7.2.1 What's that

I won't be able to explain everything, since various previous concepts are needed, but I will try to say a little about them. First, some changes:

Now we won't talk just about general facts (it rains, it's warm, etc.), but we will have a domain of known things, and we will have to say which property is true for each element.

For instance: we have the domain $\{ p,\ t,\ r\}$, which represent respectively to PROLOG (a logical programming language), a telephone, and a radio. p, t, r.

We also add a predicate letter (they're not called propositional letters anymore) $E$, which will have the following meaning: when we write $Ex$ (read ``$E$ of $x$'', but written together) we mean that $x$ is an electronic device. We will also have $Sx$ to say that $x$ is a piece of software, and $Tx$ which will mean that $x$ is a text processor.

Now we know that are true $Et$, $Er$, $Sp$ and nothing else.

Quantifiers make possible to write truths referring to some elements from the domain. There exist two quantifiers:

• Universal quantifier: $\forall$. When we put $\forall xPx$ (``for all $x$, $P$ of $x$''), we mean that all elements on the domain make true the property $P$.
• Existential quantifier: $\exists$. $\exists xPx$ (``there exists $x$ such that $P$ of $x$'') we mean that at least one element from the domain makes true the property $P$.

For instance, now are true the following formulas: $\forall x(Ex\vee Sx)$, $\neg\exists xTx$, $\forall x(Tx\Rightarrow\neg Ex)$, $\exists xEx\wedge\exists xSx$ and several more. Quantifiers have the same priority as the operator $\neg$.

The rules explained here will work only with free substitutions. Sorry for not saying what that means, but I don't want to go out of topic.