Logic

Logic Skill

Master the principles of valid reasoning: formal logic, informal logic, fallacy detection, and paradox analysis.

Fundamentals

Basic Concepts

Validity vs. Soundness

VALIDITY: If premises true, conclusion must be true
          (Logical form preserves truth)

SOUNDNESS: Valid + Actually true premises
           (Guarantees true conclusion)

EXAMPLE:
All cats are mammals.     (True)
All mammals are animals.  (True)
∴ All cats are animals.   (True) → SOUND

All fish are mammals.     (False)
All mammals can fly.      (False)
∴ All fish can fly.       (False) → VALID but not SOUND

Propositional Logic

Connectives

Valid Argument Forms

MODUS PONENS               MODUS TOLLENS
P → Q                      P → Q
P                          ¬Q
─────                      ─────
∴ Q                        ∴ ¬P

HYPOTHETICAL SYLLOGISM     DISJUNCTIVE SYLLOGISM
P → Q                      P ∨ Q
Q → R                      ¬P
─────                      ─────
∴ P → R                    ∴ Q

CONSTRUCTIVE DILEMMA       REDUCTIO AD ABSURDUM
P → Q                      Assume P
R → S                      ...
P ∨ R                      Derive contradiction
─────                      ─────
∴ Q ∨ S                    ∴ ¬P

Invalid Forms (Fallacies)

AFFIRMING THE CONSEQUENT   DENYING THE ANTECEDENT
P → Q                      P → Q
Q                          ¬P
─────                      ─────
∴ P ✗ INVALID              ∴ ¬Q ✗ INVALID

Predicate Logic

Quantifiers

Valid Inferences

UNIVERSAL INSTANTIATION    EXISTENTIAL GENERALIZATION
∀x(Fx)                     Fa
─────                      ─────
∴ Fa                       ∴ ∃x(Fx)

UNIVERSAL GENERALIZATION   EXISTENTIAL INSTANTIATION
(arbitrary a) Fa           ∃x(Fx)
─────                      ─────
∴ ∀x(Fx)                   ∴ Fa (for new constant a)

Informal Fallacies

Fallacies of Relevance

Fallacies of Presumption

Fallacies of Ambiguity

Paradoxes

Liar Paradox

"This sentence is false"

If true → It says it's false → False
If false → It says it's false, which is true → True

RESPONSES:
├── Tarskian hierarchy: No self-reference
├── Paraconsistent logic: Accept contradiction
├── Gapping: Sentence is neither true nor false
└── Contextualism: Truth conditions shift

Sorites Paradox (Heap)

1 grain is not a heap.
If n grains is not a heap, n+1 grains is not a heap.
∴ 1,000,000 grains is not a heap. ✗

RESPONSES:
├── Epistemicism: Sharp boundary, we don't know where
├── Supervaluationism: True under all precisifications
├── Degree theory: "Heap" admits degrees
└── Contextualism: Boundary shifts with context

Russell's Paradox

R = {x : x ∉ x} (Set of all sets not members of themselves)

Is R ∈ R?
If yes → By definition, R ∉ R
If no → By definition, R ∈ R

RESPONSE: Type theory, set-theoretic axioms preventing
          unrestricted comprehension

Modal Logic

Basic Modal Operators

Relations

□P ↔ ¬◊¬P   (Necessary = not possibly not)
◊P ↔ ¬□¬P   (Possible = not necessarily not)

Systems

Argument Analysis Protocol

ANALYZING ARGUMENTS
═══════════════════

1. IDENTIFY CONCLUSION
   What is being argued for?

2. IDENTIFY PREMISES
   What reasons are given?

3. SUPPLY HIDDEN PREMISES
   What's assumed but not stated?

4. EVALUATE VALIDITY
   Does conclusion follow?

5. EVALUATE SOUNDNESS
   Are premises true?

6. CHECK FOR FALLACIES
   Any reasoning errors?

Key Vocabulary

Integration with Repository

Related Skills

• argument-mapping: Visualizing argument structure
• thought-experiments: Logical analysis of scenarios