Here is a basic lexicon:

Let \(\sysm{G}\) be an abbreviation for \(\sysm{(S\fs(S\bs NP))}\)

\[\begin{align} \textit{donkey} &:=& \sysm{N} &:& \sysm{\lam x.donkey'x} &::& et\\ \textit{student}&:=& \sysm{N} &:& \sysm{\lam x.student'x} &::& et\\ \textit{John} &:=& \sysm{S\fs(S\bs NP)} &:& \sysm{\lam p.p\cnct{}john'} &::& ett\\ \quad \nonumber\\ \textit{sleeps} &:=& \sysm{S\bs NP} &:& \sysm{\lam x.sleeps'x} &::& et\\ \textit{walk(s)} &:=& \sysm{S\bs NP} &:& \sysm{\lam x.walk'x} &::& et\\ \quad \nonumber\\ \textit{lazy} &:=& \sysm{N\fs N} &:& \sysm{\lam p\lam x.p'x \land lazy'x} &::& et(et)\\ \quad \nonumber\\ \textit{slowly} &:=& \sysm{S\bs NP \bs (S\bs NP)} &:& \sysm{\lam p\lam x.slowly'px} &::& et(et)\\ \quad \nonumber\\ \textit{and} &:=& \sysm{G\bs G\fs G} &:& \sysm{\lam p\lam q\lam r.pr \land qr} &::& ett(ett(ett))\\ \quad \nonumber\\ \textit{every} &:=& \sysm{S\fs(S\bs NP)\fs N} &:& \sysm{\lam{p}\lam{q}\forall x.px\to qx} &::& et(ett)\\ \textit{a} &:=& \sysm{S\fs(S\bs NP)\fs N} &:& \sysm{\lam{p}\lam{q}\exists x.px\land qx} &::& et(et t) \end{align}\]

Derive the meaning of the following sentences specifying their order of combination, syntactic categories, semantic interpretations and semantic types in each step, based on the lexicon given in :

  1. Every donkey sleeps.
  2. John walks slowly.
  3. A lazy donkey walks.
  4. Every student and a lazy donkey walk.
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You are given the lexicon:

Let \(\sysm{G}\) be an abbreviation for \(\sysm{(S\fs(S\bs NP))}\)

\[\begin{align} \textit{donkey} &:=& \sysm{N} &:& \sysm{\lam x.donkey'x} &::& et\\ \textit{student}&:=& \sysm{N} &:& \sysm{\lam x.student'x} &::& et\\ \textit{John} &:=& \sysm{S\fs(S\bs NP)} &:& \sysm{\lam p.p\cnct{}john'} &::& ett\\ \quad \nonumber\\ \textit{sleeps} &:=& \sysm{S\bs NP} &:& \sysm{\lam x.sleeps'x} &::& et\\ \textit{walk(s)} &:=& \sysm{S\bs NP} &:& \sysm{\lam x.walk'x} &::& et\\ \quad \nonumber\\ \textit{lazy} &:=& \sysm{N\fs N} &:& \sysm{\lam p\lam x.p'x \land lazy'x} &::& et(et)\\ \quad \nonumber\\ \textit{slowly} &:=& \sysm{S\bs NP \bs (S\bs NP)} &:& \sysm{\lam p\lam x.slowly'px} &::& et(et)\\ \quad \nonumber\\ \textit{and} &:=& \sysm{G\bs G\fs G} &:& \sysm{\lam p\lam q\lam r.pr \land qr} &::& ett(ett(ett))\\ \quad \nonumber\\ \textit{every} &:=& \sysm{S\fs(S\bs NP)\fs N} &:& \sysm{\lam{p}\lam{q}\forall x.px\to qx} &::& et(ett)\\ \textit{a} &:=& \sysm{S\fs(S\bs NP)\fs N} &:& \sysm{\lam{p}\lam{q}\exists x.px\land qx} &::& et(et t) \end{align}\]

Provide the lexical entries for the undefined expressions to derive the meanings of the following sentences. Your definitions need to be like in the provided lexicon.

  1. Every donkey admires John.
  2. A donkey who admires John sleeps.
  3. If John sleeps, every donkey sleeps.
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Write the needed lexical categories (phon, syn, sem) and derive the logical forms of the following sentences:

  1. \(\interp{\text{Every man and John walk}} = (\forall x. man'x \to walk'x) \land walk'j'\)

  2. \(\text{Every woman who loves John walks} = \forall x. woman'x \land love'j'x \to walk'x\) .
  3. \(\text{Every woman whom John loves walks} = \forall x. woman'x \land love'x\,j' \to walk'x\) .
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