🧠 Deductive vs. Inductive Arguments
| Type | Support Type | Goal | Example |
|---|
| Deductive | Conclusive | Guarantee truth of conclusion | “All humans are mortal; Socrates is human → Socrates is mortal.” |
| Inductive | Probabilistic | Make conclusion likely | “All 100 squirrels seen are grey → next one will likely be grey.” |
⚙️ Key Distinctions
- Deductive Validity: Impossible for premises to be true and conclusion false.
→ If true premises, conclusion must be true.
- Inductive Strength: Unlikely for premises to be true and conclusion false.
→ Premises make conclusion probable.
✅ Evaluating Deductive Arguments
| Quality | Meaning |
|---|
| Valid | Premises → Conclusion necessarily |
| Invalid | Possible for premises true, conclusion false |
| Sound | Valid + Premises are true |
Example:
“All wines are beverages. Chardonnay is a wine → Chardonnay is a beverage.” ✅ Sound.
“Ginger ales are white wines; Vernors is ginger ale → white wine.” ❌ Invalid content – unsound premise.
🔍 Evaluating Inductive Arguments
| Quality | Meaning |
|---|
| Strong | Premises render conclusion probable |
| Weak | Premises unrelated or insufficient |
| Cogent | Strong + true premises |
| Uncogent | Weak or containing false premise |
Example:
- “Most wines made from grapes. Chardonnay is wine → likely made from grapes.” = Cogent.
- “Most ginger ales are white wines…” = Strong but false premise → Uncogent.
🧮 Sample Evaluations
- Invalid Deduction: “It rains or snows; it rains; so it doesn’t snow.” ❌
- Weak Induction: “4 of 5 dentists prefer Crest → most doctors prefer Crest.” ❌ (wrong population)
- Strong Induction: “120/135 engineering students favor longer breaks → most engineers do.” ✅
- Weak Induction (Overreach): “120/135 engineers → most students on campus.” ❌
- Strong + Cogent: Vaccine trial reported with margin of error and representative groups ✅
⚖️ Rules of Evaluation Summary
| Argument Type | Strong/Valid When | Weak/Invalid When |
|---|
| Deductive | Conclusion necessarily follows | Possible to be false with true premises |
| Inductive | Premises make conclusion likely | Insufficient evidence or false assumption |
🧩 Terms: Soundness vs. Cogency
| Deductive | Inductive |
|---|
| Valid + True Premises → Sound | Strong + True Premises → Cogent |
📦 Examples of Sound vs. Unsound
| Argument | Type | Evaluation |
|---|
| “All wines are beverages; Chardonnay is wine → beverage.” | Deductive | Sound |
| “All wines are beverages; Chardonnay is beverage → wine.” | Deductive | Unsound (Invalid form) |
| “Most wines from grapes; Chardonnay is wine → likely from grapes.” | Inductive | Cogent |
📊 Inductive Generalizations
Inductive generalizations reason from a sample to a population.
✅ Criteria for a Strong Generalization
- Representative Sample — reflects population demographics.
- Adequate Size — large enough to minimize error.
- Low Margin of Error (M.E.)
- Appropriate Confidence Level (CL) — usually 95%.
📉 Statistical Essentials
- Margin of Error (M.E.)
- Radius of confidence interval (how far result may vary).
- Depends on sample size (n) and confidence level.
- Larger sample → smaller M.E.
| Confidence Level | Formula |
|---|
| 99% | 1.29/n |
| 95% | 0.98/n |
| 90% | 0.82/n |
Example:
n = 1000, 95% CL → M.E = 0.98/31.62 ≈ 0.03 (±3%)
📏 Sample Size Guidelines (95% CL, Pop > 15,000)
| M.E. ± | Sample Size |
|---|
| 2% | ≈ 2000 |
| 3% | ≈ 1000 |
| 4% | ≈ 600 |
🧮 Interpreting Results – Examples
Example 1
Poll: 2000 students, 51% liberal (±2%, 95% CL)
- “Majority are liberals.” ❌ Weak (within ME)
- “49% ± 2% not liberal.” ✅ Strong
- “Many/some are liberals.” ✅ Strong (safe inference)
Example 2
Poll: 600 students, 54% study daily (±4%)
- “Most study daily.” ❌ Weak (may dip below 50%)
- “At least half study daily.” ✅ Strong
- “1/3 study 2 days/week.” ✅ Strong
- “Most have high GPA.” ❌ Irrelevant (not linked)
💬 Summary of Strong Inductive Generalization
- Sample → large & representative
- Error → small; confidence → high
- Conclusion matches data (no overreach)
🧩 Evaluating Analogical Arguments
💡 Definition
An analogy compares two things (A & B) that share features (X) and infers they probably share another feature (Y).
Form:
- A is like B in respect to X
- B has Y
- Things with X tend to have Y
- ⇒ A likely has Y
✅ Criteria for a Strong Analogy
- A truly has property X
- B truly has property Y
- X is reliably connected to Y
- No relevant dissimilarities between A and B
🧮 Examples
Example 1 — Government vs Family Budget
- Both have credit lines (X)
- Families → can go bankrupt (Y)
- ∴ Governments can too
✅ Shared traits true, though scale differs → moderate analogy
Example 2 — Thomson’s Violinist (Ethical Analogy)
- Woman involuntarily pregnant ≈ person involuntarily attached to violinist
- Person may detach violinist ⇒ abortion justified
⚖️ Evaluation:
- Similarity holds (both sustain another’s life)
- Disanalogy → mother’s emotional relationship with fetus
- Outcome → partially strong, morally debatable
Example 3 — Computers & Humans
- Both solve math problems (X)
- Humans think (Y)
- So computers think → ❌ weak analogy (oversimplifies)
Example 4 — Watchmaker Analogy (Paley)
- Universe & watch = designed (X)
- Watches → have purpose & maker (Y)
- ∴ Universe → has maker
⚖️ Weak: assumes design implies maker; natural phenomena show pattern w/o intent.
🧾 Diagnosing Analogies
| Question | If No → Weak Analogy |
|---|
| Do both items really share stated feature X? | ✅ |
| Is Y actually present in B? | ✅ |
| Is connection between X and Y trustworthy? | ✅ |
| Are there relevant disanalogies? | ✅ |
🪶 Summary: Analogical vs. Inductive Generalization
| Type | Evidence Base | Conclusion Type | Common Weakness |
|---|
| Generalization | Observed sample | Likely population trend | Biased / small sample |
| Analogy | Similar cases | Likely shared property | Irrelevant or weak similarity |
🧾 Combined Quick Reference
| Concept | Rule |
|---|
| Deductive Validity | If premises true → conclusion must be true |
| Inductive Strength | Evidence raises probability of conclusion |
| Soundness | Valid + true premises |
| Cogency | Strong + true premises |
| Margin of Error (95%) | 0.98/n |
| Analogical Criteria | Shared X, verified Y, reliable link, minimal disanalogy |
🧩 Takeaways
- Deductive arguments aim for certainty → test structure.
- Inductive arguments aim for probability → evaluate evidence quality.
- Sound = perfect deduction; Cogent = strong induction.
- Generalizations need representative data; Analogies need relevant similarity.
- Strong reasoning in logic depends on clarity, precision, and proportional conclusions.