Semantics

Why do semantics?

• Expressions of natural languages can contain information about the world, i.e. they have meanings.
• Speakers and listeners have (reliable) intuitions about this meaning relation:
  • The meaning of the name "Angela Merkel" is the person Angela Merkel.
  • The sentence "Angela Merkel is the first female chancellor of Germany." is true in the world.
  • The sentence "Germany lies in Asia." is false in the world.

• Speakers and listeners can infer information from the meanings of sentences:
  • From "Lilly is a cat." it follows that "Lilly is an animal."
  • From "Lilly is as old as Fido." it follows that "Fido is as old as Lilly."
  • From "Lilly is asleep." it follows that "Someone is asleep."

On the other hand:

• • From "Lilly is an animal." it does not follow that "Lilly is a cat."
  • From "Lilly is older than Fido." it does not follow that "Fido is is older than as Lilly."
  • From "Someone is asleep." does not follow that "Lilly is asleep."

But there is a problem: ambiguity

Examples:

• • The word "punch" can either refer to a drink or to a boxing movement.
  • Does it then follow from "Ethel's punch was impressive." that "Ethel's drink was impressive."?
    • Answer: The entailment relationship cannot be a relation between forms. It must be a relationship between forms depending on their meanings.
    • Thus, from "Ethel's punch was impressive." interpreted in such a way that "punch" is a drink, it does follow that "Ethel's drink was impressive."
    • But from "Ethel's punch was impressive." interpreted in such a way that "punch" is a boxing movement, it does not follow that "Ethel's drink was impressive."

• • "The police stopped a man with a gun." can describe two different situations:
    • A man with a gun was stopped by the police.
    • A man was stopped by the police with a gun.

• • • Thus, from the form "The police stopped a man with a gun." alone we cannot be certain whether it follows that the man or the policed had a gun in the stopping situation.

The solution: Logic

From Wikipedia:

Logic is the systematic study of valid rules of inference, i.e. the relations that lead to the acceptance of one proposition (the conclusion) on the basis of a set of other propositions (premises).

Logics are designed to differ from natural languages in that they are languages that impose the following crucial reqirement:

Each logical expression has at most a single meaning.

• Consequence: logical languages do not suffer from the ambiguity problem in entailment that natural languages display.

How to do inferencing with English

• To every grammatical and meaningful expression of English we assign one or more expressions of a logical language. This process is called """translation.
• To every expression of the logical language we assign one and only one meaning. This process is called interpretation or more precisely interpretation in a model.
• The interpretation of an expression is called its denotation.
• The denotations of the translations of an English expressions are (indirectly) also denotations of the English expression.
• We can now say things like the following:
  • From from "Ethel's punch was impressive." where "punch" denotes a drink, it follows that "Ethel's drink was impressive."
  • From from "Ethel's punch was impressive." where "punch" denotes a boxing movement, it does not follow that "Ethel's drink was impressive."