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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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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