Rules of inference help?

Question

I have a question here I need to solve:

1. For each of the arguments below, formalize them in propositional logic. If the argument is valid identify which inference rule was used, and formulate the tautology underlying the rule. If the argument is invalid, state whether the inverse or converse error was made.

(a) All cheaters sit in the back row. George sits in the back row. ∴ George is a cheater.

For this one I came up with the following:

$C$ = is a cheater

$B$ = sits in back row

$G$ = George

$x$ = students

$\forall$$x$($C(x)$ $\rightarrow$ $B(x)$)

$B(G)$

∴ $C(G)$

From there, I wrote that the converse error was made, since it didn't state that just because you sit in the back row you're a cheater.

The following one is harder for me.

(b) For all students x, if x studies discrete math, then x is good at logic. Dawn studies discrete math. ∴ Dawn is good at logic.

$x$ = students

$M$ = studies discrete math

$L$ = good at logic

$D$ = Dawn

$\forall$$x(M(x)$$\rightarrow$$L(x))$

$M(D)$

∴$L(D)$

I know this makes sense, but im not sure which rule of inference is made here. how do i determine that? If I got any of the statements wrong please let me know. Thanks for the help.

Answer 1

Your answer to the first question is correct.

The second argument is of the form: $$\begin{array}{} P \to Q \\ P \\ \hline \therefore Q \end{array}$$

This rule of inference is known as modus ponens. This shows up a lot in logic, and so you should memorize it.

There are many other rules of inference. Here are a few:

Modus Tollens $$\begin{array}{} P \to Q \\ \lnot Q \\ \hline \therefore \lnot P \end{array}$$

Conjunction $$\begin{array}{} P \\ Q \\ \hline \therefore P \wedge Q \end{array}$$

Here are a few websites with much more on them:

http://www.philosophypages.com/lg/e11a.htm https://en.wikipedia.org/wiki/List_of_rules_of_inference