Logic
The formal structure of reasoning: how to determine whether conclusions actually follow from premises.
Why Logic Matters
Logic is the scaffolding of all reasoning. Without it, you can't tell whether a conclusion follows from its premises. You're just guessing.
You don't need to be a logician. But understanding the basics lets you:
- Spot when someone's reasoning is broken regardless of how persuasive they sound
- Construct arguments that actually hold up
- Distinguish between "this feels right" and "this is right"
Three Types of Reasoning
| Type | Direction | Strength | Example |
|---|---|---|---|
| Deductive | General → Specific | Certain (if valid) | All mammals breathe air. Dogs are mammals. Therefore dogs breathe air. |
| Inductive | Specific → General | Probable | Every dog I've seen breathes air. Therefore all dogs probably breathe air. |
| Abductive | Observation → Best Explanation | Plausible | The grass is wet. The best explanation is that it rained. |
Deductive Reasoning
Starts with general principles and derives specific conclusions. If the premises are true and the logic is valid, the conclusion must be true.
Structure:
Premise 1: All A are B
Premise 2: C is A
Conclusion: Therefore, C is B
Strength: Guarantees truth (if premises are true and form is valid). Weakness: Can only produce conclusions already contained in the premises. Only as good as your starting premises.
Everyday examples:
- "The store closes at 9. It's 9:30. Therefore the store is closed."
- "You need a passport to travel internationally. I don't have a passport. Therefore I can't travel internationally."
Inductive Reasoning
Starts with specific observations and draws general conclusions. The conclusion goes beyond what the premises guarantee.
Structure:
Observation 1: Swan 1 is white
Observation 2: Swan 2 is white
...
Observation 1000: Swan 1000 is white
Conclusion: All swans are probably white
Strength: Generates new knowledge; the basis of scientific discovery. Weakness: Never 100% certain. One black swan disproves the conclusion.
Strong vs. Weak Induction:
| Strong Induction | Weak Induction |
|---|---|
| Large, representative sample | Small or biased sample |
| Consistent observations | Mixed results |
| No known counterexamples | Known exceptions ignored |
| Conclusion is modest | Conclusion overstates the evidence |
Abductive Reasoning
Starts with an observation and infers the most likely explanation. Also called "inference to the best explanation."
Structure:
Observation: The grass is wet
Possible explanations: Rain, sprinklers, dew, flooding
Best explanation: It rained (given the time, weather, scope)
Conclusion: It probably rained
Strength: How we actually reason most of the time. Essential for diagnosis, detective work, and everyday problem-solving. Weakness: "Best available explanation" isn't the same as "correct explanation." Better explanations may emerge.
Criteria for the "best" explanation:
| Criterion | Meaning |
|---|---|
| Simplicity | Fewer assumptions (Occam's Razor) |
| Explanatory scope | Accounts for more observations |
| Consistency | Fits with existing knowledge |
| Testability | Makes predictions that can be checked |
| Fruitfulness | Leads to new insights or predictions |
Validity and Soundness
Two critical concepts that most people confuse.
| Concept | Definition | Requires True Premises? |
|---|---|---|
| Valid | The conclusion follows logically from the premises | No |
| Sound | Valid AND all premises are true | Yes |
Valid but NOT Sound
Premise 1: All fish can fly (FALSE)
Premise 2: A salmon is a fish (TRUE)
Conclusion: A salmon can fly (LOGICALLY FOLLOWS, but false)
The logic is perfect. The conclusion follows. But the first premise is false, so the argument is unsound.
Sound
Premise 1: All humans are mortal (TRUE)
Premise 2: Socrates is a human (TRUE)
Conclusion: Socrates is mortal (TRUE and logically follows)
Invalid (Even with True Premises)
Premise 1: Some birds can fly (TRUE)
Premise 2: Penguins are birds (TRUE)
Conclusion: Penguins can fly (DOESN'T FOLLOW)
Key insight: An argument can have true premises and a true conclusion and STILL be invalid, if the conclusion doesn't actually follow from the premises.
Common Argument Structures
Modus Ponens (Affirming the Antecedent)
The most fundamental valid argument form.
If P, then Q (If it rains, the ground gets wet)
P (It's raining)
Therefore, Q (Therefore, the ground is wet)
Valid. Always.
Modus Tollens (Denying the Consequent)
If P, then Q (If it rains, the ground gets wet)
Not Q (The ground is not wet)
Therefore, not P (Therefore, it didn't rain)
Valid. Always. This is the basis of falsification in science.
Hypothetical Syllogism (Chain Reasoning)
If P, then Q (If I study, I'll pass)
If Q, then R (If I pass, I'll graduate)
Therefore, if P, then R (If I study, I'll graduate)
Valid. You can chain as many links as needed.
Disjunctive Syllogism (Process of Elimination)
P or Q (Either the battery is dead or the starter is broken)
Not P (The battery is not dead)
Therefore, Q (Therefore, the starter is broken)
Valid. But only when the "or" is genuinely exhaustive, meaning there aren't other options.
Common INVALID Forms
| Name | Structure | Why It's Invalid |
|---|---|---|
| Affirming the consequent | If P then Q; Q; therefore P | Other things could cause Q |
| Denying the antecedent | If P then Q; not P; therefore not Q | Q might still be true for other reasons |
Affirming the consequent example:
If it rains, the ground gets wet.
The ground is wet.
Therefore, it rained. ← INVALID (sprinklers could have caused it)
Denying the antecedent example:
If it rains, the ground gets wet.
It didn't rain.
Therefore, the ground isn't wet. ← INVALID (sprinklers exist)
Truth Tables
Truth tables show all possible combinations of truth values for logical operations.
Basic Operators
| P | Q | P AND Q | P OR Q | NOT P | IF P THEN Q |
|---|---|---|---|---|---|
| T | T | T | T | F | T |
| T | F | F | T | F | F |
| F | T | F | T | T | T |
| F | F | F | F | T | T |
Key Insights from Truth Tables
"If P then Q" is only false when P is true and Q is false. This confuses people. "If it rains, I'll bring an umbrella" is not violated by bringing an umbrella on a sunny day.
| Statement | What Makes It FALSE |
|---|---|
| P AND Q | Either P or Q is false |
| P OR Q | Both P and Q are false |
| IF P THEN Q | P is true but Q is false |
| P IF AND ONLY IF Q | P and Q have different truth values |
Necessary vs. Sufficient Conditions
| Condition Type | Definition | Example |
|---|---|---|
| Necessary | Must be true for the conclusion (but alone isn't enough) | Oxygen is necessary for fire |
| Sufficient | If true, guarantees the conclusion | Being a dog is sufficient for being a mammal |
| Necessary AND sufficient | The exact condition, no more, no less | Being a bachelor is necessary and sufficient for being an unmarried adult male |
How to Tell Them Apart
| Test | Question | Example |
|---|---|---|
| Necessary | "Can the conclusion be true WITHOUT this?" If no → necessary | Can you have fire without oxygen? No → oxygen is necessary |
| Sufficient | "Does this ALONE guarantee the conclusion?" If yes → sufficient | Does being a triangle guarantee having 3 sides? Yes → sufficient |
Practical Importance
Confusing necessary and sufficient conditions is one of the most common reasoning errors:
| Confusion | Example | Error |
|---|---|---|
| Treating necessary as sufficient | "Hard work is necessary for success, so if I work hard I'll succeed" | Hard work is necessary but not sufficient. Luck, talent, timing matter too |
| Treating sufficient as necessary | "A degree gets you a good job, so you need a degree for a good job" | A degree is sufficient but not necessary. Other paths exist |
Logical Equivalences
Some statements say the same thing in different ways.
| Statement | Logically Equivalent To |
|---|---|
| If P then Q | If not Q then not P (contrapositive) |
| Not (P and Q) | Not P or not Q (De Morgan's) |
| Not (P or Q) | Not P and not Q (De Morgan's) |
The contrapositive is always valid:
- "If it rains, the ground gets wet" = "If the ground isn't wet, it didn't rain"
The converse is NOT equivalent:
- "If it rains, the ground gets wet" ≠ "If the ground is wet, it rained"
The inverse is NOT equivalent:
- "If it rains, the ground gets wet" ≠ "If it doesn't rain, the ground isn't wet"
Quantifiers
Words like "all," "some," "none," and "most" change the logic of statements dramatically.
| Quantifier | To Disprove, You Need | Example |
|---|---|---|
| All X are Y | One X that isn't Y | "All politicians lie" → One honest politician disproves this |
| No X are Y | One X that is Y | "No exercise is fun" → One fun exercise disproves this |
| Some X are Y | Show that no X is Y | "Some dogs are aggressive" → Prove zero dogs are aggressive |
| Most X are Y | Show that half or fewer X are Y | "Most startups fail" → Show 50%+ succeed |
Common error: Treating "some" as "all" or "most."
"Some studies show X" does NOT mean "studies show X." It means at least one study showed X, which might be an outlier.
Putting It Together: Evaluating Real Arguments
Step-by-Step Process
- Identify the conclusion. What's being claimed?
- Identify the premises. What reasons are given?
- Check the form. Is the argument structure valid?
- Check the premises. Are the premises actually true?
- Check for hidden premises. What's assumed but not stated?
- Assess strength. How strong is the overall argument?
Worked Example
Argument: "Renewable energy is more expensive than fossil fuels. We shouldn't switch to things that cost more. Therefore we shouldn't switch to renewable energy."
| Step | Analysis |
|---|---|
| Conclusion | We shouldn't switch to renewable energy |
| Premise 1 | Renewable energy is more expensive than fossil fuels |
| Premise 2 | We shouldn't switch to things that cost more |
| Valid form? | Yes (modus ponens structure) |
| Premise 1 true? | Increasingly false: renewables are now cheaper in many cases |
| Premise 2 true? | Too simplistic: ignores externalities, long-term costs, other values |
| Hidden premises | Cost is the only relevant factor; current costs are permanent |
| Assessment | Weak argument: both premises are questionable and important factors are ignored |
Key Takeaways
- Validity ≠ Truth. An argument can be logically perfect with false premises
- Soundness is what matters. Valid logic + true premises = reliable conclusion
- Most reasoning is inductive or abductive. Deductive certainty is rare outside math
- Learn the common forms. Modus ponens, modus tollens, and the common invalid forms cover most practical cases
- Necessary ≠ sufficient. Confusing these causes enormous errors in everyday reasoning
- The contrapositive is your friend. It's always logically equivalent and often more intuitive
- Watch the quantifiers. "Some," "most," and "all" are very different claims