Logic Skill

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

Fundamentals

Basic Concepts

Term Definition

Argument Premises + Conclusion

Premise Statement offered as support

Conclusion Statement being supported

Valid Conclusion follows from premises

Sound Valid + true premises

Cogent Strong inductive + true premises

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

Symbol Name Meaning

¬ Negation Not P

∧ Conjunction P and Q

∨ Disjunction P or Q

→ Conditional If P then Q

↔ Biconditional P iff Q

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

Symbol Name Meaning

∀x Universal For all x

∃x Existential There exists x

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

Fallacy Description Example

Ad hominem Attack the person "You're wrong because you're stupid"

Appeal to authority Irrelevant authority "A celebrity says X"

Appeal to emotion Manipulate feelings Fear-mongering

Red herring Change subject Diverting attention

Straw man Misrepresent argument Attack weaker version

Fallacies of Presumption

Fallacy Description Example

Begging the question Assume conclusion Circular reasoning

False dilemma Only two options "With us or against us"

Hasty generalization Small sample "Two Xs did Y, so all Xs"

Slippery slope Unsupported chain "A leads to Z inevitably"

Fallacies of Ambiguity

Fallacy Description Example

Equivocation Shifting meaning "Light" (weight/illumination)

Amphiboly Grammatical ambiguity Headlines

Composition Parts → whole "Atoms invisible ∴ tables invisible"

Division Whole → parts "Team good ∴ each player good"

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

Symbol Meaning

□P Necessarily P

◊P Possibly P

Relations

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

Systems

System Characteristic Axiom

K Basic modal logic

T □P → P (Necessity implies truth)

S4 □P → □□P (Iterated necessity)

S5 ◊P → □◊P (Possibility is necessary)

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

Term Meaning

Entailment P logically implies Q

Tautology True under all interpretations

Contradiction False under all interpretations

Contingent Neither tautology nor contradiction

Consistent Can all be true together

Inference Moving from premises to conclusion

Deduction Conclusion follows necessarily

Induction Conclusion follows probably

Integration with Repository

Related Skills

• argument-mapping : Visualizing argument structure

• thought-experiments : Logical analysis of scenarios