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Logic

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Introduction

Logic is the study of reasoning: how a conclusion follows from reasons, and how to tell a reliable inference from a misleading one. You use it when checking a news claim, explaining a scientific result, writing a program, or deciding whether an argument really proves its point.

This course is designed for learners aged about 14–18 and adult beginners. No previous symbolic logic is needed. By the end, you can identify premises and conclusions, distinguish validity from truth, work with basic logical symbols and truth tables, and challenge common reasoning errors. A good argument is not simply one that sounds persuasive; its support must be examined.

Watch first: This short introduction places logic within philosophy and critical thinking.

Arguments, Premises and Conclusions

How arguments work

An argument consists of one or more premises offered to support a conclusion. For example: Every square is a rectangle. Every rectangle has four sides. Therefore, every square has four sides. The word therefore often signals a conclusion, while because often introduces a premise. These clues help, but the structure matters more than the wording.

A deductively valid argument is one in which it is impossible for all the premises to be true while the conclusion is false. Soundness requires both validity and actually true premises. A valid argument can have false premises. To test validity, imagine a counterexample with true premises but a false conclusion: if such a situation is possible, the argument is invalid.

A reliable rule and a tempting mistake

Modus ponens has the pattern: If P, then Q. P. Therefore, Q. For example: If a number is divisible by four, it is even. Twelve is divisible by four. Therefore, twelve is even. By contrast, concluding P merely because Q is true commits affirming the consequent; the result might have another cause.

Video lesson: Identify the stated reasons and conclusion in the examples, then decide whether the inference is valid.

Statements and Logical Connectives

From everyday sentences to symbols

In basic propositional logic, P and Q stand for statements that can be true or false. A connective builds a new statement from them. Classical logic uses two truth values; this useful model does not capture every ambiguity of everyday language.

Expression	Reading	Truth condition
¬P	not P	True exactly when P is false
P ∧ Q	P and Q	True when both statements are true
P ∨ Q	P or Q	True when at least one statement is true
P → Q	if P then Q	False only when P is true and Q is false
P ↔ Q	P if and only if Q	True when P and Q have the same truth value

In this table, or is inclusive: tea or coffee allows both unless the situation specifies otherwise. The conditional P → Q describes a truth-functional relationship; ordinary language may also suggest causation, timing or an explanation, which the symbol alone does not express.

Truth Tables and Logical Equivalence

Test all possible cases

A truth table checks every assignment of truth values. With two basic statements there are four combinations. The following table shows why P → Q is false in just one case. Use T for true and F for false.

P	Q	P ∧ Q	P ∨ Q	P → Q
T	T	T	T	T
T	F	F	T	F
F	T	F	T	T
F	F	F	F	T

A tautology is true under every assignment, as in P ∨ ¬P. A contradiction is false under every assignment, as in P ∧ ¬P. Two expressions are logically equivalent if their final columns always agree. For example, ¬(P ∧ Q) and ¬P ∨ ¬Q are equivalent by one of De Morgan's laws.

Animation: Observe how a truth table is constructed one row at a time. Pause it to predict the next result.

Animated construction of a truth table. Wikimedia Commons: Stephan Kulla, CC BY-SA 3.0.

Quantifiers, Sets and General Claims

All, some and not all

Predicate logic can describe individual objects and their properties. The universal quantifier ∀ means for every, while the existential quantifier ∃ means there exists at least one. For example, ∀x (Cat(x) → Mammal(x)) means every cat is a mammal. The expression ∃x (Cat(x) ∧ Black(x)) means at least one cat is black. Both depend on a specified domain of objects.

The denial of all students passed is 'at least one student did not pass; it is not no students passed. Similarly, some means one or more in ordinary mathematical logic, even if casual conversation sometimes suggests not all. These distinctions make policies, data claims and surveys easier to assess.

Try it: Imagine a set of three objects: two red circles and one blue square. Decide whether all objects are circles, some objects are red, and no objects are squares are true. Explain your answers using a diagram or a short sentence.

Informal Reasoning, Fallacies and Applications

Beyond certainty

Deduction aims at a conclusion that must follow if the premises hold. Induction uses observations to support a general expectation, such as predicting tomorrow's weather from previous measurements; its conclusion can be probable without being guaranteed. Abduction looks for a plausible explanation of evidence, but the best current explanation may still be wrong.

Spot the reasoning error

A false dilemma presents only two choices when more exist. An ad hominem attack targets a person instead of the claim. Affirming the consequent wrongly moves from if P then Q and Q to P. The fallacy of division assumes that what is true of a whole must also be true of each of its parts. Not every criticism of a person is automatically a fallacy; ask whether it is actually relevant to the issue.

Audio (2 minutes): Listen to this freely licensed explanation of the fallacy of division and write one original counterexample.

Audio recording: Fallacy of division. Wikimedia Commons: JohnAnkerBow, CC BY-SA 4.0.

Logic supports computer science through Boolean expressions and program conditions; it supports mathematics through proofs; and it supports everyday life through evidence-based decisions. It cannot replace checking evidence or recognizing uncertainty. A logically valid inference from false premises need not describe reality correctly.

Interactive Tasks

Quiz: Test Your Knowledge

What is a premise in an argument? (A reason offered in support of a conclusion) (!A statement that must always be false) (!The final sentence in every paragraph) (!An example with no relevance to the claim)

When is a deductive argument valid? (When true premises cannot lead to a false conclusion) (!Whenever its conclusion sounds convincing) (!Only when it contains exactly two premises) (!Whenever the audience agrees with it)

What makes a deductive argument sound? (It is valid and all its premises are true) (!It has a surprising conclusion) (!It uses only mathematical symbols) (!Its conclusion is popular)

When is the conjunction P and Q true? (When both P and Q are true) (!When at least one is true) (!Only when P is false) (!Whenever Q is false)

What does inclusive or mean in classical logic? (At least one of the statements is true) (!Exactly one statement is true) (!Both statements must be false) (!The statements must be unrelated)

In which case is if P then Q false in a truth table? (P is true and Q is false) (!P and Q are both true) (!P is false and Q is true) (!P and Q are both false)

Which reasoning pattern is modus ponens? (If P then Q and P therefore Q) (!If P then Q and Q therefore P) (!If P then Q and not P therefore not Q) (!P or Q and P therefore not Q)

What follows from the denial of all birds can fly? (At least one bird cannot fly) (!No birds can fly) (!Exactly one bird can fly) (!All animals can fly)

Which description best fits inductive reasoning? (Observations support a conclusion that may remain uncertain) (!Premises guarantee a conclusion in every case) (!A statement is true only because it is repeated) (!All conclusions are chosen at random)

What is a false dilemma? (Presenting two options while ignoring real alternatives) (!Checking whether a conclusion follows from premises) (!Writing a complete truth table) (!Using an example to clarify a concept)

Memory Game

Match each distinct concept with its definition.

Premise	A reason supplied in support of a claim
Conclusion	The claim an argument attempts to establish
Validity	Impossibility of true premises with a false conclusion
Conjunction	A connective that requires both components to be true
Tautology	A statement true under every truth assignment
Quantifier	An expression indicating all or at least one
Fallacy	A recognizable error in reasoning

Drag and Drop

Match each concept to its explanation.

Match the correct terms.	Logic concepts
Negation	Reverses the truth value of a statement
Disjunction	True if at least one component is true
Biconditional	True when both statements have matching truth values
Existential claim	Asserts that at least one example exists
Counterexample	A case showing that a proposed rule or inference fails

Crossword Puzzle

Use one-word answers to complete the clues.

PREMISE	What do you call a statement offered as a reason in an argument?
VALIDITY	What property rules out true premises and a false conclusion?
NEGATION	What connective reverses a truth value?
TAUTOLOGY	What is a statement true under every possible assignment called?
QUANTIFIER	What logical expression indicates all or at least one?
INDUCTION	What type of inference generalizes from observations without guaranteeing certainty?

LearningApps

Explore additional practice activities related to logic. Check the quality of an activity before using it as evidence of learning.

Cloze Text

An argument offers reasons called

in support of a conclusion.

An argument is deductively

when its premises cannot all be true while its conclusion is false.

A valid argument with true premises is

.

The connective AND is also called

.

A systematic chart of truth values is a truth

.

A statement true under every assignment is a

.

The symbol for the existential quantifier means that something

.

Reasoning from observations to a probable generalization is

.

A misleading choice between only two options may be a false

.

Open Tasks

Easy

Spot the argument: Find three everyday claims in fictional examples, underline each premise and circle its conclusion.
Write a deduction: Write one valid argument about shapes and explain why no counterexample works.
Make symbol cards: Create illustrated cards for NOT, AND, OR, and IF THEN with one correct everyday example each.
Build a small table: Fill in the four truth-value combinations for P AND Q and check with a partner.

Standard

Test a counterexample: Construct an invalid argument and exhibit true premises with a false conclusion.
Draw a set diagram: Represent an intersection and explain why it corresponds to logical AND.
Verify an equivalence: Create two full truth tables showing that NOT P AND Q is not equivalent to NOT P OR Q; state the correct parenthesization.
Audit a claim: Select an invented advertisement, identify a possible reasoning error, and rewrite its claim more carefully.

Advanced

Translate quantified statements: Express two claims using universal and existential quantifiers, then negate each one correctly.
Compare argument forms: Produce paired examples of modus ponens and affirming the consequent and explain their different validity.
Design a decision rule: Specify a privacy-preserving program rule using AND, OR, and NOT; test every input combination.
Create a teaching resource: Produce a short openly licensed logic lesson with original examples, source acknowledgements, and an answer key.

Learning Check

Justify validity: Explain in your own words how an argument can be valid even when one premise is false; give an original example.
Assess a conditional: Build a truth table for a conditional, identify its false row, and explain why that row is decisive.
Negate a generalization: Translate and negate the claim that every member of a set satisfies a property; explain the difference between some not and none.
Evaluate persuasion: Analyze a fictional social-media argument, separating evidence, premises, conclusion, and any possible fallacy.
Transfer to computing: Model a simple access rule with Boolean connectives and test whether an unintended case is admitted.
Compare kinds of inference: Explain how deduction differs from induction and abduction using the same real-world problem.

Learning evidence

A successful learner can present an annotated argument map with clearly identified premises and conclusion; distinguish validity, soundness and persuasiveness; complete and interpret a truth table; translate and negate an elementary quantified statement; and explain an invalid inference using a counterexample. Strong evidence also includes a reasoned analysis of a fallacy, a Boolean decision rule tested against all cases, and correct source attribution for reused media.

Assess both the answer and the explanation: can you show why it follows? Recheck unfamiliar claims against trustworthy sources rather than assuming that a valid pattern establishes the truth of its premises.

OER and Wikipedia

Open reference: The English Wikipedia overview covers formal and informal logic, argumentation, and related fields.

Further reference reading:

Stanford Encyclopedia of Philosophy: Classical Logic — definitions of validity, deduction and logical consequence.
Wikipedia: Propositional calculus — connectives and proof systems.
Wikipedia: Truth table — truth-value calculations.
Wikipedia: First-order logic — quantifiers and predicates.

Media sources and reuse rights

The images, animation and audio are hosted on Wikimedia Commons. The linked file-description pages identify each creator and licence. The items have been embedded above, not reproduced as newly created media. Credit the author and retain licence links when reusing attribution-licensed material.

Venn diagram.png — A bag of many boiled potatoes; CC0 1.0.
Modus ponens.svg — Phlsph7; CC0 1.0.
Logical connectives table.svg — Watchduck (Tilman Piesk); dedicated to the public domain by its creator.
Venn-AND-sketch.png — Marble machine; CC BY-SA 4.0.
Construction of a logical value table.gif — Stephan Kulla; CC BY-SA 3.0 (also offered under GFDL).
Existential quantifier.svg — Phlsph7; CC BY 4.0.
Fallacy of division.ogg — JohnAnkerBow; CC BY-SA 4.0.

Video sources: What Is Philosophy? and How to Argue – Philosophical Reasoning are published by CrashCourse on YouTube. They are embedded using YouTube playback; no permission to download, edit or relicense the videos is implied.