---
title: "Logic, Reason and Argument: university philosophy revision"
description: "Revise university logic: validity and soundness, informal fallacies, categorical syllogisms, propositional and predicate logic, natural deduction proofs, and inductive and probabilistic reasoning."
image: "https://www.kerfox.app/og.webp"
url: "https://www.kerfox.app/university/philosophy/logic-reason-and-argument"
---
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# Logic, Reason and Argument

Validity, fallacies, symbols and proof.

## What the course covers

Logic studies one relation: whether some claims, if true, would force another. The course builds the tools for testing it. First the vocabulary, validity, soundness and form, and the ways ordinary language hides all three. Then the fallacies, which are the patterns that persuade without supporting. Then two formal systems: categorical logic, which is Aristotle’s and still the fastest way to see a syllogism fail, and propositional and predicate logic, which is what a modern paper is set in. Truth tables decide, proofs derive, and models refute. It ends where certainty does, with induction and probability.

### What an argument is

1. What an argument is. Reasons offered, not a quarrel.
2. Finding the conclusion. Which claim is being supported.
3. What is not an argument. Explanations, reports, conditionals.
4. Validity. The premises guarantee it, or not.
5. Valid is not true. The four combinations that matter.
6. Soundness. Valid, and the premises are true.
7. Form is everything. Why the content drops out.
8. Forms worth knowing. Five patterns you will reuse.
9. Counterexamples. Same form, true premises, false end.
10. Strength and cogency. Good arguments that are not valid.
11. Reconstructing arguments. Missing premises, fairly supplied.
12. What an argument is review. Everything so far, tested.

### Logic and language

1. Saying and suggesting. What a sentence makes you feel.
2. Ambiguity. One sentence, two readings.
3. Vagueness. Borderline cases with no line.
4. Kinds of definition. Five jobs a definition can do.
5. Good and bad definitions. Too broad, too narrow, circular.
6. Definitions in arguments. Winning by fixing the terms.
7. Logic and language review. Words, and the trouble they cause.

### Informal fallacies

1. What a fallacy is. Persuades without supporting.
2. Attacking the person. The arguer is not the argument.
3. The straw man. Refuting what nobody said.
4. Appeals to feeling. Fear, pity, and what the crowd says.
5. The red herring. Changing the subject on purpose.
6. Authority and ignorance. Who said it, and what nobody knows.
7. Fallacies of ambiguity. One word quietly doing two jobs.
8. Smuggled assumptions. Assuming what was to be proved.
9. Weak inductive fallacies. Too few cases, and false causes.
10. Informal fallacies review. Naming exactly what went wrong.

### Categorical statements

1. The four forms. A, E, I and O, and no more.
2. Quantity and quality. The two dials that make four.
3. Distribution. When a term covers every member.
4. Venn diagrams. Two circles, shading and a bar.
5. The square of opposition. How four forms constrain each other.
6. Existential import. Whether nothing counts as all.
7. Immediate inferences. Conversion, obversion, contraposition.
8. Into standard form. Getting English to behave.
9. Categorical statements review. Four forms and their relations.

### Categorical syllogisms

1. Standard form. Major, minor, middle, mood, figure.
2. The Venn test. Three circles settle the question.
3. The rules. Five checks, five named failures.
4. Syllogisms with gaps. Supplying the missing premise.
5. Chains of syllogisms. Several joined end to end.
6. Categorical syllogisms review. Testing a syllogism two ways.

### Propositional logic

1. Why symbols. What eyeballing keeps getting wrong.
2. Simple and compound. Atoms, and what joins them.
3. The first three connectives. Not, and, or, with their tables.
4. Two kinds of or. The word logic has to pin down.
5. The conditional. If, and the table nobody likes.
6. The material conditional. Where the table stops matching.
7. The biconditional. If and only if, both ways.
8. Scope and grouping. Brackets, and what is in charge.
9. Other notations. The symbols your textbook uses.
10. Building a truth table. Rows, columns and discipline.
11. Three verdicts. Always, never, or sometimes true.
12. Logical equivalence. Same column, same meaning.
13. Testing arguments. Hunting for the one bad row.
14. The short way. Assume invalid, then try to build it.
15. Propositional logic review. Symbols, tables and verdicts.

### Proofs

1. Why proofs. Sixteen rows is fine, a thousand is not.
2. Modus ponens and tollens. The two rules about conditionals.
3. Chaining and eliminating. Hypothetical and disjunctive syllogism.
4. Four more rules. Splitting, joining, weakening, choosing.
5. Rules apply to whole lines. The mistake everybody makes once.
6. Finding the proof. Work backwards from the conclusion.
7. Equivalence rules. Five ways to rewrite a formula.
8. Five more equivalences. Implication, exportation and the rest.
9. Conditional proof. Assume, derive, then discharge.
10. Reductio ad absurdum. Assume the opposite and break it.
11. Proofs review. Rules, strategy and assumptions.

### Predicate logic

1. Why letters are not enough. The argument tables cannot touch.
2. Names and predicates. Looking inside a statement.
3. The universal quantifier. All, and why it takes an arrow.
4. The existential quantifier. Some, and why it takes an and.
5. Two standard mistakes. The wrong connective, every time.
6. The four forms again. A, E, I and O, symbolised.
7. Scope and binding. Free variables and bound ones.
8. Relations. Predicates with two places.
9. Order matters. Everyone loves someone, two ways.
10. Interpretations. Building a world to break it.
11. Quantifier rules. Stripping and restoring quantifiers.
12. Identity and descriptions. Only, at most one, and the.
13. Predicate logic review. Quantifiers, models and identity.

### Inductive reasoning

1. Strength, not validity. Support that comes in degrees.
2. Statistical syllogism. From most of them to this one.
3. Generalisation. From this sample to all of them.
4. Argument from analogy. Alike here, so alike there.
5. Correlation and cause. Four things a correlation can mean.
6. Mill’s methods. Five ways to isolate a cause.
7. Best explanation. The hypothesis that would explain it.
8. Inductive reasoning review. Evidence that does not compel.

### Probability

1. What probability is. Three ways to read a number.
2. The axioms. Between nought and one, always.
3. And. Multiplying, and when you cannot.
4. Or. Adding, minus the overlap.
5. Conditional probability. Given that, what are the odds.
6. Bayes theorem. Updating a belief on evidence.
7. Base rates. Why the accurate test misleads.
8. Probability fallacies. Conjunction, gambler and the rest.
9. Expected value. What a bet is actually worth.
10. Probability review. Rules, updating and traps.

## One lesson in full

**Valid is not true.** The four combinations that matter.

All fish are birds. All birds have wheels. So all fish have wheels. Every statement in that is false and the reasoning is perfect. Validity survives being wrong about absolutely everything.

### Two different words

Statements are true or false. Arguments are valid or invalid. Calling an argument true, or a statement valid, is not a slip of vocabulary but a sign that the distinction has not landed.

### One forbidden cell

Four combinations of premise truth and conclusion truth are conceivable. Validity rules out exactly one of them. That is why a valid argument may have false premises, a false conclusion, or both.

### Right answer, bad argument

An invalid argument can land on a true conclusion by pure luck. Grass is green, therefore two plus two is four. Both statements are true and the second owes nothing whatever to the first.

### Why logic accepts this

Because it buys generality. A test that ignored content works on any subject matter at all, which is why the same rules serve mathematics, law and physics without being rewritten for each.

## A question to try

All fish are birds. All birds have wheels. So all fish have wheels. This argument is:

1. invalid but sound
2. invalid
3. valid but unsound
4. sound

**The answer, and whyThe answer: valid but unsound.**

Every statement in it is false and the reasoning is perfect, which is exactly what validity permits.

- invalid but sound. Nothing can be sound without being valid, since soundness is validity plus true premises.
- invalid. You judged the argument by the truth of its statements. The link is faultless, and that is what validity is about.
- sound. Soundness also needs true premises, and these are plainly false.

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