*5.3. Basic Properties*

In this subsection, we list some basic properties of the logical operators, such as the linearity and monotonicity of the integral. We use them in an exemplary reasoning.

The DC4F calculus has the following logical properties:


Point (1) concerns integration of constants. Point (2) expresses the fact that integrals are linear operators, i.e., they commute with multiplication by constants and are distributive in respect to addition. Point (3) expresses the monotonicity of integral operators. If the antecedent inequality holds for all non-negative reals, then the consequent inequality holds for all time intervals. Property (4) can be expressed as ∫ *ba f dt* + ∫ *cb f dt* = ∫ *ca f dt*. The above-mentioned points are specific to DC4F because it allows integrable functions, not only propositional ones as in the case of DC. The following points are common with DC (cf. [4], Section 2). Property (5) states that every interval can be split into an interval of length 0 and the rest. Point (6) specifies three exemplary propositional tautologies (cf., e.g., [27]) that we use later in an exemplary derivation. The first tautology says that truth is implied by any formula. The second tautology expresses the monotonicity of the conjunction operator. The third one expresses the commutativity of conjunction. It should be noted that all propositional tautologies hold in case of DC as well as DC4F. Point (7) specifies the monotonicity of the chop operator. Property (8) is the modus ponens reasoning rule. It says that if the antecedent of an implication is true and the implication is true, then the consequent of the implication is true as well. The rule is used in many kinds of logic (cf., e.g., [2,27]).
