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

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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

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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)

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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"

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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

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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)

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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?

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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

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Integration with Repository

Related Skills

• argument-mapping

  : Visualizing argument structure

• thought-experiments

  : Logical analysis of scenarios