Law of syllogism

Another pillar of deductive reasoning is contained in the following argument:

$ $$\begin{array}{l} \text{If Socrates is a man, then he is an animal.} \ \text{If Socrates is an animal, then he is mortal.} \\hline \text{Therefore, if Socrates is a man, then he is mortal.} \end{array}$$ $

Here we are working with conditionals only: we are saying that given two conditionals with a "middle" proposition, we can skip it altogether. Formally,

$ $$\begin{array}{l} A \rightarrow B \ B \rightarrow C \\hline \therefore A \rightarrow C \end{array}$$ $

Remember, this is valid regardless from A, B or C being true or false.