Exercise 8.10

This exercise uses the function ${MapColor}$ and predicates ${In}(x,y)$, ${Borders}(x,y)$, and ${Country}(x)$, whose arguments are geographical regions, along with constant symbols for various regions. 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. Paris and Marseilles are both in France.

    1. ${In}({Paris} \land {Marseilles}, {France})$.

    2. ${In}({Paris},{France}) \land {In}({Marseilles},{France})$.

    3. ${In}({Paris},{France}) \lor {In}({Marseilles},{France})$.

  2. There is a country that borders both Iraq and Pakistan.

    1. ${\exists\,c\;\;}$ ${Country}(c) \land {Border}(c,{Iraq}) \land {Border}(c,{Pakistan})$.

    2. ${\exists\,c\;\;}$ ${Country}(c) {:\;{\Rightarrow}:\;}[{Border}(c,{Iraq}) \land {Border}(c,{Pakistan})]$.

    3. $[{\exists\,c\;\;}$ ${Country}(c)] {:\;{\Rightarrow}:\;}[{Border}(c,{Iraq}) \land {Border}(c,{Pakistan})]$.

    4. ${\exists\,c\;\;}$ ${Border}({Country}(c),{Iraq} \land {Pakistan})$.

  3. All countries that border Ecuador are in South America.

    1. ${\forall\,c\;\;} Country(c) \land {Border}(c,{Ecuador}) {:\;{\Rightarrow}:\;}{In}(c,{SouthAmerica})$.

    2. ${\forall\,c\;\;} {Country}(c) {:\;{\Rightarrow}:\;}[{Border}(c,{Ecuador}) {:\;{\Rightarrow}:\;}{In}(c,{SouthAmerica})]$.

    3. ${\forall\,c\;\;} [{Country}(c) {:\;{\Rightarrow}:\;}{Border}(c,{Ecuador})] {:\;{\Rightarrow}:\;}{In}(c,{SouthAmerica})$.

    4. ${\forall\,c\;\;} Country(c) \land {Border}(c,{Ecuador}) \land {In}(c,{SouthAmerica})$.

  4. No region in South America borders any region in Europe.

    1. $\lnot [{\exists\,c,d\;\;} {In}(c,{SouthAmerica}) \land {In}(d,{Europe}) \land {Borders}(c,d)]$.

    2. ${\forall\,c,d\;\;} [{In}(c,{SouthAmerica}) \land {In}(d,{Europe})] {:\;{\Rightarrow}:\;}\lnot {Borders}(c,d)]$.

    3. $\lnot {\forall\,c\;\;} {In}(c,{SouthAmerica}) {:\;{\Rightarrow}:\;}{\exists\,d\;\;} {In}(d,{Europe}) \land \lnot {Borders}(c,d)$.

    4. ${\forall\,c\;\;} {In}(c,{SouthAmerica}) {:\;{\Rightarrow}:\;}{\forall\,d\;\;} {In}(d,{Europe}) {:\;{\Rightarrow}:\;}\lnot {Borders}(c,d)$.

  5. No two adjacent countries have the same map color.

    1. ${\forall\,x,y\;\;} \lnot {Country}(x) \lor \lnot {Country}(y) \lor \lnot {Borders}(x,y) \lor {}$\ $\lnot ({MapColor}(x) = {MapColor}(y))$.

    2. ${\forall\,x,y\;\;} ({Country}(x) \land {Country}(y) \land {Borders}(x,y) \land \lnot(x=y)) {:\;{\Rightarrow}:\;}{}$\ $\lnot ({MapColor}(x) = {MapColor}(y))$.

    3. ${\forall\,x,y\;\;} {Country}(x) \land {Country}(y) \land {Borders}(x,y) \land {}$\ $\lnot ({MapColor}(x) = {MapColor}(y))$.

    4. ${\forall\,x,y\;\;} ({Country}(x) \land {Country}(y) \land {Borders}(x,y) ) {:\;{\Rightarrow}:\;}{MapColor}(x\neq y)$.

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