Formal SemanticsHomework 51 of 3

1.Semantic Composition With Case-Marking

1.Cases

(1)cases are encoded by non-negative integers: 0, 1, 2, 3, …

2.Types

(1)Dis a type;[uninflected definite-noun-phrase (DNP)]

(2)if  is a case, then D is a type;[-marked D]

(3)Sis a type;[sentence]

(4)if A and B are types, then (AB) is a type;

(5)nothing else is a type.

3.Common Nouns and Bare-Adjectives

We treat CNPs and Bare-Adjectives as having type C, but defineCas follows.

CD0S

Bare-adjectives can stand-alone. By contrast, modifier-adjectives (e.g., ‘former’, ‘imitation’, ‘putative’)
can't stand alone, and have type CC.

4.Rules of Composition

1.Categorial Logic

Items of typesA, Bcombine to form an item of type Cif

A ; BC i.e., Cfollows fromA,B in Categorial Logic[Linear Logic without Identity].

according to CL.

2.Conjunction

If two items have the same type, which is S-ending, then they combine to make an item of the same type, via conjunction.

ex.1: Fa ; Gb  Fa & Gb
ex.2: 0F ; 0G0{F & G}
ex.3: 21R ; 21K21{R & K}

3.No other combinations are allowed.

5.S-Ending Types

1.S isS-ending;

2.if B is S-ending, and A is a type, then AB is S-ending;
3.nothing else is S-ending.

2.Lexicon

Translate each of the following into expanded-lambda-calculus.
Treat genitive-nouns as inherently indefinite. Use def to make them definite,but only when necessary.

Morpheme / Type / Translation / Gloss
every
some
no
who
the, def
is [cop]
is [id]
nom, +1
acc, +2
by, +5
's, of, +6
-ed [pass]
respects
knows
nextto
woman
man
mother
father
friend
Jay
Kay

3.Semantic Trees

(1)Construct the most natural tree-structure; every node should be grammatically-sensible.

(2)Assign to each node its translation into the currently active logical calculus.

(3)Include every morpheme (appropriately spelled) above its terminal node.

(4)Make sure that each composition follows the currentrules of composition.

(5)Produce the exact sentence in question, not one you merely stipulateto be "equivalent".

However:

(1)you can take for granted morphologicalvariations, including irregular forms,
e.g., the various irregular forms of ‘be’.

(2)you can hypothesize elided (unpronounced) morphemes.

(6)If a tree doesn't fit nicely and neatly on the page, reformat it so that it does!
For example, use multiple-tables [for example, split NP and VP],
with appropriate notation connecting them.

(7)Use Table-Style Trees (illustrated below).

Jay's mother is virtuous

Jay / 's / mother / def / +1 / is / virtuous
j / .6 / 60M / P0P / .1 / P0.P1 / 0V
j6 / 1V
0Mj
Mj
[Mj]1
V[Mj]

4.Assignment

Required

  1. Jay respects every man who respects Kay.
  2. Every woman [whom] Jay respects is next-to Jay.
  3. Kay's mother is respected by every friend of Kay.

Optional

  1. Every woman respected by Jay is respected by Kay.
  2. Every woman who respects Kay respects Kay's mother.
  3. Jay knows some woman whose mother respects Kay.
  4. No man's mother respects every man's father.
  5. Every man who respects every man is respected by every man.
  6. Every man [whom] Kay respects is a man respected by Kay.
  7. Every friend of every virtuous man is virtuous.