Philosophy of Language: Projection and Negation

Presupposition and Projectivity

• Sentences like "My shirt is on fire" and its negation "My shirt is not on fire" presuppose the existence and uniqueness of the shirt.
• Projection Definition: If a sentence φ expresses meaning p, and p persists under various logical operators (¬, □, ♢, →), then p is termed projective.
• Presuppositions tend to maintain their truth even when sentences are negated or modified by operators.

Diagnosing Presupposition

• The Family of Sentences Test checks for projectivity, which aids in identifying presuppositions.
• Steps for testing presupposition:
  • Identify sentence φ and its meaning p.
  • Use different operators to embed φ.
  • If the meaning p remains, then it is projective; if not, it isn’t.

Example of Presupposition Testing

• For "Allison’s dog is cute," the presupposition is that Allison has a dog.
• Variants tested:
  • "Allison’s dog isn’t cute" (¬φ)
  • "Allison’s dog might be cute" (♢φ)
  • "If Allison’s dog is cute, I want to meet it" (φ → ψ)
  • "Is Allison’s dog cute?" (?φ)
• All variants assume Allison has a dog, indicating that "Allison has a dog" is a projective presupposition.

Non-Presuppositional Examples

• For "Rex is a dog," the implication is that Rex is a mammal.
• Variants tested:
  • "Rex isn’t a dog" (¬φ)
  • "Rex might be a dog" (♢φ)
  • "If Rex is a dog, he’ll want this bone" (φ → ψ)
  • "Is Rex a dog?" (?φ)
• These variants do not necessitate belief that Rex is a mammal, demonstrating that "Rex is a mammal" is not a presupposition.

Implicature

• Implicature involves meanings conveyed through context rather than strict semantic entailment.
• It is systematic and predictable, contrary to misunderstanding it as illogical.
• Differences between implicature and implication must be noted.

Understanding Entailment

• Entailment: A logical relationship where a sentence φ entails another sentence ψ (φ ⊨ ψ); if φ is true, ψ must also be true.
• Example sentences demonstrate entailment:
  • "Rex is a dog" entails "Rex is a mammal."
  • "Rex is a dog" and "Rex is a dog" entail each other.
  • Contradictions, such as "Rex could be a mouse," highlight that entailment is about logical truth.

Tautologies and Logical Relations

• Tautologies are necessarily true statements (e.g., "It is snowing or it is not snowing").
• They imply that any variation must hold true across logical interpretations, affecting how entailment operates.

Cases of Implicature

• Typical examples of implicature include:
  • "Carlo ate some of the cookies" implies he didn’t eat all of them.
  • "Dana has two cats" implies she has exactly two.
  • "Can you pass the salt?" conveys a request rather than a question about ability.
• These cases indicate meanings beyond the literal interpretations of the sentences and require deeper analytical exploration.