logical-reasoning

# Logical Reasoning

Logic is the study of what follows from what. A valid logical argument has a form that preserves truth — if the premises are true, the conclusion cannot be false. This skill covers the core machinery of deductive and inductive reasoning: the rules of inference, the standard argument forms, the common errors, and the boundary between the two styles of reasoning.

**Agent affinity:** paul (chair-level framing), elder (inference-pattern drills), tversky (inductive strength)

**Concept IDs:** crit-deductive-reasoning, crit-inductive-reasoning, crit-argument-structure

## The Reasoning Toolbox at a Glance

| # | Pattern | Form | Type |
|---|---|---|---|
| 1 | Modus ponens | If P then Q; P; therefore Q | Deductive, valid |
| 2 | Modus tollens | If P then Q; not Q; therefore not P | Deductive, valid |
| 3 | Hypothetical syllogism | If P then Q; if Q then R; therefore if P then R | Deductive, valid |
| 4 | Disjunctive syllogism | P or Q; not P; therefore Q | Deductive, valid |
| 5 | Constructive dilemma | (P or Q); (if P then R); (if Q then S); therefore (R or S) | Deductive, valid |
| 6 | Universal instantiation | All A are B; x is an A; therefore x is a B | Deductive, valid |
| 7 | Existential generalization | a has property P; therefore something has property P | Deductive, valid |
| 8 | Affirming the consequent | If P then Q; Q; therefore P | Deductive, INVALID |
| 9 | Denying the antecedent | If P then Q; not P; therefore not Q | Deductive, INVALID |
| 10 | Enumerative induction | Every observed A has been B; therefore all A are B | Inductive, probable |
| 11 | Statistical generalization | n% of sampled A are B; therefore about n% of all A are B | Inductive, probable |
| 12 | Inference to the best explanation | H explains the observations better than alternatives; therefore H | Inductive, probable |
| 13 | Analogical reasoning | A and B share features F1..Fn; A has Fm; therefore B has Fm | Inductive, probable |

## Deductive Reasoning: Form Preserves Truth

A deductive argument is valid when the premises entail the conclusion. If the form is valid and all premises are true, the conclusion must be true. This is the only style of reasoning that guarantees its conclusions.

### Pattern 1 — Modus Ponens

**Form:** If P then Q. P. Therefore Q.

**Worked example.**
```
P1. If it is raining, then the street is wet.
P2. It is raining.
C.  The street is wet.
```

Modus ponens is the engine of deductive reasoning. Most chained arguments reduce to sequences of modus ponens applications.

### Pattern 2 — Modus Tollens

**Form:** If P then Q. Not Q. Therefore not P.

**Worked example.**
```
P1. If the theory is correct, the experiment will show effect X.
P2. The experiment did not show effect X.
C.  The theory is not correct (or is incomplete).
```

Modus tollens is the engine of scientific falsification. Popper built his philosophy of science on it.

### Pattern 3 — Hypothetical Syllogism (Chain Rule)

**Form:** If P then Q. If Q then R. Therefore if P then R.

**Worked example.**
```
P1. If interest rates rise, then borrowing becomes more expensive.
P2. If borrowing becomes more expensive, then business investment slows.
C.  If interest rates rise, then business investment slows.
```

Long conditional chains can be built by repeated hypothetical syllogism. Breaks in the chain (any false sub-implication) invalidate the whole.

### Pattern 4 — Disjunctive Syllogism

**Form:** P or Q. Not P. Therefore Q.

**Worked example.**
```
P1. The problem is either the cable or the router.
P2. We tested the cable and it works fine.
C.  The problem is the router.
```

**Caution.** Requires that the disjunction be genuinely exhaustive. If the problem could also be the ISP, the modem, or the wall jack, P1 is false and the argument is unsound.

### Pattern 5 — Constructive Dilemma

**Form:** P or Q. If P then R. If Q then S. Therefore R or S.

**Worked example.**
```
P1. Either we cut spending or we raise revenue.
P2. If we cut spending, services will degrade.
P3. If we raise revenue, taxes will increase.
C.  Either services will degrade or taxes will increase.
```

Constructive dilemma shows how binary choices push the consequence forward.

### Pattern 6 — Universal Instantiation

**Form:** All A are B. x is an A. Therefore x is a B.

**Worked example.**
```
P1. All mammals have a four-chambered heart.
P2. A platypus is a mammal.
C.  A platypus has a four-chambered heart.
```

This is the most basic application of quantified reasoning to particular cases.

### Pattern 7 — Existential Generalization

**Form:** a has property P. Therefore there exists something with property P.

**Worked example.**
```
P1. Kepler-452b is an exoplanet.
C.  There exists at least one exoplanet.
```

Existential generalization moves from specific evidence to existence claims. It is logically weaker than universal generalization but is always valid when the premise is.

## Invalid Deductive Forms (Formal Fallacies)

### Fallacy 8 — Affirming the Consequent

**Form:** If P then Q. Q. Therefore P. (INVALID)

**Worked example.**
```
P1. If it is raining, then the street is wet.
P2. The street is wet.
C.  It is raining.  [DOES NOT FOLLOW]
```

The street could be wet because a truck sprayed it, a pipe broke, or the street cleaners came by. Q can hold for reasons other than P.

### Fallacy 9 — Denying the Antecedent

**Form:** If P then Q. Not P. Therefore not Q. (INVALID)

**Worked example.**
```
P1. If you study hard, you will pass the exam.
P2. You did not study hard.
C.  You will not pass the exam.  [DOES NOT FOLLOW]
```

Not studying is not the only route to failing, and studying is not the only route to passing. The conditional tells us what one path to Q looks like; it does not say all paths must go through P.

**The valid/invalid pair.** Modus ponens and modus tollens are valid; affirming the consequent and denying the antecedent are not. Learning these four patterns together — and being able to name each one — is the