# Modus Ponens Proof

I have written the truth table for all of the forms of $P$ and $Q$.
Then maintained the table to find $P \rightarrow Q$ and $[(P \rightarrow Q) \wedge P]$.As we know, we can write arguments in forms of $[Expression_{one} \wedge Expression_{two}]\rightarrow Result$
Right now, I have the table with all possible forms of $P$ and $Q$ which leads to their conclusions.How can I prove that Modus Ponens is a valid argument according to the table I have written?

• Which definition of "valid argument" are you using? Sep 24, 2014 at 14:22
• @HenningMakholm Valid means if you consider the premises to be true, then the conclusion must be true.However, I don't know any other definition of a valid argument and I appreciate if you could possibly explain all possible cases. Sep 24, 2014 at 14:44
• Why nobody answers my question? Sep 24, 2014 at 22:50

Modus ponens rule is :

from $$P \rightarrow Q$$ and $$P$$, infer $$Q$$.

This rule correspond to the soundness of the "argument" :

$$P \rightarrow Q, P \vDash Q$$

where an argument is sound when, from true premises, licences the derivation of a true conclusion.

This means that modus ponens is equivalent to :

$$\vDash ((P \rightarrow Q) \land P) \rightarrow Q$$,

i.e. $$((P \rightarrow Q) \land P) \rightarrow Q$$ is a tautology.

Thus, as said in the above answer, you can check it with a truth table.

When you have a truth table for $(P \to Q) \wedge P$ note that whenever the whole statement is true, $Q$ is also true.

• Did you mean, whenever the whole statement is true $P$ is true? Sep 26, 2014 at 17:38
• @FreeMind Modus Ponens states that if we have $(P \to Q) \land P$, then we can get $Q$. A rule is valid if the truth of its conclusion follows from the truth of its premises. So you have to show that if the premise $(P \to Q) \land P$ is true, you can get the truth of the conclusion $Q$. Sep 30, 2014 at 8:41

It is simple: Basically, if Modes Ponens is a valid inference rule, then whenever we know some P implies Q, and at the same time we know that P happened to be true, then Q must be true. So, basically,

the Modes Ponens is this statement: (((P => Q) & P) => Q)

And that statement should, at the end (i.e., in the last column of the last Q, be TRUE for all truth values of P and Q, and it will). Do the truth table for the above statement

There's another branches of mathematics that also verify the validity of Modes Ponens (e.g, probability theory and set theory). if you like I can supply the simple proof there too.