| title |
date |
lastmod |
First Order Logic |
2022-11-08 |
2022-11-21 |
Propositional Logic can only deal with a finite number of propositions:
T: Tommy is faithful
J: Jimmy is faithful
L: Laika is faithful
$All\ dogs\ are\ faithful\iff T\land J\land L$
What if there is an infinite/unknown number of dogs?
“All dogs are mammals"
General form: $\forall xDog(x)\implies Mammal(x)$
Use conjunction? $\forall x Dog(x) \land Mammal(x)$ : this is means everything is a dog and a mammal!
"John owns a dog"
General form: $\exists x Dog(x)\land Owns(John,x)$
Use implication? $\exists xDog(x)\implies Owns(John,x)$: this can mean that John owns things which are not dogs as well
Using substitutions is also called Generalized Modus Ponens.
The substitution used is called the unifier.
$$\begin{align}
\exists xStudent(x)\land \neg Takes(x,AI)\tag1\equiv Student(K)\land\neg Takes(K,AI) \\
\tag2 \exists xStudent(x)\land Takes(x,AI)\land\neg pass(x,AI)\equiv \\
Student(F)\land Takes(F,AI)\land \neg pass(F,AI)\tag3\\
\forall x,y \neg Student(x)\lor\neg pass(x,y)\lor\neg hard(y)\lor diligent(x)\models\\tag4\neg Student(x)\lor\neg pass(x,y)\lor\neg hard(y)\lor diligent(x) \\
3+4:\neg pass(x,y)\lor\neg hard(y)\lor diligent(x)\tag5 \\
\tag6 3+4+5: Takes(x,AI)\lor\neg hard(AI)\\
6+Subst{x/K}\ Takes(K,AI)\lor\neg hard(y)\tag7\\
1+7: \neg hard(AI)\tag8\\
8+iv:\emptyset
\end{align}$$
