ions on $\mathcal{L}_{Q}^{P}$.

A$_{1}$. *The propositional letters* $p$*,* $q$*, ... are substituted by the symbols* $E(x)$*,* $F(x)$*, ..., with* $E$*,* $F$*, ...* $\in \mathcal{E}$*.*

A$_{2}$. *The set* $\psi _{R}^{Q}$* of all rfs of* $\mathcal{L}_{Q}^{P}$* reduces to the set of all atomic rfs of* $\mathcal{L}_{Q}^{P}$* (in different words, if* $\alpha $* is a rf of* $\mathcal{L}_{Q}^{P}$*, then* $\alpha =E(x)$*, with* $E\in 
\mathcal{E}$*).*

A$_{3}$. *Only the logical-pragmatic signs* $\vdash $*,* $N$*,* $K$* and* $A$* appear in the afs of* $\mathcal{L}_{Q}^{P}$*.*

The substitution in A$_{1}$ aims to suggest the *intended interpretation* that we adopt in the following. To be precise, the rfs $E(x)$, $F(x)$, ... are interpreted as sentences stating that the physical object $x$ has the properties $E$, $F$, ..., respectively (Sec. 2.4).

The restriction in A$_{2}$ aims to select rfs that are interpreted as *testable* sentences, i.e., sentences stating testable physical properties (Sec. 2.1), so that physical procedures exist for testing their truth values (which may not occur in the case of a rf of the form, say, $E(x)\vee F(x)$; note that a similar restriction has been introduced in Ref. 27 when recovering intuitionistic propositional logic within $\mathcal{L}^{P} $).

The restriction in A$_{3}$ is introduced for the sake of simplicity, since only the pragmatic connectives $N$, $K$ and $A$ are relevant for our goals in this paper.

Because of A$_{1}$, A$_{2}$ and A$_{3}$, the set $\psi _{A}^{Q}$ of afs of $\mathcal{L}_{Q}^{P}$ is made up by all formulas constructed by means of the following recursive rules.

\(i) *Let* $E(x)$* be a rf. Then* $\vdash E(x)$* is an af.*

\(ii) *Let* $\delta $* be an af. Then,* $N\delta $* is an af.*

\(iii) *Let* $\delta _{1}$* and* $\delta _{2}$* be afs. Then,* $\delta _{1}K\delta _{2}$* and* $\delta _{1}A\delta _{2}$* are afs.*

Let us come now to the semantics of $\mathcal{L}_{Q}^{P}$. We introduce the following assumption on $\mathcal{L}_{Q}^{P}$.

A$_{4}$. *E