Consider the sentence: 'John stopped smoking.' You negate it to get: 'John didn't stop smoking.' Both the original and the negation carry the implication that John used to smoke. What does this behavior tell you about the proposition 'John used to smoke' relative to the original sentence?
AIt is asserted by the original sentence, since it is clearly communicated
BIt is presupposed — it is background content that projects through negation rather than being canceled by it
CIt is a conversational implicature that can be canceled in the right context
DIt is a logical entailment, since 'stopping' logically requires a prior state
The diagnostic test for presupposition is exactly this: content that *survives negation* is presupposed, not asserted. Asserted content is canceled by negation — 'It is raining' becomes false when negated. But 'John used to smoke' is carried by both 'John stopped smoking' and 'John didn't stop smoking,' so it is presupposed background content, not the main claim. Option D (logical entailment) is partially correct in that there is a logical dependency, but presupposition is distinguished as the specific content that projects not just through negation but also through questions ('Did John stop smoking?') and conditional embeddings.
Question 2 Multiple Choice
Under the partial functions formalization of presupposition, what is the truth value of 'The king of France is bald' in the actual world, given that France has no king?
AFalse — the sentence makes a claim about a king of France, and since no such person exists, the claim is false
BTrue — the sentence's presupposition being unfulfilled doesn't make it false, so it defaults to true
CUndefined — the sentence's presupposition fails, so the sentence falls outside the domain of true and false
DMeaningless — sentences with failed presuppositions have no semantic content
The partial functions formalization says that a sentence with a presupposition denotes a function that is only *defined* (returns T or F) in worlds where the presupposition holds. In worlds where the presupposition fails, the function is undefined — neither true nor false. This is importantly different from falsehood. 'The king of France is bald' is *false* only if there is a king of France who is not bald. Without a king of France, the sentence is like asking whether a number you haven't chosen is odd — the question doesn't have a truth value. This three-valued logic (true / false / undefined) directly formalizes the intuition that presupposition failure is different from falsehood.
Question 3 True / False
A sentence that presupposes content P and a sentence that asserts content P differ formally in that presupposed content survives embedding under negation, questions, and conditionals, while asserted content does not.
TTrue
FFalse
Answer: True
This projection behavior is the formal signature of presupposition and the key diagnostic test. If you negate a sentence and a piece of content disappears (the sentence no longer implies it), that content was asserted. If the content survives — appears in both the sentence and its negation — it was presupposed. The same test applies to questions: 'Is the king of France bald?' still presupposes France has a king, even though the assertion (baldness) is being questioned. And to conditionals: 'If the king of France is bald...' still assumes France has a king. Projection is the operational criterion that distinguishes presupposition from assertion.
Question 4 True / False
The sentence 'The king of France is not bald' is false in the actual world, because there is no king of France and the sentence makes a claim about him.
TTrue
FFalse
Answer: False
This is the classic confusion between presupposition failure and falsehood. The sentence is *undefined* (lacks a truth value) in the actual world, not false. It is false only in worlds where France has a king who has hair. When the presupposition fails (no king exists), the sentence cannot be evaluated as true or false — it is like asking whether the current king of France is bald: the question doesn't have an answer, it has a defective presupposition. Under the partial functions formalization, the sentence's truth value function is simply not defined for the actual world. Three-valued logic captures this with a third value (⊥, or 'undefined') that is distinct from false.
Question 5 Short Answer
What is the formal diagnostic test for distinguishing whether a piece of content is presupposed or asserted, and why does it work?
Think about your answer, then reveal below.
Model answer: The diagnostic test is projection under negation (and other operators like questions and conditionals): negate the sentence and check whether the content survives. Asserted content is canceled by negation — it appears in the positive but not the negative version. Presupposed content projects through negation — it appears in both. The test works because negation targets the *asserted* claim of a sentence, flipping its truth value, while leaving background presuppositions intact. A sentence and its negation share the same presuppositions but have opposite assertions.
This projection test is what makes presupposition a formal, testable category rather than just an informal intuition about 'background knowledge.' It allows linguists to distinguish presupposed content from entailments (which don't project through all operators the same way), conversational implicatures (which are cancelable), and direct assertions (which are negation-reversible). The test can be extended to other operators — questions, conditionals, attitude verbs — to refine the analysis further.