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