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If i have the following arguments :

\begin{align} & a \to (b \lor c)\\ & \lnot b \lor \lnot c \\ & c \lor a \\ & --- \\ & b \end{align}

How do i prove that its valid. My thought was that if the conclusion is false, that is to say b = F, and we also assume that all the premises are True,

then $\lnot b$ is T. This leads us to say that ($\lnot b \lor \lnot c$) makes $\lnot c$ to be T and hence c = F

But to make $c \lor a$ True so must a = T. But this says otherwise in $a \to (b \lor c)$. That led me to think the argument is invalid. But my co-partner says its valid. Am a little confused here what is the meaning of being or not being valid.

Hakim Marley
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1 Answers1

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Your approach will lead to finding a counterexample in this case.

For an argument to be valid, all possible truth assignments of the propositions ($a$, $b$, $c$) that lead to the premises being true must also lead to the conclusion(s) being true. That is, $p_0 \land p_1 \land \ldots \land p_n \Rightarrow q_0 \land \ldots \land q_m$, where the $p_i$ are the premises and the $q_j$ are the conclusions, must hold for all truth values of the underlying propositions.

In the worst case, you can always just construct a truth table to decide the validity of a propositional argument (however, this doesn't provide much insight into why the argument is or isn't valid and you can't do this once quantifiers enter into the mix). You can also use a deductive system (https://en.wikipedia.org/wiki/Formal_system#Deductive_system) to prove an argument valid. On the other hand, a counterexample suffices to prove that an argument isn't valid.

As barak manos points out above, the counterexample in this case is $a = F$ and $c = T$, since then we have that the first and third premises are true, and the second is true if and only if the conclusion is false.

Chris Dugan
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  • So what you basicall mean is that if i can prove that all the premises are true and their preceeding conclusions then the argument is valid....right? – Hakim Marley Jan 04 '16 at 17:00
  • Right, if whenever the premises are true, the conclusions are true, then you've proved it valid. – Chris Dugan Jan 04 '16 at 17:03
  • Ok. but one other thing, the truth table of $p \to q$ shows 3 possible outcomes for True and one for False. If i follow this table and get the premises final value and conclusions to lead to true, have i proved the validity of the arguments? – Hakim Marley Jan 04 '16 at 17:06
  • Not sure I follow, but I think what you're saying is right. The truth table for $p \rightarrow q$ is false only when $p$ is true and $q$ is false. This case corresponds to invalid arguments. The other three cases apply to valid arguments. – Chris Dugan Jan 04 '16 at 17:08
  • For example, $a = T$ and $b = c = F$ corresponds to $p = F$ and $q = F$ in the $p \rightarrow q$ truth table, since the first premise is false but so is the conclusion. – Chris Dugan Jan 04 '16 at 17:11
  • The truth table for $p \to q$ is only false if p = F and q = T. So with this in mind, i was thinking if all premises values and conclusions give me a True, according to the truth table, is it an appropriate methodology to prove validity? Or should i stick to only if all premises are true, and conclusion is true, then the argument is valid? – Hakim Marley Jan 04 '16 at 17:14