Problem-solving recipes

A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.

What are you being asked to do?

Start with what you've been handed and what you're asked to produce.

A paragraph of everyday reasoning, and you need to see its argument — Find the conclusion of a short passage for a few sentences. For a longer passage, Reconstruct and diagram a passage numbers every claim and draws how they support one another.

An everyday argument, and you need a verdict on it — Evaluate an argument with three questions is the first stop: are the premises true, do they support the conclusion, and how strongly? Then pick the tool for the part in doubt:

The pattern looks like a named form → Identify the form of an everyday argument.
You suspect it is invalid and want a check anyone can follow → Build a counterexample by substitution.
The form looks fine but something still feels wrong → Check a suspect argument for an informal fallacy. Then Repair a fallacious argument if the point is worth saving.
It is about groups ("all," "no," "some") → Put a statement in standard form. Then check the syllogism with the five rules or with a Venn diagram (each checks the other).
You want a fully symbolic verdict → Take an English argument all the way to a verdict, which chains the symbolic recipes below.

An English sentence to put into symbols — Symbolize an if-then sentence for a single conditional. Translate an English sentence into symbols for anything with "only if," "unless," "neither," or several joining words. Symbolize a whole argument when every premise and the conclusion must share one key.

A formula, and you need its truth value or its type — Find the main connective first. Then compute its value on one row, build its complete truth table, or classify it as a tautology, contradiction, or contingency. Check a prediction with a partial truth table if you already have a guess.

Two formulas: are they equivalent? — Check whether two sentences are equivalent with a table for a yes/no verdict. Build a chain of equivalences when you must show it step by step with named laws. Read a normal form off a truth table when you need a DNF or CNF.

A symbolized argument: is it valid? — Few letters → Test an argument with a truth table. Many letters, or a conclusion that is easy to make false → Test validity with the short method. Both hand you a countermodel if the argument is invalid.

A symbolized argument you believe is valid, and you must prove it — Plan a natural deduction proof reads the goal's main connective and picks the rule. If the goal is a conditional, Prove an if–then with a subproof. If nothing direct works, Prove a sentence by indirect proof. Check or write one line of a proof whenever you doubt a single step.

A truth-teller-and-liar puzzle — Solve a knights-and-knaves puzzle.

Finding the argument in a passage

Find the conclusion of a short passage

Use when: a passage gives reasons for something and you need to say which claim is being supported

Goal: the conclusion, its premises, and any sentences that play no part

Indicator words

Reconstruct and diagram a passage

Use when: you're handed a paragraph of reasoning with no numbers and need to see how its claims support one another

Goal: a numbered list of the passage's distinct claims and an argument diagram with the main conclusion at the bottom

Diagramming longer arguments

The full version for longer passages, with intermediate conclusions and joint premises.

Evaluating an everyday argument

Evaluate an argument with three questions

Use when: someone offers you an argument and you need to decide whether it gives you a good reason to accept its conclusion

Goal: a verdict of sound or unsound (deductive) or cogent or uncogent (inductive), with the reason for it and a precise statement of what it shows about the conclusion

Three questions for any argument

The starting point. The recipes below each answer one of its questions more sharply.

Identify the form of an everyday argument

Use when: an everyday passage argues for something and you want to know whether its pattern is one of the named valid forms or a look-alike

Goal: the argument's skeleton in letters, matched to a named form or fallacy (or marked as matching none)

Recognizing forms in the wild

Build a counterexample by substitution

Use when: you suspect a deductive argument is invalid and want a demonstration anyone can check, instead of a story about its own subject

Goal: a new argument with the same pattern word for word, whose premises are obviously true and whose conclusion is obviously false

Proving invalidity with counterexamples

Shows invalidity without any symbols. Its symbolic counterpart is the countermodel from the truth-table test or the short method.

Spotting and fixing informal fallacies

Check a suspect argument for an informal fallacy

Use when: an argument feels wrong but its form looks fine, or you need to pin down what is wrong with it

Goal: the first of the four questions the argument fails, plus one sentence saying what its premises fail to do

What makes a fallacy

Repair a fallacious argument

Use when: an argument's conclusion might be worth defending, but its support has a recognizable defect

Goal: a rewritten argument that keeps the arguer's point as far as the evidence allows and has no fallacy

Repairing fallacious arguments

Run the four-question check first, so you know which defect you are repairing.

Arguments about groups (categorical logic)

Put a statement in standard form

Use when: an everyday sentence makes a claim about how two groups relate, but it is not yet written with all, no, or some, a group, are or are not, and a second group

Goal: a standard-form statement, true in just the same situations as the original, with its type (A, E, I, or O)

Putting statements in standard form

Always first. Both syllogism tests need standard-form statements.

Check a syllogism with the five rules

Use when: an argument has two premises and a conclusion, each an all, no, or some statement, sharing exactly three groups

Goal: a verdict, valid or invalid, and for an invalid one the name of the rule it breaks

Syllogisms, mood, figure, and distribution

Fast, and it names the rule an invalid syllogism breaks.

Test a syllogism with a Venn diagram

Use when: you have a syllogism in standard form and want a verdict you can see, or want to double-check a verdict from the five rules

Goal: the verdict, valid or invalid, read off a diagram of the premises alone

Testing syllogisms with Venn diagrams

Slower, but you can see the verdict. Use it to double-check the rules.

From English to symbols

Symbolize an if-then sentence

Use when: a sentence sets up a condition and a result with “if,” “provided that,” “when,” or “whenever,” in either order

Goal: a conditional with the condition before the arrow and the result after it

The conditional

Translate an English sentence into symbols

Use when: you're given an English sentence with one or more joining words (not, and, but, or, if, only if, unless, if and only if) and asked to write it in symbols

Goal: a formula, with a symbolization key, that makes the same claim the sentence makes, bracketed so no two different connectives stand side by side

If and only if, and longer sentences

Covers every joining word in the course, including "only if," "unless," and "neither … nor."

Symbolize a whole argument

Use when: you have an English argument and want to test it with a truth table

Goal: the premises and conclusion as formulas, all written with one shared key

Symbolizing whole arguments

One shared key for the whole argument, ready for a truth table.

Truth values, tables, and classifying one sentence

Find the main connective

Use when: you have a well-formed formula and need to know what kind of sentence it is, or which connective holds the whole thing together

Goal: the one connective whose scope is the whole formula, and the formula's name (negation, conjunction, disjunction, conditional, or biconditional)

Finding the main connective

The first step of almost every recipe in this group.

Compute a compound's value on one row

Use when: you're given a sentence and one valuation, and asked whether the sentence is true or false there

Goal: the sentence's truth value on that valuation, with every intermediate part's value written down

Computing the value of a compound · extended later in this course

Build a complete truth table

Use when: you're asked for a sentence's full truth table, or its value on every valuation

Goal: a table with every valuation in standard order, a helper column for each compound part, and the sentence's own column last

Building complete truth tables

Extends the one-row recipe to every valuation, in standard order.

Classify a sentence as a tautology, a contradiction, or a contingency

Use when: you are given one sentence of sentential logic and asked whether it is always true, always false, or sometimes true and sometimes false

Goal: one of the three labels, backed by rows of the truth table that settle it

Classifying with truth tables

Check a prediction with a partial truth table

Use when: you have guessed that one sentence is a tautology, a contradiction, or a contingency, and want to confirm or refute the guess without building every row

Goal: either one row that breaks the guess, or a clash showing that no such row exists

Predict, then check with partial truth tables

The shortcut when you already have a guess to confirm or break.

Equivalence

Check whether two sentences are equivalent

Use when: you are given two sentences and asked whether they are logically equivalent, and they use at most four or five letters

Goal: a verdict: equivalent, because every row matches, or not equivalent, with one row where they disagree

Checking equivalence with truth tables

Build a chain of equivalences

Use when: you must show two sentences are logically equivalent step by step, with a law named at each step, or simplify a sentence as far as possible

Goal: a numbered list of sentences from the start to the target, each equivalent to the one before, each step labeled with its law

Chains of equivalences

Use when a named law must justify each step, or to simplify a sentence.

Read a normal form off a truth table

Use when: you have (or can build) a sentence's truth table and need an equivalent sentence in disjunctive or conjunctive normal form

Goal: a DNF built from the true rows, or a CNF built from the false rows, equivalent to the sentence

Normal forms

Testing validity

Test an argument with a truth table

Use when: an argument is written in symbols and uses few enough letters to list every row

Goal: a verdict, valid or invalid, and a countermodel row if it is invalid

The truth table test

Always works. It gets long fast, since each new letter doubles the rows.

Test validity with the short method

Use when: an argument has too many letters for a comfortable full table, or its conclusion is a conditional, a disjunction or a negation that forces values quickly

Goal: a verdict: a contradiction (valid) or a checked countermodel (invalid)

The short method

Assume the premises true and the conclusion false, then look for a clash.

Take an English argument all the way to a verdict

Use when: you're handed an argument in ordinary English and asked whether its conclusion really follows

Goal: a verdict of valid or invalid, backed by a proof or by a countermodel read back into English

From English to verdict

The whole pipeline, from an English passage to a verdict read back in English.

Writing formal proofs

Check or write one line of a proof

Use when: you are reading someone's proof and need to confirm a line, or writing a line of your own and need its citation

Goal: a line whose citation names a rule, cites earlier available lines of the right shapes, and produces exactly the sentence written

Proof format: lines, premises, and citations

Prove an if–then with a subproof

Use when: the goal, or a line you need, has an if–then arrow as its main connective

Goal: the conditional written at the outer level, justified by conditional introduction on a closed subproof

Subproofs and conditional introduction

Prove a sentence by indirect proof

Use when: the goal is not a negation you can get with negation introduction, and no direct route from the premises reaches it

Goal: the goal sentence, written at the outer level of the proof and justified by IP

Indirect proof

Plan a natural deduction proof

Use when: you have premises and a goal and must find a proof, not just check one

Goal: a complete proof whose last line is the goal at the outer level

Proof strategy

The strategy recipe. It tells you when to reach for the two above.

Logic puzzles

Solve a knights-and-knaves puzzle

Use when: every speaker either always tells the truth or always lies, and you must decide who is which from what they say

Goal: the one assignment of knight or knave to each person that makes every statement consistent with its speaker

Knights and knaves