Basic Formal Logic

What is Logic

• Based on rules that are assumed correct
• Formal Logic focuses on rules and 100% correctness = mathematics
• Not concerned with meaning, only with formal validity

“Edifice” (Descartes)

Descartes described logic as an edifice (building):

• Foundation consists of basic laws of logic
• Built upon it are deductive and inductive reasoning
• Formal logic follows strict rules to ensure correctness

Four Laws of Logic

1. Law of Identity (同一的): A proposition is itself, no more and no less
   • $p \to p$
2. Law of No-Contradiction: A claim and its direct denial cannot both be true
   • $\lnot (p \land \lnot p)$
3. Law of Excluded Middle: Either a proposition or its negation must be true
   • $p \lor \lnot p$
4. Law of Double Negation: A double negative equals the positive
   • $p \Leftrightarrow \lnot \lnot p$

Principle of Sufficiency

• “Nothing is without reason” (存在即有理由)
• Sufficient: $p\to q$
• Necessary: $p\leftarrow q$
• Raises questions about uncaused events (e.g., Big Bang) and infinite regress

Note: If any one of the four laws is broken, the argument is “invalid”

Statements/Propositions/Claims

• A sentence with a truth value (true/false)
• Atomic Proposition: Simplified statement/proposition
  • Subject + Predicate
    • Linking item: affirmation/negation
  • Universal (all S are P) vs particular (some S are P) proposition

Propositional Logic Classification

Syllogism (三段式) - Aristotle

A syllogism consists of:

• Major premise (e.g., “All humans are mortal”)
• Minor premise (e.g., “I am a human”)
• Conclusion (e.g., “I am mortal”)

Terms:

• “I”: Subject/major term (S)
• “mortal”: Predicate/minor term (P)
• “human”: middle term (M)

Example Formalized:

M A P    (All humans (M) are mortal (P))
S I M    (I (S) am a human (M))
-----
S I P    (I (S) am mortal (P))

4 Figures of Syllogism

Figure 1:

M   P
S   M
-----
S   P

Figure 2:

P   M
S   M
-----
S   P

Figure 3:

M   P
M   S
-----
S   P

Figure 4:

P   M
M   S
-----
S   P

• Out of 256 possible specific figures, only 15 are valid

Compound Propositions

• Compound propositions are formed by linking 2 or more atomic propositions.

Types of Compound Propositions

• Conjunctive Proposition (AND): $p \land q$
• Disjunctive Proposition (OR): $p \lor q$ (either A or B)
• Negation (NOT): $\lnot p$
• Conditional/Hypothetical Proposition (IF…THEN): $p \to q$
  • Modus Ponens (肯定前件式): Affirming the antecedent
    1. If A, then B: $A \to B$
    2. A: $A$
    3. Therefore B: $B$
  • Modus Tollens (否定后件式): Denying the consequent
    1. If A, then B: $A \to B$
    2. Not B: $\lnot B$
    3. Therefore not A: $\lnot A$
       • Equivalent form: $(p\to q)\equiv(\lnot q\to\lnot p)$

Variables

• Propositional variables: p, q, r, s…
• Used to represent compound statements abstractly

Deduction Rules for Compound Propositions

• Hypothetical syllogism: $(p\to q, q\to r) \equiv (p \to r)$
• Disjunctive elimination: $(p\lor q, p\to r, q\to s) \equiv (r\lor s)$

De Morgan’s Laws

• $\lnot(p\land q) \Leftrightarrow (\lnot p\lor\lnot q)$
• $\lnot(p\lor q) \Leftrightarrow (\lnot p\land\lnot q)$

Note: In this context, $\lor$ represents inclusive OR, not XOR