Assessment

The following question was given to ChatGPT

Determine what's correct, fix all the mistakes, and assign a letter grade: A$,$B$,$C$,$D$,$F$.$

Question: Find two interpretations $I_1$ and $I_2$ such that $I_1()=T$ and $I_2()=F$

where $$ is the FOL statement EEx$(QxPxc).$

Answer:

Q Let's analyze the given first$-$order logic $($FOL$)$ statement $,$ which is $),$ and

find two interpretations $I_1$ and $I_2$ such that $I_1()=T($true$)$ and $I_2()=F($false$).$

FOL Statement:

$=$EEx$(Q(x)P(x,c))$

This statement asserts that there exists some $x$ in the domain such that if $Q(x)$ is true, then

$P(x,c)$ must also be true.

Interpretation $I_1($where $$ is true$)$:

Domain $D$:$\{d_1,d_2\}$

Interpretation $I_1$ :

$c$ refers to $d_2$

Predicate $Q$ :

$Q(d_1)$ is true

$Q(d_2)$ is false

Predicate $P$ :

$P(d_1,d_2)$ is true

$P(d_2,d_2)$ is irrelevant since $Q(d_2)$ is false

Under this interpretation:

EEx$(Q(x)P(x,c))$ holds because we can choose $x=d_1.$ For $x=d_1$ :

$Q(d_1)P(d_1,d_2)$ is true because $Q(d_1)$ is true and $P(d_1,d_2)$ is true.

Therefore, $$ is true $(I_1()=T)$ under this interpretation.

Interpretation $I_2($where $$ is false$)$:

Domain $D$:$\{d_1,d_2\}$

Interpretation $I_2$ :

$c$ refers to $d_2$

Predicate $Q$ :

$Q(d_1)$ is true

$Q(d_2)$ is false

Predicate $P$ :

$P(d_1,d_2)$ is false

$P(d_2,d_2)$ is irrelevant since $Q(d_2)$ is false

Under this interpretation:

EEx$(Q(x)P(x,c))$ does not hold because for $x=d_1$ :

$Q(d_1)P(d_1,d_2)$ is false because $Q(d_1)$ is true and $P(d_1,d_2)$ is false.

Therefore, $$ is false $(I_2()=F)$ under this interpretation.

Summary:

Interpretation $I_1$ :

Domain $D=\{d_1,d_2\}$

$c$ refers to $d_2$

$Q(d_1)=$ true, $Q(d_2)=$ false

$P(d_1,d_2)=$ true

$$ is true $(I_1()=T).$

Interpretation $I_2$ :

Domain $D=\{d_1,d_2\}$

$c$ refers to $d_2$

$Q(d_1)=$ true, $Q(d_2)=$ false

$P(d_1,d_2)=$ false

$$ is false $(I_2()=F).$