For instance, if someone says “substutution of a term into a predicate is pullback”, then we clearly need a pullback square, so four arrows. The two arrows we start with are $t : B \to A$ (the substituted term) and the subset inclusion $i_P : P \to A$ (and you can figure this out because of the three forms of understanding “$P$ is a predicate on $A$” only this one is an arrow). The other two arrows are a subset inclusion $i_Q : Q \to B$ where $Q = \lbrace y \in B \mid t(y) \in P\rbrace$ and the arrow $Q \to P$ which is the restriction of $t$ to $Q$, i.e., it takes $y \in Q$ to $t(y) \in P$. Does that help?

]]>`Prop`

, for instance. In a topos it is $\Omega$, and it is isomorphic to $\mathcal{P}(\lbrace * \rbrace)$.
You are mistaken about excluded middle. It is not expressed as $\forall S \in \mathcal{P}(\lbrace * \rbrace) . S \lor \lnot S$ but rather as \mathcal{P}(\lbrace * \rbrace) . \star in S \lor \lnot (\star in S)$. In any case, this is a technicality which can be dealt with in many ways. For instance, one can observe that $\mathcal{P}(\lbrace * \rbrace)$ is a complete Heyting algebra under the $\subseteq$ ordering, and that logical formulas denote elements of this Heyting algebra. Then, once can really write $S \lor \lnot S$, and this would mean the same thing as $S \cup (\lbrace{\star\rbrace \setminus S)$. I recommend that you consult a textbook such as Scott & Lambek’s “Introduction to Higher-Order Categorical Logic”. They explain these things there. You will also learn from that textbook about internal languages, which allow us to mix syntactic forms with semantic objects.

]]>When you say: “Q is the pullback of the inclusion P \hookrightarrow B along t”, do you say it in the same why as “p2 is a pullback or fiber product of the pair. We also say that p2 is the pullback of f along g” from [http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf page 273]; if yes, then I must conclude that Q is an arrow, t(x) is an arrow, the inclusion from P to B is an arrow and the objects are the sets A, P & B; but then I’m missing the arrow from A to P and from A to B.

My attempt: https://postimg.org/image/dvw8j0moh/

I’m a noob and I’m going nuts trying to figure it out

]]>You say that I can imagine $\mathsf {Prop}$ to be the set $\mathcal P({\star})$. But then the law of excluded middle

$\forall\phi\in\mathsf{Prop}. \, \phi\lor\neg\phi$.

becomes

$\forall S\in \mathcal P({\star}).\, S\lor\neg S$.

But this latter statement does not make sense, since $S$ is a mathematical object and not a statement. Thus one can’t use $S$ to express a statement like “$S\lor\neg S$”. That’s where my confusion comes from. I hope you can clarify.