Unit 1: Propositions and Truth

Learning Objectives

Distinguish propositions from non-propositions
Assign truth values (true, false, indeterminate) to propositions
Explain three theories of truth — correspondence, coherence, and pragmatic
Recognise the key terms premise, conclusion, reasoning, and argument
Use basic logical form notation (P, Q)

What is Applied Logic?

Logic is the study of correct reasoning. It provides tools for distinguishing good arguments from bad ones — not on the basis of whether we agree with the conclusion, but on the basis of whether the reasoning is valid.

Applied logic focuses on using these tools in real contexts: evaluating news articles, spotting weak arguments in debates, identifying errors in everyday reasoning, and constructing clearer arguments of your own.

This unit introduces the most fundamental concept in logic: the proposition — a statement that has a truth value. Everything else in the course builds on this idea.

What is a Proposition?

A proposition is a declarative statement that is either true or false. It asserts something about the world that can, in principle, be evaluated. Not every sentence is a proposition — questions, commands, and exclamations are not propositions because they do not make a claim that can be true or false.

Propositions

Water boils at 100°C at sea level.
The Earth is older than 4 billion years.
Paris is the capital of Germany.
All prime numbers greater than 2 are odd.

Not Propositions

Is Paris the capital of France? (question)
Close the door! (command)
What a beautiful sunset. (exclamation)
Hello, how are you? (greeting)

Notice that a proposition can be false — "Paris is the capital of Germany" is a proposition even though it is factually wrong. The defining feature is not that the statement is true, but that it could be true or false.

Logical form notation

In formal logic, propositions are represented by letters — typically P, Q, and R. This allows us to reason about the structure of an argument without being distracted by its content.

P: It is raining.
Q: The ground is wet.
If P, then Q — If it is raining, then the ground is wet.

You will use this notation throughout the course. For now, simply note that P and Q are placeholders for any proposition.

Thinking About Truth

Before we study the concept formally, consider these questions. Think about each one, and if possible discuss your answers with someone nearby.

What does it mean to say that something is true?
Can you always tell whether a statement is true or false? Give an example where you cannot.
Is it logical to say 1 + 1 = 2? Why — or why not?
Is the statement "This sentence is false" a proposition? What is its truth value?
If two people sincerely disagree about whether a statement is true, can they both be right?

There are no simple right or wrong answers here. These questions are designed to surface your intuitions about truth so you can compare them with the formal theories in the next activity.

Truth Values

Every proposition has a truth value: it is either true (T), false (F), or — in some logics — indeterminate (we cannot currently assign a definite truth value).

Truth Value	Meaning	Example
True (T)	The proposition corresponds to facts about the world.	Water is composed of hydrogen and oxygen.
False (F)	The proposition does not correspond to facts about the world.	The Moon is larger than the Sun.
Indeterminate	We cannot assign T or F — due to vagueness, incomplete evidence, or paradox.	There is intelligent life elsewhere in the universe.

Case study: How many colours are in a rainbow?

Consider this statement: "There are seven colours in a rainbow."

Is this true? Is it false? How confident are you? The number of colours we perceive depends on how we draw the boundaries between colour categories — and different cultures and languages draw those boundaries differently. What looks like a straightforward factual question reveals something important: many propositions that appear clear-cut turn out to be more complicated on closer inspection. The truth value depends partly on how we define the terms in the proposition.

Three Theories of Truth

Philosophers have proposed several theories to explain what it means for a proposition to be true. Three of the most influential are described below.

Correspondence Theory

A proposition is true if it corresponds to the facts — if it accurately describes a state of affairs in the world. This is the most intuitive view.

"Snow is white" is true if and only if snow is, in fact, white. — Tarski

Coherence Theory

A proposition is true if it coheres — fits consistently — with a set of other accepted propositions. Truth is a matter of internal consistency within a system of beliefs.

A statement is true if it is part of a coherent web of beliefs.

Pragmatic Theory

A proposition is true if believing it works — if it leads to successful outcomes or useful predictions. Truth is defined by its practical consequences.

"True ideas are those that we can assimilate, validate, corroborate, and verify." — William James

Each theory captures something important, and each has weaknesses. In this course we will mostly work within the correspondence theory — the default in formal logic — but being aware of the alternatives helps you recognise when arguments rest on different assumptions about what counts as true.

Introduction to Logic — Video

Watch this introductory lecture on logic. As you watch, pay attention to how the presenter defines the following key terms — you will need them throughout the course.

Premise — a statement offered as evidence or a reason
Conclusion — the claim that the premises are intended to support
Reasoning — the process of moving from premises to conclusion
Argument — premises + reasoning + conclusion
Validity — does the conclusion follow from the premises?
Soundness — is the argument valid and are the premises true?

Introduction to Logic and Language

Plato's Allegory of the Cave — Optional

Plato's Allegory of the Cave is one of the most famous thought experiments in philosophy. It explores how our perception shapes what we believe to be true — and raises the question of whether the world we experience is the same as the world as it really is.

Watch this short animated explanation (6 min 28 sec). Consider: what does the cave tell us about the relationship between truth, perception, and knowledge?

Plato's Allegory of the Cave

The allegory connects to the correspondence theory of truth: the prisoners mistake shadows for reality. What does this imply about the difficulty of establishing correspondence between our beliefs and the world?

Critical Analysis: Professor X

The following passage is a short argument. Read it carefully. Each sentence is numbered for ease of reference.

Professor X

Professor X is an efficient and effective teacher.
All his students enjoy his classes according to the feedback given on the student feedback questionnaires.
Every student who attended the course in full received a grade A, which is testimony of his expertise in teaching.
The professor not only holds a doctorate in physics but is also a polyglot and a polymath.
His course is always popular with students.
Every course offered in the previous two years has seen enrolments meeting or exceeding the minimum number of students.
To ensure he has enough energy, he always brings a cup of coffee to the classroom.
This is yet more evidence of his dedication to his students.
Finally, Professor X's Facebook page has received thousands of "likes", a clear indication of votes of confidence in his teaching.

Work through these questions:

What is the main conclusion of the passage?
Which sentences provide genuine evidence for that conclusion?
Which sentences are irrelevant to the conclusion?
Which sentences make claims that are not adequately supported?
Are any of the sentences themselves propositions with an indeterminate truth value?

This exercise is a preview of argument analysis skills covered in depth in Unit 2. For now, focus on identifying what is being claimed and what evidence is offered.

Review

Expand each concept to check your understanding before moving on.

A proposition is a declarative statement that has a truth value — it is either true or false (or indeterminate). Questions, commands, and exclamations are not propositions. A proposition can be false and still be a proposition: the key criterion is that it could be true or false.

True (T): the proposition corresponds to the facts. False (F): it does not. Indeterminate: we cannot assign a definite truth value — due to vagueness, paradox, or incomplete evidence. Classical (two-valued) logic uses only T and F; some logics add a third value.

Correspondence: a proposition is true if it corresponds to facts about the world. Coherence: a proposition is true if it coheres with a system of accepted beliefs. Pragmatic: a proposition is true if acting on it produces successful outcomes. Formal logic predominantly uses the correspondence view.

An argument consists of one or more premises (the evidence offered) and a conclusion (the claim being supported). Reasoning is the process by which we move from premises to conclusion. Not every passage with opinions is an argument — there must be an attempt to provide reasons for a conclusion.

P and Q are propositional variables — letters that stand for any proposition. Writing If P, then Q expresses a conditional relationship between two propositions without specifying their content. This notation lets us analyse the structure of an argument separately from its subject matter.

Key concepts covered in this unit: declarative statement, proposition, truth value (true, false, indeterminate), correspondence theory, coherence theory, pragmatic theory, Plato's allegory of the cave, premise, conclusion, reasoning, argument, propositional variable (P, Q).

Proceed to Unit 2: Arguments and Inference when ready.