Questions on Truth-Functional Logic

Question

I would really appreciate and answer to the three questions and an explanation, as I don't fully understand them. I personally think they are all false.

Use the basic definition of validity in truth-functional logic to answer if the following is true or false:

(i) ‘From a contradiction anything follows’ (Or more precisely: from a contradictory set of premises any conclusion can be validly derived.) Is this true or false? (Hint: think about the associated conditional)

(ii) ‘Any invalid inference can be made valid by adding extra premises’. True or false?

(iii) ‘Any valid inference remains valid no matter what extra premises you may add to it.’ True or false?

Thank you so much!

Answer 1

An argument is just an implication of the form $(p_1\land p_2\land...\land p_n)\implies q$ , where $p_i (i=1,2,...,n)$ are premises and $q$ is conclusion. An argument is valid if the above implication is true ($T$).(which is possible for $T\implies T$, $F\implies T$ and $F\implies F$).

$(1)$ If $(p_1\land p_2\land...\land p_n)\equiv F$(contradiction), then $(F\implies q)\equiv T$, no matter whether $q$ is true or false.

$(2)$ Suppose $(p_1\land p_2\land...\land p_n)\implies q$ is invalid i.e. it has truth value $F$ which implies $q$ definitely has truth value $F$ and $(p_1\land p_2\land...\land p_n)$ has truth value $T$. By just adding a false premise $p_{n+1}$ in the antecedant part of the conditional, you get $(F\implies F)\equiv T$.

$(3)$ Suppose $(p_1\land p_2\land...\land p_n)\implies q$ is valid then the following cases arise:

$a$. $(p_1\land p_2\land...\land p_n)$ is $T$ and $q$ is $T$.

$b$. $(p_1\land p_2\land...\land p_n)$ is $F$ and $q$ is $T$.

$c$. $(p_1\land p_2\land...\land p_n)$ is $F$ and $q$ is $F$.

In cases $b$ and $c$, adding $p_{n+1}$ ($T$ or $F$) will not change the overall truth value of the conditional as $(p_1\land p_2\land...\land p_n\land p_{n+1})$ is still $F$. In case $a$ too; the overall truth value of the conditional remains $T$, no matter whether $(p_1\land p_2\land...\land p_n\land p_{n+1})$ is $T$ or $F$.

Answer 2

All three of them are true. For the first, for any formula $\phi$, we have that $$ \bot \to \phi $$ is a tautology. To see this, look at the truth table for $\to$.

For the second, for any $\phi,\psi$, even if $\phi \to \psi$ does not hold, we always have $$ \phi \land \psi \to \psi $$ is a tautology.

Finally, for any $\phi,\psi,\chi$, if $\phi\to\psi$ holds, then so does $$ \phi \land \chi \to \psi. $$

Answer 3

i) True. Remember that an argument is valid when it is impossible for the premises to be true and the conclusion to be false. Well, if the premises contain a contradiction, then they cannot be all true, so it certainly can't be the case either that all premises are true and the conclusion is false. So, it is always a valid inference, no matter what the conclusion is.

ii) True. Just add the premise 'if all of the other premises are true, then the conclusion is true'

iii) True. Think about it this way. Once the argument is valid, then that means that there is enough information in the premises in order for the conclusion to be true. Adding more information to the premises is not going to take anything away from that.

So: sorry to inform you, but they are all true, not all false!