# Find the argument form for the argument and determine whether it is valid

Q: Find the argument form for the argument and determine whether it is valid. Explain your reasoning.

• If I get a bonus, I will buy the painting.
• If I sell my books, I will buy the painting.

• ∴ If I get a bonus or sell my books, I will buy the painting.

We have to build and solve this argument using the rules of inference, however, I have not found a way to do it so far.

What I have done: List of premises: p --> q, r --> q, Conclusion: (p V r ) ---> q

1. p --> q (if p then q )
2. r --> q
3. 3.

The question should be solved like this example:

Ex: Show that the premises "It is not sunny this afternoon and it is colder than yesterday," "We will go swimming only if it is sunny," "If we do not go swimming, then we will take a canoe trip," and "If we take a canoe trip, then we will be home by sunset" lead to the conclusion "We will be home by sunset."

Solution:

1. ¬p ^ q (Premise)
2. ¬p (Simplification using 1)
3. r --> p (premise)
4. ¬r (Modus tollens using 2 and 3)
5. ¬r --> s (Premise)
6. s (Modus ponens using 4 and 5)
7. s --> t (Premise)
8. t (Modus ponens using 6 and 7)

List of available inferences:

modus ponens modus tollens hypothetical syllogism disjunctive syllogism addition simplification conjunction resolution

We can also use logical equivalences including the ones with conditional statements

• Sometimes validity is easier shown in a truth table. All you need to do is show that whenever you have $p \to q$ true and $r \to q$ true, you always have $(p \lor r) \to q$ true – WaveX Jan 28 at 23:02
• Your example shows some of the rules that are apparently available for you to use, but what other rules do you have? – Bram28 Jan 28 at 23:09
• Check the updated question at the very end for list of inferences – DreamVision2017 Jan 28 at 23:21
• Youre right about the truth table but the question asks to present it in an argument form such as the example above – DreamVision2017 Jan 28 at 23:23
• How is the resolution rule defined? – Bram28 Jan 29 at 3:48

$$\def\fitch#1#2{\quad\begin{array}{|l} #1\\\hline #2\end{array}}$$
$$\fitch{\text{Premises}}{\fitch{\text{Antecedent}}{\vdots\\\text{Consequent}}\\\text{Antecedent}\to\text{Consequent}}$$
So in this case, you wish to show $$p\to q, r\to q \vdash (p\vee r)\to q$$.   As the assumption you need to make is a disjunction, you need so rule of inference that will eliminate a disjunction...aka a proof by cases argument.
$$\fitch{1. ~p\to q\\2.~r\to q}{\fitch{3.~p\vee r}{\vdots\\X.~q\qquad\qquad\text{... disjunction elimination}}\\Y.~(p\vee r)\to q\quad\text{3-X,conditional introduction}}$$