# Construct a truth table for the following sentence to determine whether the argument is valid or invalid P ∨ Q, P → R, ¬R ∴ Q

Proving Validity of a Symbolic Argument Using Truth Tables

I am looking to determine the validity of this argument using the truth table method:

P ∨ Q, P → R, ¬R ⊨ Q

I cannot yet embed the image of my truth table on here because I haven't earned emough points of my profile yet, so I will try to explain it the best I can.

I have used the column headers | P | Q | R |¬R | P ∨ Q | P → R | Q |

After filling out this truth table I have found that in row 3, the premises 'P ∨ Q' and 'P → R' are true and yet the conclusion is false, which would indicate that the argument is invalid.

However I am wondering whether I should also take into account the truth value of the premise '¬R'. If I do this, then I find only one critical row (row 6), and here I find the conclusion to be true. As there are no rows where the premises are true and the conclusion is false, I read from this that the argument is valid.

However, I am apprehensive to draw this conclusion; for one because I am not sure whether I have used the correct column headers, and for another because I am not sure whether I should be be looking at '¬R' as a separate premise and considering it's truth value to determine the validity of the conclusion.

As you can probably tell, I am a beginner in logic, so I would appreciate any help to clarify this. Thank you.

• – Dan Christensen Nov 29 '17 at 19:08
• @Dan Christensen Do you really think that WA is the absolute answer for this question ? – Jean Marie Nov 29 '17 at 19:14
• You have to connect your different expressions by "&" connector, i.e., build the truth table of $(P ∨ Q)&(P → R)&(¬R)→Q$ and verify it's a tautology. Remark: Mapping ⊨ onto → is the only thing you can do in the context of boolean logic. – Jean Marie Nov 29 '17 at 19:20
• @JeanMarie Thank you for your advice, that is really helpful. – user508231 Nov 29 '17 at 19:48
• A truth table is more bother than it is worth when there is a simple proof of the statement. – William Elliot Nov 30 '17 at 2:24

So yes, you do have to take into account all of the premises, including $\neg R$. And since you found that in the only row where all of the premises are true, the conclusion is also true, the argument is valid.