof From Logic Primer E. Barrett Turnstile Sequent Proof Annotation Assumption Set Turnstile Line Number Line of Proof I Definition. The TURNSTILE is the symbol `. Proof for a Given Argument Primitive Rules Final Comment Primitive Rules of Primitive Rules of Proof - Sequent Proof From Logic Primer E. Barrett Sequent Turnstile Sequent I Definition. A SEQUENT consists of a number of Proof sentences separated by commas (corresponding to the Annotation premises of an argument), followed by a turnstile, Assumption Set followed by another sentence (corresponding to the Line Number conclusion of the argument). Line of Proof Proof for a Given I Example. Argument (P & Q) ! R; :R & P ` :Q Primitive Rules Final Comment I Comment. Sequents are nothing more than a convenient way of displaying arguments in the formal notation. The turnstile symbol may be read as 'therefore'. Primitive Rules of Primitive Rules of Proof - Proof Proof From Logic Primer E. Barrett Turnstile Proof Sequent Proof I Definition. A PROOF is a sequence of lines Annotation containing sentences. Each sentence is either an Assumption Set assumption or the result of applying a rule of proof to Line Number earlier sentences in the sequence. The primitive rules Line of Proof of proof are stated below. Proof for a Given Argument I Comment. The purpose of presenting proofs is to Primitive Rules demonstrate unequivocally that a given set of premises Final Comment entails a particular conclusion. Thus, when presenting a proof we associate three things with each sentence in the proof sequence: Primitive Rules of Primitive Rules of Proof - Annotation Proof From Logic Primer E. Barrett Turnstile Sequent Proof Annotation Annotation Assumption Set I On the right of the sentence we provide an Line Number ANNOTATION specifying which rule of proof was Line of Proof Proof for a Given applied to which earlier sentences to yeild the given Argument sentence. Primitive Rules Final Comment Primitive Rules of Primitive Rules of Proof - Assumption Set Proof From Logic Primer E. Barrett Turnstile Sequent Proof Annotation Assumption Set Assumption Set Line Number I On the far left we associate with each sentence an Line of Proof ASSUMPTION SET containing the assumption on Proof for a Given which the given sentence depends. Argument Primitive Rules Final Comment Primitive Rules of Primitive Rules of Proof - Line Number Proof From Logic Primer E. Barrett Turnstile Sequent Proof Annotation Line Number Assumption Set Line Number I Also on the left, we write the current LINE Line of Proof NUMBER of the proof. Proof for a Given Argument Primitive Rules Final Comment Primitive Rules of Primitive Rules of Proof - Line of Proof Proof From Logic Primer E. Barrett Turnstile Line of Proof Sequent Proof I Definition. A sentence of a proof, together with its anotation, its assumption set and the line number, is Annotation called a LINE OF THE PROOF. Assumption Set Line Number I Example. Line of Proof Proof for a Given Argument Primitive Rules 1; 2 (7)P ! Q & R 6 ! I (3) Final Comment ""%- (AssumptionSet)(LineNumber)(Sentence)(Annotation) Primitive Rules of Primitive Rules of Proof - Proof for a Given Proof From Logic Primer Argument E. Barrett Turnstile Sequent Proof Annotation Proof for a Given Argument Assumption Set Line Number I Definition. A PROOF FOR A GIVEN ARGUMENT Line of Proof is a proof whose last sentence is the argument's Proof for a Given Argument conclusion depending on nothing other than the Primitive Rules argument's premises. Final Comment Primitive Rules of Primitive Rules of Proof - Primitive Rules Proof From Logic Primer E. Barrett Turnstile Sequent Primitive Rules Proof Annotation I Definition. The ten PRIMITIVE RULES OF Assumption Set PROOF are the rules assumption, Line Number ampersand-introduction, ampersand-elimination, Line of Proof wedge-introduction, wedge-elimination, Proof for a Given Argument arrow-introduction, arrow-elimination, reductio ad Primitive Rules absurdum, double-arrow-introduction, and Assumption Ampersand-Intro double-arrow-elimination, as described below. Ampersand-Elim Wedge-Intro Wedge-Elim Arrow-Intro Arrow-Elim Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Assumption Proof From Logic Primer E. Barrett Turnstile Sequent Assumption Proof Assume any sentence. Annotation I Annotation: A Assumption Set Line Number I Assumption Set: The current line number. Line of Proof I Comment: Anything may be assumed at any time. Proof for a Given However, some assumptions are useful and some are Argument Primitive Rules not! Assumption Ampersand-Intro I Example. Ampersand-Elim Wedge-Intro 1 (1) P _ QA Wedge-Elim Arrow-Intro Arrow-Elim Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Ampersand-Intro Proof From Logic Primer E. Barrett Turnstile Ampersand-Intro Sequent Given two sentences (at lines m and n), conlcude a Proof Annotation conjunction of them. Assumption Set I Annotation: m, n & I Line Number I Assumption Set: The union of the assumption sets at Line of Proof Proof for a Given lines m and n. Argument I Comment: The order of lines m and n in the proof is Primitive Rules Assumption irrelevant. The lines referred to by m and n may also Ampersand-Intro Ampersand-Elim be the same. Wedge-Intro Wedge-Elim Arrow-Intro I Also known as: Conjunction (CONJ). Arrow-Elim Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Ampersand-Intro Proof From Logic Primer E. Barrett Turnstile Sequent Examples. Proof Annotation 1 (1)PA Assumption Set 2 (2)QA Line Number Line of Proof 1; 2 (3)P & Q 1; 2 & I Proof for a Given Argument 1; 2 (4)Q & P 1; 2 & I Primitive Rules Assumption 1 (5)P & P 1; 1 & I Ampersand-Intro Ampersand-Elim Wedge-Intro Wedge-Elim Arrow-Intro Arrow-Elim Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Ampersand-Elim Proof From Logic Primer E. Barrett Turnstile Sequent Proof Ampersand-Elim Annotation Given a sentence that is a conjunction (at line m), conclude Assumption Set either conjunct. Line Number Line of Proof I Annotation: m & E Proof for a Given Argument I Assumption Set: The same as at line m. Primitive Rules I Also known as: Simplification (S). Assumption Ampersand-Intro Ampersand-Elim Wedge-Intro Wedge-Elim Arrow-Intro Arrow-Elim Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Ampersand-Elim Proof From Logic Primer E. Barrett Examples. Turnstile Sequent (a) Proof 1 (1)P & QA Annotation Assumption Set 2 (2)Q 1 & E Line Number 1; 2 (3)P 1 & E Line of Proof Proof for a Given Argument Primitive Rules Assumption Ampersand-Intro (b) Ampersand-Elim Wedge-Intro 1 (1)P &(Q ! R) A Wedge-Elim Arrow-Intro Arrow-Elim 2 (2)Q ! R 1 & E Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Wedge-Intro Proof From Logic Primer E. Barrett Turnstile Sequent Proof Wedge-Intro Annotation Given a sentence (at line m), conclude any disjunction Assumption Set having it as a disjunct. Line Number Line of Proof I Annotation: m _ I Proof for a Given Argument I Assumption Set: The same as at line m. Primitive Rules I Also known as: Addition (ADD). Assumption Ampersand-Intro Ampersand-Elim Wedge-Intro Wedge-Elim Arrow-Intro Arrow-Elim Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Wedge-Intro Proof From Logic Primer E. Barrett Examples. Turnstile Sequent (a) Proof 1 (1)PA Annotation Assumption Set 1 (2)P _ Q 1 _ I Line Number 1 (3)(R $ :T ) _ P 1 _ I Line of Proof Proof for a Given Argument Primitive Rules Assumption Ampersand-Intro (b) Ampersand-Elim Wedge-Intro 1 (1)Q ! RA Wedge-Elim Arrow-Intro Arrow-Elim 1 (2)(Q ! R) _ (P & :S) 1 _ I Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Wedge-Elim Proof From Logic Primer E. Barrett Turnstile Wedge-Elim Sequent Given a sentence (at line m) that is a disjunction and Proof another sentence (at line n) that is a denial of one of its Annotation disjuncts conclude the other disjunct. Assumption Set Line Number I Annotation: m, n _ E Line of Proof I Assumption Set: The union of the assumption sets at Proof for a Given lines m and n. Argument Primitive Rules I Comment: The order of lines m and n in the proof is Assumption Ampersand-Intro irrelevant. Ampersand-Elim Wedge-Intro Wedge-Elim I Also known as: Modus Tollendo Ponens (MTP), Arrow-Intro Arrow-Elim Disjunctive Syllogism (DS). Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Wedge-Elim Proof From Logic Primer Examples. E. Barrett (a) Turnstile 1 (1)P _ QA Sequent 2 (2):PA Proof Annotation 1; 2 (3)Q 1; 2 _ E Assumption Set Line Number (b) Line of Proof 1 (1)P _ (Q ! R) A Proof for a Given Argument 2 (2):(Q ! R) A Primitive Rules Assumption 1; 2 (3)P 1; 2 _ E Ampersand-Intro Ampersand-Elim Wedge-Intro Wedge-Elim (c) Arrow-Intro Arrow-Elim Reductio ad Absurdum 1 (1)P _:RA Double-Arrow-Intro 2 (2)RA Double-Arrow-Elim Final Comment 1; 2 (3)P 1; 2 _ E Primitive Rules of Primitive Rules of Proof - Arrow-Intro Proof From Logic Primer Arrow-Intro E.