Arguments

We can now define propositions, which are either true or false, but we don’t care about their values. What’s the point, then?

Let’s look at the Socrates example once again.

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

Here we have three propositions. The above form is logic’s way of saying: If we hold the two first propositions true, then the third one is also true. Note that this is a conditional: we don’t know, or care, if the first two propositions are intrinsically true (for all we know, Socrates could be my pet rock), but we posit that if we consider the first two propositions true, the third one is also automatically true.

Modus ponens

In natural language, the above argument looks self-evident: of course, you can create the third one from the two first. Now take a look at the formal notation of the same argument:

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

If you don’t know what the symbols mean, you may have some doubt about the validity of the reasoning. A particular logic is defined by the arguments you claim as valid. In deductive logic, the first basic valid argument is the above one, called modus ponens.

Also known as law of detachement, or affirming the antecedent, modus ponens expresses formally what we usually mean by “if A then B”. If the conditional holds, and A holds, then we can say that B holds.

Validity

An argument has two parts: the hypotheses (the propositions above the line), and the conclusion (below the line).

An argument declared valid in a particular logic if you can apply a chain of valid arguments to the hypotheses to reach the conclusion. The modus ponens example is valid by definition of deductive logic, but several chained applications of modus ponens build a valid argument:

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

Here we have three hypotheses, and by applying modus ponens two times we can reach the conclusion: the argument is valid in deductive reasoning.