How to express an invalid argument such as "affirming the consequent" in propositional logic?

Question

For example, say you want to write a symbolic statement for affirming the consequent. You could write it in the following two ways $$ (P \to Q, Q) \to P $$ or $$ \frac{P \to Q, Q}{P} $$ Which one should be used? Is it mostly a difference in taste/preference, or is one more favourable than another, or perhaps are there circumstances where one is used instead of the other?

Answer 1

I have never seen the first one … and for good reason: it uses the symbol $\to$ both as a logic symbol (a symbol of the language of logic) as well as a meta-logic symbol (a symbol that is used to say something about logic), and you really want to separate those two.

But the second way is a pretty standard to express an inference, sure! Another common notation is $P \to Q, Q \therefore P$

Answer 2

You seem to want a means of expressing an argument without actually asserting it is valid or invalid. So, if you want to express that $P$ allegedly follows from the premises $Q, P \to Q$, then the convention is to use a symbol such as the colon ":" or therefore sign "$\therefore$" or even the double slash "//"as follows

$$ Q, P \to Q : P \\ Q, P \to Q \therefore P \\ Q, P \to Q \text{ // } P \\ $$

I've seen these symbols often used to express arguments whose validity is not yet decided. Such arguments may be referred to as sequents. Tomassi expounds on this idea in his book Logic beginning on page $43$.

I do not suggest using the turnstile "$\vdash$" to formally express an argument unless your intent is to assert the conclusion is provable from the premises in a given system. This is because the turnstile "$\vdash$" has a very specific meaning, that is, a sentence of the form

$$ P_1,P_2,...,P_n \vdash C $$

not only alleges that $C$ follows from the premises $P_1,P_2,...,P_n$ but also makes the stronger claim that $C$ is in fact a logical consequence of the premises, provable from the formulas $P_1,P_2,...,P_n$ using the inference rules of the given system.

So, if you cannot prove $P$ is a logical consequence of the premises $Q, P \to Q$ in the system you're working with, then using the turnstile to express

$$ Q, P \to Q \vdash P $$

would be patently incorrect. In fact, the argument $ Q, P \to Q : P $ is an invalid argument form referred to as "affirming the consequent" in all propositional systems I'm aware of (although it could certainly be a valid argument in other systems). In such cases, writing

$$ Q, P \to Q \nvdash P $$

would be correct, stating that $Q, P \to Q$ does not prove $P$.

I also agree with the comments made about using the arrow and horizontal bar.

Answer 3

How to express an invalid argument such as "affirming the consequent" in propositional logic?

We can say that $((P\to Q)\land Q) \to P$ is a contingency, i.e. it is sometimes true, and sometimes false.

The truth table:

Specifically, we can formally state that $(((P\to Q)\land Q) \to P)\iff (Q\to P)$

The truth table: