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19 Cards in this Set

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X → Y


X


——————


Y

Modus ponens (MP)- If X → Y and X are both lines in your proof, you can add a new line with the statement Y. This lineis justified by citing the modus ponens rule, MP, and the lines containing X → Y and X.

X → Y


~Y


——————


~X

Modus tollens (MT)- If X → Y and ~Y are both lines in your proof, you can add a new line with the statement ~X. Thisline is justified by citing the modus tollens rule, MT, and the lines containing X → Y and ~Y.

X ∨ Y


~X


——————


Y


or


X∨Y


~Y


——————


X

Disjunctive syllogism (DS)- If X ∨ Y and ~Y are both lines in your proof, you can add a new line with the statement X. This lineis justified by citing the disjunctive syllogism rule, DS, and the lines containing X ∨ Y and ~Y.

X & Y


——————


Y


or


X & Y


——————


X

Simplification (SIMP)- If X & Y is a line in your proof, you can add a new line with the statement X or a new line with thestatement Y. This new line is justified by citing the simplification rule, SIMP, and the line containingX & Y.

X → Y


Y → Z


——————


X→Z

Hypothetical syllogism (HS)- If X → Y and Y → Z are both lines in your proof, you can add a new line with the statement X → Z.This line is justified by citing the hypothetical syllogism rule, HS, and the lines containing X → Y andY → Z.

X


——————


X ∨ Y

Addition (ADD)- If X is a line in your proof, you can add a new line with the statement X ∨ Y. This line is justified byciting the addition rule, ADD, and the line containing the statement X.

X


Y


——————


X&Y or Y&X



Conjunction (CONJ)- If X and Y are both lines in your proof, you can add a new line with the statement X & Y or a newline with the statement Y & X. This line is justified by citing the conjunction rule, CONJ, and thelines containing X and Y.

X∨Y


——————


Y∨X


or


X&Y


——————


Y&X

The Commutation Rule(COMM)- If X ∨ Y is a line in your proof, you can add a new line with the statement Y ∨ X. If X & Y is a line in your proof, you can add a new line with the statement Y & X. Justify this new line by citing (by number) the line used and the rule, COMM.

W → X


Y → Z


W ∨ Y


——————


X∨Z

Constructive dilemma (CD)- If W → X, Y → Z, and W ∨ Y are all lines in your proof, you can add a new line with the statementX ∨ Z. This line is justified by citing the constructive dilemma rule, CD, and the lines containing W→ X, Y → Z, and W ∨ Y.

~~X


——————


X

The Double Negation Rule (DN)- If X is a line in your proof, you can add a new line with the statement ~~X, and vice versa: If ~~X isa line in your proof, you can add a new line with the statement X. Justify this new line by citing (by number) the line used and the rule, DN.

(X ∨ Y) ∨ Z


——————


X ∨ (Y ∨ Z)


or


(X & Y) & Z


——————


X & (Y & Z)

The Association Rule(ASSOC)- If (X ∨ Y) ∨ Z is a line in your proof, you can add a new line with the statement X ∨ (Y ∨ Z), andvice versa; If X ∨ (Y ∨ Z) is a line in your proof, you can add a new line with the statement (X ∨ Y)∨ Z. If (X & Y) & Z is a line in your proof, you can add a new line with the statement X & (Y & Z), andvice versa; If X & (Y & Z) is a line in your proof, you can add a new line with the statement (X & Y)& Z. Justify this new line by citing (by number) the line used and the rule, ASSOC.

X ∨ (Y & Z)


——————


(X ∨ Y) & (X ∨ Z)


or


X & (Y ∨ Z)


——————


(X & Y) ∨ (X & Z)

The Distribution Rule(DIST)- If X ∨ (Y & Z) is a line in your proof, you can add a new line with the statement (X ∨ Y) & (X ∨Z), and vice versa; If (X ∨ Y) & (X ∨ Z) is a line in your proof, you can add a new line with thestatement X ∨ (Y & Z). If X & (Y ∨ Z) is a line in your proof, you can add a new line with the statement (X & Y) ∨ (X &Z), and vice versa; If (X & Y) ∨ (X & Z) is a line in your proof, you can add a new line with thestatement X & (Y ∨ Z). Justify this new line by citing (by number) the line used and the rule, DIST.

X → Y


——————


~Y → ~X

The Transposition Rule(TRANS)- If X → Y is a line in your proof, you can add a new line with the statement ~Y → ~X, and viceversa: If ~Y → ~X is a line in your proof, you can add a new line with the statement X → Y. Justify this new line by citing (by number) the line used and the rule, TRANS.

X → Y


——————


~X ∨ Y

The Implication Rule (IMP)- If X → Y is a line in your proof, you can add a new line with the statement ~X ∨ Y, and vice versa:If ~X ∨ Y is a line in your proof, you can add a new line with the statement X → Y. Justify this new line by citing (by number) the line used and the rule, IMP.

X↔Y


——————


(X → Y) & (Y →X)

The Equivalence Rule (EQUIV)- If X ↔ Y is a line in your proof, you can add a new line with the statement (X → Y) & (Y → X),and vice versa: If (X → Y) & (Y → X) is a line in your proof, you can add a new line with thestatement X ↔ Y. Justify this new line by citing (by number) the line used and the rule, EQUIV.

(X & Y) → Z


——————


X → (Y → Z)

The Exportation Rule (EXP)- If (X & Y) → Z is a line in your proof, you can add a new line with the statement X → (Y → Z), andvice versa: If X → (Y → Z) is a line in your proof, you can add a new line with the statement (X &Y) → Z. Justify this new line by citing (by number) the line used and the rule, EXP.

~(X & Y)


——————


~X ∨ ~Y


or


~(X ∨ Y)


——————


~X & ~Y

DeMorgan's Law (DM)- If ~(X & Y) is a line in your proof, you can add a new line with the statement ~X ∨ ~Y, and viceversa: If ~X ∨ ~Y is a line in your proof, you can add a new line with the statement ~(X & Y). If ~(X ∨ Y) is a line in your proof, you can add a new line with the statement ~X & ~Y, and viceversa: If ~X & ~Y is a line in your proof, you can add a new line with the statement ~(X ∨ Y). Justify this new line by citing (by number) the line used and the rule, DM.

~X


Z & ~Z


——————


X

Indirect Proof (IP)- If from the assumption ~X (and possibly other assumptions), you can prove a contradiction, i.e. anystatement of the form Z & ~Z, then you can add a new line X. Justify this new line by citing theindirect proof rule (IP) and all the lines in your subproof.

X


Y


——————


X → Y

Conditional Proof (CP)- If, from the assumption X (and possibly other assumptions), you can prove Y, then you can add anew line X → Y. This line is justified by citing the conditional proof rule, CP, and all the lines used inyour subproof.