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logic skill

/.claude/skills/logic

This skill helps you analyze arguments and detect fallacies using formal and informal logic to assess validity, soundness, and deductive structures.

npx playbooks add skill chrislemke/stoffy --skill logic

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---
name: logic
description: "Master formal and informal logic - valid reasoning, fallacies, paradoxes. Use for: argument analysis, validity, soundness, logical form, fallacies, paradoxes. Triggers: 'valid', 'validity', 'fallacy', 'premise', 'conclusion', 'modus ponens', 'modus tollens', 'paradox', 'logical', 'syllogism', 'deduction', 'induction', 'inference', 'argument form', 'soundness', 'entailment', 'contradiction'."
---

# 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

Overview

This skill teaches formal and informal logic so you can evaluate reasoning, detect fallacies, and analyze paradoxes. It covers propositional and predicate logic, valid argument forms, common informal fallacies, modal operators, and protocols for structured argument analysis. Use it to improve clarity, rigor, and argumentative reliability.

How this skill works

The skill inspects argument structure by identifying premises, conclusions, and hidden assumptions, then tests validity using standard inference rules (e.g., modus ponens, modus tollens, universal instantiation). It flags informal fallacies (ad hominem, straw man, false dilemma, equivocation) and analyzes paradoxes (liar, sorites, Russell) with available logical responses. It also supports modal and quantifier reasoning to evaluate necessity, possibility, and existence claims.

When to use it

  • Assess whether conclusions truly follow from premises (validity checks)
  • Evaluate whether premises are true and the argument is sound
  • Detect and label informal fallacies in essays, debates, or drafts
  • Translate natural-language arguments into logical form (propositional or predicate)
  • Analyze paradoxes and compare formal responses (type theory, paraconsistent logic, contextualism)

Best practices

  • Start by stating the conclusion succinctly, then list premises explicitly
  • Translate ambiguous natural language into clear logical form before testing validity
  • Look for hidden premises and make them explicit for evaluation
  • Use formal inference rules for deductive reasoning and probability/induction rules for inductive claims
  • When diagnosing fallacies, explain why the move is mistaken and offer a corrected form

Example use cases

  • Proofread an op-ed to identify equivocation and false dilemmas
  • Convert a multi-premise policy argument into predicate logic to test validity
  • Explain why an argument is valid but unsound, and suggest premise revisions
  • Analyze a reported paradox and outline formal responses (e.g., supervaluationism vs paraconsistent logic)
  • Teach students common valid forms (modus ponens, hypothetical syllogism) with examples

FAQ

How do I tell validity from soundness?

Validity means the conclusion must follow from the premises; soundness means the argument is valid and its premises are actually true. A valid argument can be unsound if premises are false.

What if an argument uses vague language?

Translate vague terms into explicit predicates or note multiple precisifications. Use supervaluation or degree approaches when appropriate, and flag equivocation as a fallacy.