basic constructions:
strong axioms
further
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
In logic, a proposition is intended to be interpreted semantically as having a truth value. In modern logic, it’s cleanest to start by specifying a context and considering the propositions in that context.
If (in a given context $\Gamma$) we have a type $A$, then we may extend $\Gamma$ to a context $\Delta \coloneqq \Gamma, x\colon A$ (assuming that the variable $x$ is not otherwise in use). We may then think of any proposition in $\Delta$ as a predicate $P$ in $\Gamma$ with the free variable $x$ of type $A$; this generalises to more complicated extensions of contexts (say by several variables).
If $P$ is a predicate with free variable $x$ of type $A$ and $t$ is a term of type $A$, then we get a proposition $P[t/x]$ by substituting $t$ for every instance of $x$ in $P$. Conversely, any proposition $Q$ may be interpreted as a predicate $Q[\hat{x}]$ in which the free variable $x$ simply doesn’t appear. (We have $Q[\hat{x}][t/x] = Q$ for every term $t$.)
There is a more traditional approach of viewing a predicate as a function from terms to propositions, a propositional function. Then $P[t/x]$ is written $P(t)$, while $P$ itself from above is written $P(x)$ (since a variable is a term). In this approach, less care is usually taken with the context, so that $Q[\hat{x}]$ may be conflated with $Q$ (since $Q[\hat{x}](x) = Q$, or this would be so if $x$ were a term in $\Gamma$ instead of only in $\Delta$).
From a higher-categorical point of view, predicates could be viewed as an indexed family of propositions with index object $I$ and function $P:I \rightarrow Prop$, the (-1)-groupoid version of the indexed family of sets found in set theory with index object $I$ and functor $S:I \rightarrow Set$ and the indexed family of groupoids found in groupoid theory? with index object $I$ and 2-functor $G:I \rightarrow Grpd$.
In categorial logic/categorical semantics, we have a category $\mathcal{C}$ and a class of monomorphisms? (often all monomorphisms) $\mathcal{M}$ in $\mathcal{C}$. Then a context is an object of $\mathcal{C}$ and a proposition in the context $\Gamma$ is an $\mathcal{M}$-subobject of $\Gamma$. We also have a class of display maps (often all morphisms in $\mathcal{C}$) such that $\mathcal{M}$ is closed under pullbacks both along display maps and along sections of display maps. These two ways of pulling back propositions in one context to propositions in another context correspond (respectively) to forming $Q[\hat{x}]$ and $P[t/x]$.
More specifically, if $\mathcal{C}$ is a finitely complete category, then the objects of $\mathcal{C}$ may equivalently be viewed as contexts and as types in the internal language of $\mathcal{C}$; a morphism from $\Gamma$ to $A$ is a term of type $A$ in context $\Gamma$. The extension of $\Gamma$ by a variable $x$ of type $A$ is the product $\Gamma \times A$, and the display map to $\Gamma$ is simply the projection. Every term $t\colon \Gamma \to A$ defines a section of this display map, and we may literally construct $Q[\hat{x}]$ and $P[t/x]$ as pullbacks.
If $\mathcal{C}$ is even a topos, then a proposition $Q$ in $\Gamma$ may be identified with a term whose type is the subobject classifier $\Omega$, and the predicate $Q[\hat{x}]$ is the composite $\Gamma \times A \to \Gamma \to \Omega$. Given a term $t\colon \Gamma \to A$ and a predicate $P\colon \Gamma \times A \to \Omega$, the proposition $P[t/x]$ is the composite $\Gamma \to \Gamma \times A \to \Omega$. Internalising a bit (by currying), we may view $Q$ as a global element $1 \to \Omega^\Gamma$ and $P$ as a morphism $A \to \Omega^\Gamma$, recovering the view that predicates are proposition-valued ‘functions’ (morphisms).
In general, we may intuitively think of an object $A$ in the slice category $\mathcal{C}/\Gamma$ as the ‘set’ (object) of possible values of terms $t$ of type $A$ in context $\Gamma$, and think of a predicate $P$ with a free variable of type $A$ (in the same context) as being the ‘subset’ (subobject) on those $t$ for which the statement $P(t)$ is true.
In type theory under the propositions as types paradigm, every type represents the proposition that it is inhabited. Hence the types which have at most one term may be identified with propositions (“propositions as some types”). In homotopy type theory these are the (-1)-types. The reflection that sends types to their underlying proposition qua (-1)-truncation is the n-truncation modality for $n = (-1)$, also called bracket type-formation.
In propositional logic, we fix a single context (considered the empty context) and consider the logic of propositions in that context. In predicate logic, we fix the empty context but work also in extensions of that context by free variables. Predicate logic uses quantifiers as a way to move between contexts, more specifically to move from a predicate $P$ in a given context $\Gamma$ (which is a proposition in some extension of $\Gamma$) to a proposition in $\Gamma$. The free variables in the predicate still appear in the written form of the proposition, but they are now bound variables and are not free in the proposition's context; some logicians prefer to systematically replace bound variables with numbered placeholders (especially when defining Gödel number?s and the like).