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In each of the following we give an English sentence and a number of candidate logical expressions. For each of the logical expressions, state whether it (1) correctly expresses the English sentence; (2) is syntactically invalid and therefore meaningless; or (3) is syntactically valid but does not express the meaning of the English sentence.

  1. Every cat loves its mother or father.

    1. ${\forall,x;;} {Cat}(x) {:;{\Rightarrow}:;}{Loves}(x,{Mother}(x)\lor {Father}(x))$.

    2. ${\forall,x;;} \lnot {Cat}(x) \lor {Loves}(x,{Mother}(x)) \lor {Loves}(x,{Father}(x))$.

    3. ${\forall,x;;} {Cat}(x) \land ({Loves}(x,{Mother}(x))\lor {Loves}(x,{Father}(x)))$.

  2. Every dog who loves one of its brothers is happy.

    1. ${\forall,x;;} {Dog}(x) \land (\exists y\ {Brother}(y,x) \land {Loves}(x,y)) {:;{\Rightarrow}:;}{Happy}(x)$.

    2. ${\forall,x,y;;} {Dog}(x) \land {Brother}(y,x) \land {Loves}(x,y) {:;{\Rightarrow}:;}{Happy}(x)$.

    3. ${\forall,x;;} {Dog}(x) \land [{\forall,y;;} {Brother}(y,x) {;;{\Leftrightarrow};;}{Loves}(x,y)] {:;{\Rightarrow}:;}{Happy}(x)$.

  3. No dog bites a child of its owner.

    1. ${\forall,x;;} {Dog}(x) {:;{\Rightarrow}:;}\lnot {Bites}(x,{Child}({Owner}(x)))$.

    2. $\lnot {\exists,x,y;;} {Dog}(x) \land {Child}(y,{Owner}(x)) \land {Bites}(x,y)$.

    3. ${\forall,x;;} {Dog}(x) {:;{\Rightarrow}:;}({\forall,y;;} {Child}(y,{Owner}(x)) {:;{\Rightarrow}:;}\lnot {Bites}(x,y))$.

    4. $\lnot {\exists,x;;} {Dog}(x) {:;{\Rightarrow}:;}({\exists,y;;} {Child}(y,{Owner}(x)) \land {Bites}(x,y))$.

  4. Everyone’s zip code within a state has the same first digit.

    1. ${\forall,x,s,z_1;;} [{State}(s) \land {LivesIn}(x,s) \land {Zip}(x){{,=,}}z_1] {:;{\Rightarrow}:;}{}$
      $[{\forall,y,z_2;;} {LivesIn}(y,s) \land {Zip}(y){{,=,}}z_2 {:;{\Rightarrow}:;}{Digit}(1,z_1) {{,=,}}{Digit}(1,z_2) ]$.

    2. ${\forall,x,s;;} [{State}(s) \land {LivesIn}(x,s) \land {\exists,z_1;;} {Zip}(x){{,=,}}z_1] {:;{\Rightarrow}:;}{}$
      $ [{\forall,y,z_2;;} {LivesIn}(y,s) \land {Zip}(y){{,=,}}z_2 \land {Digit}(1,z_1) {{,=,}}{Digit}(1,z_2) ]$.

    3. ${\forall,x,y,s;;} {State}(s) \land {LivesIn}(x,s) \land {LivesIn}(y,s) {:;{\Rightarrow}:;}{Digit}(1,{Zip}(x){{,=,}}{Zip}(y))$.

    4. ${\forall,x,y,s;;} {State}(s) \land {LivesIn}(x,s) \land {LivesIn}(y,s) {:;{\Rightarrow}:;}{}$
      ${Digit}(1,{Zip}(x)) {{,=,}}{Digit}(1,{Zip}(y))$.