When a set of propositions is said to be consistent?

Question

Suppose I am given a set of propositions, and I am asked to find whether they are consistent of not? Can someone explain with an example. What does it actually mean when we say that this set of proposition is consistent?

Answer 1

See Consistency:

A set of formulas $\Gamma$ in a specified logical language (with negation: $\lnot$) is consistent if there is no formula $\varphi$ such that $\Gamma \vdash \varphi$ and $\Gamma \vdash \lnot \varphi$.

In more simple terms, a system $S$ is consistent if there is no statement $\varphi$ such that $S$ proves both $\varphi$ and its negation.

An example regarding propositional calculus is the system with some of the usual axioms systems for propositional logic with Modus Ponens as rule of inference and the "additional axiom": $\vdash p$.

Compare with Rosen, page 18:

System specifications should be consistent, that is, they should not contain conflicting requirements that could be used to derive a contradiction.

"Conflicting requirements" means two requirements $\text {req}$ and $\lnot \text {req}$.

If a system contains a pair of "conflicting requirements", we can derive - using the Conjunction rule (see page 72) - the contradiction $\text {req} \land \lnot \text {req}$.