Symbolic Logic - Inference and Replacement Rules

[Fringe Pioneer]

There are various rules of inference and replacement that one may use during the course of a proof, and some of them differ by differing philosophers.

Rules of Inference

Simplification (Simp.)
p · q
/∴ p

p · q
/∴ q

If the conjunction of two truth functional statements are true, then each of the truth functional statements must individually be true.

Conjunction (Conj.)
p
q
/∴ p · q

If two truth functional statements are individually true, then their conjunction must be true.

Addition (Add.)
p
/∴ p ∨ q

q
/∴ p ∨ q

If a truth functional statement is individually true, then the disjunction of that statement and whatever other truth functional statement must be true (regardless of whether that other statement is true or false).

Disjunctive Syllogism (D. S.)
p ∨ q
~p
/∴ q

p ∨ q
~q
/∴ p

If the disjunction of two truth functional statements is true, and the negation of one of those truth functional statements is true, then the other truth functional statement must individually be true.

Modus Ponens (M. P.)
p ⊃ q
p
/∴ q

If the material implication of an antecedent truth functional statement and the consequent truth functional statement is true, and the antecedent is individually true, then the consequent must individually be true.

Modus Tollens (M. T.)
p ⊃ q
~q
/∴ ~p

If the material implication of an antecedent truth functional statement and the consequent truth functional statement is true, and the negation of the consequent is individually true, then the negation of the antecedent must individually be true.

Hypothetical Syllogism (H. S.)
p ⊃ q
q ⊃ r
/∴ p ⊃ r

If the material implication of the first truth functional statement as an antecedent and a second truth functional statement as a consequent is true, and the material implication of the second truth functional statement as an antecedent and a third truth functional statement as a consequent is true, then the material implication of the first truth functional statement as an antecedent and the third truth functional statement as a consequent must be true.

Dilemma (Dil.)
p ⊃ q
r ⊃ s
p &or r
/∴ q ∨ s

If the material implication of an antecedent and a consequent is true, the material implication of another antecedent and another consequent is true, and the disjunction of the antecedents is true, then the disjunction of the consequents must also be true.

Rules of Inference

Double Negation (D. N.)
p :: ~~p

A truth functional statement is logically equivalent to its double negation.

Duplication (Dup.)
p :: p ∨ p
p :: p · p

A truth functional statement is logically equivalent to the disjunction or the conjunction of that same truth functional statement and itself.

Commutation (Comm.)
p ∨ q :: q ∨ p
p · q :: q · p

The disjunction of two truth functional statements is logically equivalent to the disjunction of the same two truth functional statements in reverse order. The same is true of conjunctions.

Association (Assoc.)
(p ∨ (q ∨ r)) :: ((p ∨ q) ∨ r)
(p · (q · r)) :: ((p · q) · r)

The disjunction of the first truth functional statement with the disjunction of the second and third truth functional statements is logically equivalent to the disjunction of the disjunction of the first and second truth functional statements and the third truth functional statement. The same is true of conjunctions.

Contraposition (Contrap.)
p ⊃ q :: ~q ⊃ ~p

The material implication of an antecedent with its consequent is logically equivalent to the material implication of the negation of the consequent with the negation of the antecedent.

DeMorgan's (DeM.)
~(p ∨ q) :: ~p · ~q
~(p · q) :: ~p ∨ ~q

The negation of a disjunction of two truth functional statements is logically equivalent to the conjunction of the negations of the two truth functional statements, and the negation of a conjunction of two truth functional statements is logically equivalent to the disjunction of the negations of the two truth functional statements.

Biconditional Exchange (B. E.)
p ≡ q :: (p ⊃ q) · (q ⊃ p)

The material equivalence of two truth functional statements is logically equivalent to the conjunction of the material implication of the two truth functional statements and the material implication of the two truth functional statements in reverse order.

Conditional Exchange (C. E.)
p ⊃ q :: ~p ∨ q

The material implication of an antecedent with its consequent is logically equivalent to the disjunction of the negation of the antecedent and the consequent.

Distribution (Dist.)
p · (q ∨ r) :: (p · q) ∨ (p · r)
p ∨ (q · r) :: (p ∨ q) · (p ∨ r)

The conjunction of a first truth functional statement with the disjunction of two other truth functional statements is logically equivalent to the disjunction of the conjunction of the first truth functional statement with one of the disjuncts and the conjunction of the first truth functional statement with the other disjunct.

The disjunction of a first truth functional statement with the conjunction of two other truth functional statements is logically equivalent to the conjunction of the disjunction of the first truth functional statement with one of the conjuncts and the disjunction of the first truth functional statement with the other conjunct.

Exportation (Exp.)
(p · q) ⊃ r :: p ⊃ (q ⊃ r)

The material implication of a conjunction as the antecedent and a truth functional statement as the consequent is logically equivalent to the material implication of the first conjunct as the antecedent with the material implication of the second conjunct as the antecedent with the consequent, all as the consequent.

[Qwerty]

This is actually pretty useful to know in Calculus.

[Fringe Pioneer]

Ironically, then, I never consciously used these rules in calculus, or even geometry when making proofs...

[Qwerty]

I mean for notetaking. It saves me a ton of time.

[Fringe Pioneer]

Oh, well yes, using a dot instead of writing an ampersand or the word "AND" and using a triple bar instead of writing out "if and only if" must be time saving, although in math I was taught to use "iff" as the shorthand to "if and only if," but even then drawing three straight lines should be shorter than making at least 4 lines of a particular shape recognizable as text...

Well, I should probably make a thread showing how to make a proof...

[Qwerty]

Er, time consuming? I think you mean time saving. Just making sure, you know, before someone screenshots the post. Like me.

[Lily]

Anyone able to solve this?
Using truth-functional inference rules, truth-functional replacement rules,
and the four new quantifier rules, supply proof for the following argument.

1. (3x) (Gx) > (x)(Hx>Sx)
2. (3x)(Gx c Sx)
3. (x) (Sx > Gx) / (x) (~Gx > ~Hx)

[noodlesoup]

No