x̄ - > Logic and proof - Statement and argument

Chapter 1 — Statements & Arguments

Short poetic preface: A sentence is a flute-note; a statement is the tune that can be true or false. An argument is the slow gathering of notes into a chorus that aims to make the tune inevitable.

“A sentence is a flute-note; a statement is the tune that can be true or false. An argument is the slow gathering of notes into a chorus that aims to make the tune inevitable.”

What is a statement?

A statement (or proposition) is something that is true or false (but not both). A sentence is the grammatical vehicle — it sometimes carries a statement, sometimes other things (questions, commands, exclamations).

Sentence: “The moon is made of green cheese.”
Is it a statement? Yes

It asserts a claim that is false. So it expresses the statement: The moon is made of green cheese.

Sentence: “Is the moon made of green cheese?”
Is it a statement? No

This is a question — not truth-apt. It does not express a statement.

Sentence: “Close the window.”
Is it a statement? No

Imperatives are commands — they are not truth-apt and do not express statements.

Sentence: “x + 2 = 5.” (with x unspecified)
Is it a statement? Not yet

This contains a variable: an open formula. It's not truth-apt until the variable is fixed (e.g., "3 + 2 = 5" is a true statement).

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Relation between sentences and statements

Sentences are vehicles (syntax); statements are semantic contents. The same statement can be expressed by different sentences; some sentences express no statement at all. Always be suspicious of casual speech — translation into a crisp statement is an act of craftsmanship.

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What is an argument?

An argument is a set of statements (premises) offered to support another statement (the conclusion). A mere collection of sentences is not yet an argument — there must be at least one claim offered as evidence for another.

Passage: “It rained last night. The streets are wet. Therefore, the streets are wet because it rained.”

This can be read as an argument if the first sentence is a premise supporting the conclusion. A clearer standard form makes the implicit premise explicit.

Standard form:

1. It rained last night. (Premise)
2. If it rained last night, then the streets would be wet. (Implicit premise — make it explicit)
3. ∴ The streets are wet. (Conclusion)

Identify premises & conclusion

Click to reveal the labels for the passage above.

Premises: (1) and (2). Conclusion: ∴ The streets are wet.

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Quick interactive quiz — Statement or not?

Choose whether each sentence is a statement. Be skeptical of your first instinct.

A: “There are monsters under the bed.”

B: “Either it rains or it doesn’t.”

C: “Please, be quiet.”

Criteria for a good argument

Validity

If the premises are true, the conclusion must be true. Ensures the logical link.

Soundness

Valid + true premises → guarantees the conclusion.

Clarity

Clear premises and conclusion prevent equivocation.

Relevance

Premises must bear on the conclusion; otherwise they do no work.

Acceptability

Premises should be credible to the intended audience.

Sufficiency

Premises together supply enough support for the conclusion.

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Tips — the skeptical craftsman

• Always translate casual speech into an explicit statement before reasoning.
• Be suspicious of implicit premises — make them explicit.
• Test validity by imagining premises true: can conclusion still be false?

Passage: "If the lamp is on, then someone is home. The lamp is on. Therefore, someone is home."

Is this an argument? (Yes) Identify premise(s) and conclusion. Try to write it in standard form.

Premises: (1) If the lamp is on, then someone is home. (2) The lamp is on. Conclusion: Therefore, someone is home. This is an instance of modus ponens.

Practice suggestion: Take ordinary conversations and ask: which sentences are statements? Which passages try to argue? Make implicit premises explicit. Be traditional in method — careful translation — and skeptical in spirit.