The validity of modus ponens and modus tollens is guaranteed by the definition of p→q. The meaning of this symbol again is defined by the following truth table (a “truth table” defines the meaning of a logical connective by assigning a truth value—true or false—to the connected term for each of all possible combinations of the propositions involved, i.e. “p” and “q” in our case. The first two columns in the truth table below list all possible combinations of truth values for the involved propositions, and the third column assigns the truth values for the whole term p→q):

p	q	p→q
T	T	T
T	F	F
F	T	T
F	F	T

This truth table shows, that the term p→q is only false, if p is true and q false (see the 2^nd row). In all the other possible cases, the term is true. Based on this definition of p→q, we can formulate the proof for the validity of modus ponens as follows. Since there are only four possible combinations of p and q (see the first and second column in the table below), we can put the two premises of the modus ponens in columns three and four, and its conclusion in column 5:

p	q	p→q	p	q
T	T	T	T	T
T	F	F	T	F
F	T	T	F	T
F	F	T	F	F

Since we defined the validity of an argument scheme by the fact that the conclusion is necessarily true if the premises are true, we only have to check those rows in our truth table in which all the premises are true. In the truth table for modus ponens, this is the case only in the first row. Since in this row also the conclusion is true (last column; the truth values here are simply the truth values as defined in the 2^nd column), the argument scheme is valid.

By contrast, the so-called “affirming the consequent” is an invalid argument scheme, which again can be demonstrated by means of a truth table:

p	q	p→q	q	p
T	T	T	T	T
T	F	F	F	T
F	T	T	T	F
F	F	T	F	F

In this case, we get true premises in the 1^st and 3^rd row, but in the 3^rd row the conclusion is false. Therefore, this is an invalid argument scheme. In a similar way, the validity of modus tollens can be proved.