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