Chapter 10 Notes: Proofs II

• Rules of replacement represent logical equivalences, while rules of implication do not

• Rules of replacement can be applied to PARTS of lines in a proof (as well as to whole lines)

  • Rules of implication can only be applied to WHOLE lines

What is the difference between rules of replacement and rules implication?

double negation (definition)

An even number of tildes is equal to NONE

double negation

∼∼p is equal to p

commutativity (definition)

the order in which the conjuncts/disjuncts are written makes no logical difference

commutativity

p ● q is equal to q ● p 

AND

p ∨ q is equal to q ∨ p

associativity (definition)

When both operators are dots or wedges, the parentheses can be moved without changing the meaning or truth value of the statements

associativity

p ● (q ● r) is equal to (p ● q) ● r  

AND

p ∨ (q ∨ r) is equal to (p ∨ q) ∨ r

all the operators are dots or all the operators are wedges

The associativity rule can only be used when…

demorgan’s rule (definition)

Denying that two statements are both true is equivalent to saying that at least one is false, AND denying that at least one of the statements is true is equivalent to saying that they’re both false

demorgan’s rule

∼(p ● q) is equal to ∼p ∨ ∼q 

AND

∼(p  ∨ q) is equal to ∼p ● ∼q

double negation

In the case of Demorgan’s Rule, when what’s inside the parentheses is already negated, you get a _____ ______

replacement line

You CAN apply a __________ _____ to a part of a line, as well as to the whole line.

logical equivalences

Rules of replacement represent _______ _____

• Because rules of replacement represent logical equivalences

• Only replacing part of a line will not change the actual meaning of the statement

  • Replacing ∼∼B with B could not change the truth value of ∼A ∨ ∼∼B because B and ∼∼B are equivalent statements

Why can you apply a replacement rule to part of a line as well as to the whole line? 

distribution

p ● (q ∨ r) is equal to (p ● q) ∨ (p ● r) 

AND 

p ∨ (q ● r) is equal to (p ∨ q) ● (p ∨ r)

• The operator that STARTS INSIDE the parentheses ENDS up outside (and vice versa)

• The leftmost letter stays on the left

What should you remember when applying distribution?

transposition

p ⊃ q is equal to ∼q ⊃ ∼p

transposition example (words)

If being a collie implies being a dog, then if something isn’t a dog, it’s not a collie.

tilde, there wasn’t one there to begin with

delete, tilde, one was already there

In transposition, when you switch the antecedent and consequent around, you should add a _____ if ________________ and _____ the ______ if __________.

material implication

p ⊃ q is equal to ∼p ∨ q

material implication example (words)

If continuing to overeat means you’ll gain weight, then either you’re not going to continue overeating, or you’re going to gain weight (and vice versa).

exportation

(p ● q) ⊃ r is equal to p ⊃ (q ⊃ r)

exportation example (words)

If being valid and having true premises implies that an argument is sound, then if an argument is valid, its having true premises implies that it’s sound (and vice versa).

tautology

p is equal to p ∨ p

AND

p is equal to p ● p

addition rule

p entails p ∨ p by the ______ ___

simplification rule

p entails p ● p by the _______ ___

material equivalence

(p ≡ q) is equal to (p ⊃ q) ● (q ⊃ p)

AND

(p ≡ q) is equal to (p ● q) ∨ (∼p ● ∼q)

material equivalence definition

• A biconditional statement is equivalent to two conditions, with the antecedent and consequent switched in the two (explains first half of material equivalence)

• A biconditional statement is only true when each side has the same truth value.

So, if P ≡ Q, then either P and Q are both the case or neither the case.

two conditions, with the antecedent and consequent switched in the two

A biconditional statement is equivalent to… 

• Double Negation (DN)

• Commutativity (Com)

• Associativity (Assoc)

• DeMorgan’s Rule (DM)

• Distribution (Dist)

• Transposition (Trans)

• Material Implication (Impl)

• Exportation (Exp)

• Tautology (Taut)

• Material Equivalence (Equiv)

Rules of Replacement (10)

• Modus Ponens (MP)

• Modus Tollens (MT)

• Disjunctive Syllogism (DS)

• Hypothetical Syllogism (HS)

• Simplification (Simp)

• Conjunction (Conj)

• Constructive Dilemma (CD)

• Addition (Add)

Rules of Implication (8)

modus ponens

1. p ⊃ q

2. p

3. So, q

modus tollens

1. p ⊃ q

2. ∼q

3. So, ∼p

disjunctive syllogism

1. p ∨ q

2. ∼p

3. So, q

AND

1. p ∨ q

2. ∼q

3. So, p

hypothetical syllogism

1. p ⊃ q

2. q ⊃ r

3. So, p ⊃ r

simplification

1. p ● q

2. So, p

AND

1. p ● q

2. So, q

conjunction

1. p

2. q 

3. So, p ● q

constructive dilemma

1. (p ⊃ q) ● (r ⊃ s)

2. p ∨ r

3. So, q ∨ s

addition

1. p

2. So, p ∨ q