  We present a logic for the reasoning about necessity and justifications which
is independent from relational semantics. We choose the concept of
justification -- coming from a class of "Justification Logics" (Artemov 2008,
Fitting 2009) -- as the primitive notion on which the concept of necessity is
based. Our axiomatization extends Suszko's non-Fregean logic SCI (Brown, Suszko
1972) by basic axioms from Justification Logic, axioms for quantification over
propositions and over justifications, and some further principles. The core
axiom is: $\varphi$ is necessarily true iff there is a justification for
$\varphi$. That is, necessity is first-order definable by means of
justifications. Instead of defining purely algebraic models in the style of
(Brown, Suszko 1972) we extend the semantics investigated in (Lewitzka 2012) by
some algebraic structure for dealing with justifications and prove soundness
and completeness of our deductive system. Moreover, we are able to restore the
modal logic principle of Necessitation if we add the axiom schema
$\square\varphi \to \square\square\varphi$ and a rule of Axiom Necessitation to
our system. As a main result, we show that the modal logics S4 and S5 can be
captured by our semantics if we impose the corresponding modal logic principles
as additional semantic constraints. This will follow from proof-theoretic
considerations and from our completeness theorems. For the system S4 we present
also a purely model-theoretic proof.

Logic in Computer Science