1. State whether each atomic formula is true or false in :

    \[\begin{gather*} \con{student}\con{anna}\\ \con{book}\con{anna}\\ \con{linguist}\con{cara}\\ \con{admire}\con{ben}\con{anna}\\ \con{admire}\con{anna}\con{ben}\\ \end{gather*}\]

  2. Evaluate the following formulas in .

    \[\begin{gather*} \exists x.\con{student}x\\ \forall x.\con{student}x \rightarrow \con{linguist}x\\ \exists x.\con{student}x \land \con{linguist}x\\ \forall x.\con{book}x \rightarrow \con{student}x\\ \exists x\forall y. \con{admire}y\cnct x\\ \con{student}x\\ \con{linguist}x\\ \exists x.\con{admire}x\cnct y\\ \end{gather*}\]
  3. Translate the following sentences into FOL, and evaluate in :
    1. Anna is a student.
    2. Ben read a book.
    3. Every student read a book.
    4. Some linguist recommended every book.
    5. No student recommended Ben.
    6. Not every student read a book.
    7. No book is read by every student.
    8. Some book is read by every student.

  4. You have two predicates \(p\) and \(q\). Express the following situations in first order logic:

    1. there is no \(p\) that is not also \(q\).
    2. there is exactly one \(p\).
    3. there are exactly two \(p\)s.
    4. there is at most one \(p\).
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For each pair, determine the logical relation between the left and right formulas (implies, contradicts, or independent):

\[\begin{gather} \forall x.\con{p}x\rightarrow \con{q}x& \exists x.\con{p}x\\ \forall x.\con{p}x\rightarrow \con{q}x& \exists x.\con{q}x\\ \exists x.\con{p}x\wedge \con{q}x& \exists x.\con{p}x\\ \forall x.\con{p}x &\exists x.\con{p}x\\ \neg\exists x\,\con{r}x& \forall x.\neg\con{r}x\\ \forall x.\con{p}x\land\con{r}x& \forall x.\con{p}x\rightarrow \con{r}x\\ \forall x.\con{p}x\land\con{r}x& \forall x.\con{p}x :\land. \forall x.\con{r}x\\ \forall x.\con{p}x\lor\con{r}x& \forall x.\con{p}x :\lor. \forall x.\con{r}x\\ \exists x.\con{p}x\land\con{r}x& \exists x.\con{p}x :\land. \exists x.\con{r}x\\ \exists x.\con{p}x\lor\con{r}x& \exists x.\con{p}x :\lor. \exists x.\con{r}x\\ \end{gather}\] 🗝️

Translate to first-order logic:

  1. No one speaks French. (Note: you can take \emph{speak French} as a single lexical item. But, if you like, try to take the two words separately.)
  2. No one speaks French or Arabic.
  3. Every student is happy.
  4. Every student is happy and tired.
  5. Every student is happy, and every student is tired.
  6. Every student is happy or tired.
  7. Every student is happy, or every student is tired.
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