4

I'm trying to wrap my head around this new subject. I have to determine the validity of this argument (using a truth table):

"If Steve went to the movies then Maria's sister would not have stayed home. Either Steve went to the movies or Maria or both. If Maria went, then Maria’s sister would have stayed home. Both Maria's sister and Steve’s sister stayed home. So, Steve did not go to the movies."

So far, I've come up with this, as a first step. I doubt it's correct. Please correct and guide me, as I'm still new to all of this. Thank you

P = Steve went to the movies
Q = Maria's sister stayed home
R = Maria went to the movies
S = Steve's sister stayed home

$(P \rightarrow \lnot Q), ((P \lor R) \lor (P \land R)), (R \rightarrow Q), (Q \land S) ∴ \lnot P$

Robert Shore
  • 23,332
explorer
  • 77
  • 2
    The phrase $P \lor R$ means either or both of $P$ and $R$ are true, so you don't need to separately account for the possibility of $P \land R$. So that's a redundancy rather than an error. Otherwise you've gotten off to a correct start. You should find that you only need the first and fourth statements to evaluate whether the conclusion necessarily follows. – Robert Shore Nov 28 '19 at 07:10
  • If you are satisfied with your answer, then please click the green check mark to close the post. Thanks! – RyRy the Fly Guy Nov 30 '19 at 14:17

2 Answers2

2

A reasoning with premises P1, P2, P3, etc. and conclusion C is valid iff its corresponding conditional is valid ( = is aa tautology).

By " corresponding conditional" I mean : " P1 & P2 & P3... --> C"

( For this kind of problem, this definition is perfectly OK).

So build the corresponding conditional of your reasoning by putting an "&" between the premises, adding an arrrow and finally your conclusion.

Build a truth table for this conditional.

Note : here you need a 2 to the 4th power = 16 lines truth table.

In case this conditional has truth value " true" on all lines of the truth table, the reasoning is valid.

enter image description here

Note : apparently, premises 2 and 3 are useless, since the conclusion can be proved using only premises 1 and 4

(1) P --> ~ Q

(2) Q & S

(3) Since Q and S is true, Q is true.

(4) Since P --> ~Q is true , Q --> ~P is true ( by contraposition)

(5) Since Q --> ~P and Q are true, ~ P is true ( as desired).

1

Your translation into propositional logic looks great! Note that the inclusive disjunction $\vee$ is true whenever one or both operands are true, so the statement "either Steve went to the movies or Maria or both" may be written as $P \vee R$ as opposed to $(P \vee R) \vee (P \wedge R)$. However, the latter is not incorrect; it is actually logically equivalent, so if you wish to leave it the way it is, then you may do so.

Now that you have correctly translated your premises and conclusion into propositional logic, you may evaluate the validity of the argument with a truth table using either of the following approaches:

APPROACH I

$1$. Construct a truth table including all premises as well as the conclusion. Be sure to demarcate the premises and conclusion from one another. For example, you may place a slash $/$ between premises and a double slash $//$ to separate the conclusion from the premises.

$2$. Fill in the truth table with its truth values, and compute the truth value of each statement.

$3$. Scan each line of the truth table to ensure the following: if the truth values of all the premises are true, then the truth value of the conclusion is also true. If this is the case, then the argument is valid. If this is not the case, then the argument is not valid. In other words, if there exists at least one line in the truth table such that the truth values of all the premises are true and the truth value of the conclusion is false, then the argument is not valid. This criteria follows by definition of validity.

APPROACH II

Alternatively, you may check and see whether the corresponding conditional of the argument is a tautology. In other words, simply create a truth table for the corresponding conditional and determine whether the statement is always true. If so, the argument is valid; otherwise, it is not valid.

RyRy the Fly Guy
  • 5,950
  • 1
  • 11
  • 27