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Exercise 8.12

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

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