Semantics week 2

LING2100

What is a set in formal semantics?

A set is an unordered collection of distinct elements. Sets are defined solely by their members, meaning {a, b, c} is equal to {c, a, b}, and duplicates are ignored.

How do you denote set membership?

With the symbol ∈. For example, if x is an element of set A, we write x ∈ A. If x is not in the set, we write x ∉ A.

What is the difference between a subset and a proper subset?

A subset A ⊆ B means every element of A is also in B. A set is a subset of itself.

A proper subset A ⊂ B means A is a subset of B but A ≠ B (i.e., B has at least one element not in A).

What are the common operations on sets?

Union (A ∪ B): Elements in A or B or both

Intersection (A ∩ B): Elements in both A and B

Difference ( A - B): Elements in A but not in B

Power set (p (A): The set of all subsets of A, including the empty set and A itself.

What is the cardinality of a set?

Cardinality is the number of elements in a set. It is denoted as | A | . For example, |{a, b, c} |= 3. The empty set ø has a cardinality of 0.

What is a function?

A function is a special kind of relation where every input maps to exactly one output. Formally, for all a ∈ A, there is a unique b ∈ B, such that (a, b)∈f

What are injective, surjective, and bijective functions?

Injective (one to one): no two input map to the same output.

Surjective (onto): every element in the output set is covered.

Bijective: both injective and surjective - every input maps to a unique output, and every output has a corresponding input.

What is an NPI?

An NPI is a word or phrase that inly appears in negative or downward-entailing contexts.

Example: ¨ever¨

GOOD: ‘I haven’t ever read it'

BAD: ‘I have ever read it’

What is downward entailment?

An environment is downward entailing if it licences inference from general to specific.

Example:

‘no sportsman sang’ → ‘no tennis player sang’

Since ‘tennis player’ ⊆ ‘sportsman’, the entailment holds

What is the Fauconnier-Ladusaw Generalisation?

An expression licenses NPI's exactly where it licenses downward entailments. This connects syntactic environments to the logical properties of expressions.

Compare the NPI-licensing behaviour of ‘no’, ‘every’, and ‘some’.

No: downward entailment in both restrictor and scope; licences NPIs in both.

Every: Downward entailing in the restrictor, but upward in the scope; only licenses NPIs in the restrictor

Some: Upward entailing in both; does not license NPIs.

What is the direction of entailment in simple positive sentences like ‘Casper is a tennis player’?

This is an upward-entailing context. From ‘casper is a tennis player’, we can infer ‘Casper is a sportsman’ because ‘tennis player’ is a subset of ‘sportsman’.

Why does negation reverse entailment direction?

Negation turns upward entailments into downward ones.

Example:

‘Casper is not a sportsman’ → ‘Casper is not a tennis player’

Here, inference flows from general to specific.