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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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. Barrett Given a sentence (at line n), conlcude a conditional having Turnstile it as the consequent and whose antecedent appears in the Sequent proof as an assumption (at line m). Proof Annotation

I Annotation: n → I(m) Assumption Set I Assumption Set: Everything in the assumption set at Line Number line n excepting m, the line number where the Line of Proof antecedent was assumed. Proof for a Given Argument

I Comment: The antecedent must be present in the Primitive Rules Assumption proof as an assumption. We speak of Ampersand-Intro Ampersand-Elim DISCHARGING this assumption when applying this Wedge-Intro Wedge-Elim rule. Placeing the number m in parentheses indicates Arrow-Intro Arrow-Elim it is the discharged assumption. The lines m and n Reductio ad Absurdum Double-Arrow-Intro may be the same. Double-Arrow-Elim Final Comment I Also known as: Conditional Proof (CP). Primitive Rules of Primitive Rules of Proof - Arrow-Intro 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 1 (4)P → Q 3 → I (2) Line Number Line of Proof (b) Proof for a Given Argument

1 (1)RA Primitive Rules Assumption 2 (2)PA Ampersand-Intro Ampersand-Elim Wedge-Intro 1 (3)P → R 1 → I (2) Wedge-Elim Arrow-Intro Arrow-Elim Reductio ad Absurdum (c) Double-Arrow-Intro 1 (1)PA Double-Arrow-Elim Final Comment (2)P → P 1 → I (1) Primitive Rules of Primitive Rules of Proof - Arrow-Elim Proof From Logic Primer

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Arrow-Elim Turnstile Sequent Given a conditional sentence (at line m) and another Proof

sentence that is its antecedent (at line n), conclude the Annotation

consequent of the conditional. Assumption Set I Annotation: m,n → E Line Number Line of Proof I Assumption Set: The union of the assumption sets at Proof for a Given lines m and n. Argument

I Comment: The order of m and n in the proof is Primitive Rules Assumption irrrelevant. Ampersand-Intro Ampersand-Elim Wedge-Intro I Also known as: Modus Ponendo Ponens (MPP), Wedge-Elim Arrow-Intro Modus Ponens (MP), Detachment, Affirming the Arrow-Elim Reductio ad Absurdum Antecedent. Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Arrow-Elim Proof From Logic Primer

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Proof Examples. Annotation Assumption Set 1 (1)P → QA Line Number 2 (2)PA Line of Proof Proof for a Given 1, 2 (3)Q 1, 2 → E Argument Primitive Rules 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 - Reductio ad Proof From Logic Primer

Absurdum E. Barrett

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Reductio ad Absurdum Sequent Given both a sentence and its denial (at lines m and n), Proof conclude the denial of any assumption appearing in the Annotation proof (at line k). Assumption Set Line Number I Annotation: m,n RAA(k) Line of Proof

I Assumption Set: The union of the assumption sets at Proof for a Given lines m and n, excluding k (the denied assumption). Argument Primitive Rules I Comment: The sentence at line k is the assumption Assumption Ampersand-Intro discharged (a.k.a. the REDUCTIO ASSUMPTION) Ampersand-Elim Wedge-Intro and the conclusion must be a denial of the discharged Wedge-Elim Arrow-Intro assumption. The sentences at lines m and n must be Arrow-Elim Reductio ad Absurdum denials of each other. Double-Arrow-Intro Double-Arrow-Elim I Also known as: Indirect Proof (IP), ∼ Intro/ ∼ Elim. Final Comment Primitive Rules of Primitive Rules of Proof - Reductio ad Proof From Logic Primer

Absurdum E. Barrett

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Sequent

Proof Examples. Annotation (a) Assumption Set Line Number

1 (1)P → QA Line of Proof 2 (2)¬QA Proof for a Given Argument

3 (3)PA Primitive Rules Assumption 1, 3 (4)Q 1, 3 → E Ampersand-Intro Ampersand-Elim Wedge-Intro 1, 2 (5)¬P 2, 4RAA(3) Wedge-Elim Arrow-Intro Arrow-Elim Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Reductio ad Proof From Logic Primer

Absurdum E. Barrett Examples. Turnstile (b) Sequent 1 (1)P ∨ QA Proof Annotation

2 (2)¬PA Assumption Set 3 (3)¬P → ¬QA Line Number 2, 3 (4)¬Q 2, 3 → E Line of Proof Proof for a Given 1, 2, 3 (5)P 1, 4 ∨ E Argument Primitive Rules 1, 3 (6)P 2, 5RAA(2) Assumption Ampersand-Intro Ampersand-Elim (c) Wedge-Intro Wedge-Elim Arrow-Intro 1 (1)PA Arrow-Elim Reductio ad Absurdum 2 (2)QA Double-Arrow-Intro Double-Arrow-Elim 3 (3)¬QA Final Comment 2, 3 (4)¬P 2, 3RAA(1) Primitive Rules of Primitive Rules of Proof - Double-Arrow-Intro Proof From Logic Primer

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Sequent Double-Arrow-Intro Proof Given two conditional entences having the forms φ → ψ Annotation and ψ → φ (at lines m and n), conclude a biconditional Assumption Set with φ on one side and ψ on the other. Line Number Line of Proof I Annotation: m,n ↔ I Proof for a Given I Assumption Set: The union of the assumption sets at Argument lines m and n. Primitive Rules Assumption Ampersand-Intro I Comment: The order of m and n in the proof is Ampersand-Elim Wedge-Intro irrelevant. Wedge-Elim Arrow-Intro Arrow-Elim Reductio ad Absurdum Double-Arrow-Intro Double-Arrow-Elim Final Comment Primitive Rules of Primitive Rules of Proof - Double-Arrow-Intro Proof From Logic Primer

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Sequent

Proof

Examples. Annotation

Assumption Set

1 (1)P → QA Line Number 2 (2)Q → PA Line of Proof Proof for a Given 1, 2 (3)P ↔ Q 1, 2 ↔ I Argument 1, 2 (4)Q ↔ P 1, 2 ↔ I Primitive Rules 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 - Double-Arrow-Elim Proof From Logic Primer

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Sequent

Proof

Double-Arrow-Elim Annotation

Given a biconditional sentence φ ↔ ψ (at line m), conclude Assumption Set

either φ → ψ or ψ → φ. Line Number

I Annotation: m ↔ E Line of Proof Proof for a Given I Assumption Set: The same as at line m. Argument

I Also known as: Sometimes the rules ↔ I and ↔ E are Primitive Rules Assumption subsumed as Definition of Biconditional (df. ↔). 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 - Double-Arrow-Elim Proof From Logic Primer

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Proof Examples. Annotation Assumption Set 1 (1)P ↔ QA Line Number 1 (2)P → Q 1 ↔ E Line of Proof Proof for a Given 1 (3)Q → P 1 ↔ E Argument Primitive Rules 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 - Final Comment Proof From Logic Primer

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Annotation These ten rules of proof are truth-preserving. Given true Assumption Set premises, they will always yield true conclusion. This entails Line Number that if a proof can be constructed for a given argument, Line of Proof Proof for a Given then the argument is valid. Argument

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Final Comment