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Chapter 4

PHIL 10 Chapter 4: Replacement Rules, Indirect Proof, and Tautologies

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Rick Grush

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PHIL 10 Textbook Notes Chapter 4: Replacement Rules, Indirect Proof, and
4.2 Replacement Rules: CE
Inference rules go from one or more statements to one new statement that is implied by
o Embody implications
Replacement rules go from one statement to one equivalent statement
o Embody equivalences
o Anything that is equivalent is an implication as well
Inference rules can be used in the same way as replacement rules, and
vice versa
Conditional Exchange (CE)
(X Y) :: (~X v Y)
(X v Y) :: (~X Y)
Can apply replacement rules to parts of lines
o Rest of the line stays intact
Replacement rules are truth-preserving
4.3 DN and Comm
Double Negation (DN)
X :: ~~X
Commutation (Comm)
(X v Y) :: (Y v X)
(X • Y) :: (Y • X)
4.4 Last Three Replacement Rules: DeM, Contra, Assoc
DeMorgan’s (DeM)
~(X v Y) :: ~X • ~Y
~(X • Y) :: ~X v ~Y
Contraposition (Cont)
(X v Y) v Z :: X v (Y v Z)
(X • Y) • Z :: X • (Y • Z)
1. A • ~(B v ~C) / ~(C ~A)
2. A • (~B • C) 1 DeM
3. A • (C • ~B) 2 Comm
4. (A • C) • ~B 3 Assoc
5. A • C 4 Simp
6. ~(~A v ~C) 5 DeM
7. ~(A ~C) 6 CE
8. ~(C ~A) 7 Contra
4.5 Why You Can Use Replacement Rules on Parts of a Line
Implication does not guarantee truth preservation if used on parts of a statement, but
equivalence does
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