This is part 2 of my series on logic and critical thinking.
 Introductory Logic, Part 1—Introducing both logic in general (such as the difference between a deductive and inductive argument) and propositional logic in particular
 Introductory Logic, Part 2—More propositional logic
 Introductory Logic, Part 3—A defense of the material conditional
Bait
To bait both atheists and theists into reading this article on logic (though I hope you already have an interest in learning logic), consider this argument from evil:
 If God exists, then gratuitous evil does not exist.
 Gratuitous evil does exist.
 Therefore, God does not exist.
 If God does not exist, then objective morality does not exist.
 Objective morality does exist.
 Therefore, God exists.
Review
In my previous entry I talked about the connectives and various rules of logic. To recap the connectives:
Connective  Symbolic Logic  English Meaning  Notes 

∧ (conjunction)  p ∧ q  p and q  The p and q parts are called conjuncts. 
∨ (disjunction)  p ∨ q  p or q  The p and q parts are called disjuncts. 
→ (conditional)  p → q  If p, then q  The p part is called the antecedent and the q part is called the consequent. Sometimes p → q is read as “p implies q.” 
↔ (biconditional)  p ↔ q  p, if and only if q  This means the same thing as “p → q and q → p.” 
¬ (negation)  ¬p  Notp  The negation of P is ¬P, and ¬P means “notP” or “P is false.” 
To recap the rules of inference:
modus ponens  

In English  In Symbolic Logic 
If p then q p Therefore, q 
p → q p ∴ q 
modus tollens  

In English  In Symbolic Logic 
If p then q Notq Therefore, notp 
p → q ¬q ∴ ¬p 
Disjunctive Syllogism  

In English  In Symbolic Logic 
p or q Notp Therefore, q 
p ∨ q ¬p ∴ q 
p or q Notq Therefore, p 
p ∨ q ¬q ∴ p 
simplification  

In English  In Symbolic Logic 
p and q Therefore, p 
p ∧ q ∴ p 
p and q Therefore, q 
p ∧ q ∴ q 
conjunction 

p q ∴ p ∧ q 
constructive dilemma 

(p → q) ∧ (r → s) p ∨ r ∴ q ∨ s 
hypothetical syllogism 

p → q q → r ∴ p → r 
addition 

p ∴ p ∨ q 
absorption 

p → q ∴ p → (p ∧ q) 
Some equivalences:
equivalence  name of equivalence 

p ⇔ ¬¬p  double negation 
p → q ⇔ ¬q → ¬p  transposition (also called contraposition) 
p → q ⇔ ¬p ∨ q  material implication 
p ↔ q ⇔ (p → q) ∧ (q → p)  biconditional equivalence 
¬(p ∧ q) ⇔ ¬p ∨ ¬q  De Morgan’s laws 
¬(p ∨ q) ⇔ ¬p ∧ ¬q  
p ⇔ p ∧ p  idempotence 
p ⇔ p ∨ p 
And some more equivalences:
equivalence  name of equivalence 

p ∧ q ⇔ q ∧ p  commutation 
p ∨ q ⇔ q ∨ p  
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ r  association 
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r  
p → (q → r) ⇔ (p ∧ q) → r  exportation 
p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r)  distribution 
p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r) 
Next I’ll introduce some cool new rules of inference: conditional proof and indirect proof (also known as proof by contradiction, among other names). I’ll also show how to use symbolic logic to prove a statement without using any premises.
Conditional Proofs
Recall that the conditional is symbolized as p → q where p is called the antecedent and q is called the consequent. The conditional proof aims to prove that a conditional is true, with the antecedent of the conditional being the conditional proof assumption which is often used to help show that if the antecedent is true then the consequent is true also. The structure of a conditional proof takes the following form below:
conditional proof  


For example, suppose we want to prove A → (B ∧ C) from premises 1 and 2 below:
 A → B
 A → C
 A conditional proof assumption
 B 1, 3, modus ponens
 C 2, 3, modus ponens
 B ∧ C 4, 5, conjunction
 A → (B ∧ C) 36, conditional proof
 A → B
 A → C
 A conditional proof assumption
 B 1, 3, modus ponens
 C 2, 3, modus ponens
 B ∧ C 4, 5, conjunction
 A → (B ∧ C) 36, conditional proof
 (A → B) ∧ A 1, 3, conjunctionMistake!
 (A → B) ∧ C 1, 5, conjunctionMistake!
reiteration 

p ∴ p 
At first blush reiteration might not seem like a very handy rule, but its use comes from being able to put one proposition from the “outside” to the “inside,” as illustrated below
 B
 A conditional proof assumption
 B 1 reiteration
 A → B 23, conditional proof
 A → B
 A → C
 A conditional proof assumption
 B 1, 3, modus ponens
 C 2, 3, modus ponens
 B ∧ C 4, 5, conjunction
 A → (B ∧ C) 36, conditional proof
 A 3, reiterationMistake!
 C 5, reiterationMistake!
Indirect Proof
Another proof method known by various names as indirect proof and proof by contradiction begins by assuming the opposite of what you want to prove and then obtaining a logical contradiction, i.e. a contradiction of the p ∧ ¬p or ¬p ∧ p sort. The structure of an indirect proof:
indirect proof  

 

For example, suppose we wanted to prove ¬(H ∧ R) from premise 1 below:
 (H ∧ R) → ¬(H ∧ R)
 H ∧ R indirect proof assumption
 ¬(H ∧ R) 1, 2, modus ponens
 (H ∧ R) ∧ ¬(H ∧ R) 2, 3, conjunction
 ¬(H ∧ R) 24, indirect proof
 ¬B → ¬D
 ¬B → E
 [¬B → (¬D ∧ E)] → D
 ¬B indirect proof assumption
 ¬B conditional proof assumption
 ¬D 1, 5, modus ponens
 E 2, 5, modus ponens
 ¬D ∧ E 6, 7, conjunction
 ¬B → (¬D ∧ E) 58, conditional proof
 D 3, 9 modus ponens
 ¬D 1, 4, modus ponens
 D ∧ ¬D 10, 11 conjunction
 B 412, indirect proof
 ¬B → ¬D
 ¬B → E
 [¬B → (¬D ∧ E)] → D
 ¬B indirect proof assumption
 ¬B conditional proof assumption
 ¬D 1, 5, modus ponens
 E 2, 5, modus ponens
 ¬D ∧ E 6, 7, conjunction
 ¬B → (¬D ∧ E) 58, conditional proof
 D 3, 9 modus ponens
 ¬D 1, 5, modus ponensMistake! 5 is inaccessible!
 E 7, reiterationMistake!
 D ∧ ¬D 10, 11 conjunction
 B 413, indirect proof
 D ∨ Z 10, additionMistake!
Demonstrating Inconsistent Premises
One proves that the premises are inconsistent with each other by deriving a logical contradiction. This has an interesting application for modus ponens and modus tollens. First, consider the representative modus ponens argument below:
 P → Q
 P
 Q 1, 2, modus ponens
 ¬(P → Q)
 ¬P
 ¬Q conditional proof assumption
 ¬P 2, reiteration
 ¬Q → ¬P 34, conditional proof
 P → Q 5, transposition
 ¬(P → Q) ∧ (P → Q) 1, 6, conjunction
 P → Q
 ¬Q
 ¬P 1, 2, modus tollens
 ¬(P → Q)
 ¬¬Q
 P conditional proof assumption
 Q 2, double negation
 P → Q 34, conditional proof
 ¬(P → Q) ∧ (P → Q) 1, 5, conjunction
Symbolic Logic Proofs Without Premises
So how do you prove things in propositional logic without premises? Easy: you can use conditional proofs and indirect proofs. For example, suppose we want to prove (A ∧ ¬A) → (K ∧ I):
 A ∧ ¬A conditional proof assumption
 ¬A 1, simplification
 A 1, simplification
 A ∨ (K ∧ I) 3 addition
 K ∧ I 2, 4, disjunctive syllogism
 (A ∧ ¬A) → (K ∧ I) 15, conditional proof
In propositional logic, proofs without premises are called theorems. As you might guess, not every statement is a theorem, but the famous law of logic the law of noncontradiction is. A famous rule of logic called the law of noncontradiction says that for any proposition p, it is impossible for p and notp to be true at the same time and in the same context. In symbolic logic, the law of noncontradiction is expressed as ¬(p ∧¬p). Here’s how one can prove it:
 P ∧¬P indirect proof assumption
 P ∧¬P 1, reiteration
 ¬(P ∧ ¬P) 12, indirect proof
 ¬(P ∨¬P) indirect proof assumption
 ¬P ∧¬¬P 1, De Morgan’s laws
 P ∨ ¬P 12, indirect proof
No, I Won’t Bait and Switch
First, for the atheist:
 If God exists, then gratuitous evil does not exist.
 Gratuitous evil does exist.
 Therefore, God does not exist.
 G → ¬E
 E
 ¬(G → ¬E)
 ¬E
 G conditional proof assumption
 ¬E2, reiteration
 G → ¬E 34, conditional proof
 ¬(G → ¬E) ∧ (G → ¬E) 1, 5, conjunction
 If God does not exist, then objective morality does not exist.
 Objective morality does exist.
 Therefore, God exists.
 ¬G → ¬M
 M
 ¬(¬G → ¬M)
 ¬M
 ¬G conditional proof assumption
 ¬M ∧ ¬G2, 3, conjunction
 ¬M 4, simplification
 ¬G → ¬M 35, conditional proof
 ¬(¬G → ¬M) ∧ (¬G → ¬M) 1, 6, conjunction
Symbolic Logic Summary
To summarize the rules of logic learned in this entry:
conditional proof  


reiteration 

p ∴ p 
indirect proof  

 
