Proving that a hypotheses implies a proposition

Question

I am trying to work on predicate logic and I couldn't really understand the following question (Sorry for the screenshot. I couldn't figure out how to type the notation):

How do I tie these hypotheses to one variable among them? I was thinking of giving each variable a proposition but I think I'm being asked to solve this algebraically. To make the question more concise, how do I solve something like this?

Answer 1

Indeed there is no way to associate some kind of concrete statement to $P,Q,R,S,T$ since they're meant to be any logical statement.

We need to prove $\overline T$. The only one of the hypotheses $H_1,H_2,H_3,H_4$ related to $T$ in any way is $H_4 \equiv T \wedge P \to R$. Since $H_4$ is true, we have two cases, given the truth table for implication ($\to$):

1. If $R$ is false, then $T\wedge P$ is false. Then either $T$ or $P$ is false.
2. If $R$ is true, then $T\wedge P$ can be either true or false.

So, if we prove that $R$ is false and $P$ is true, then necessarily $T$ must be false; i.e. $\overline T$ is true. You should be able to prove $P$ and $\overline R$ from the first three hypotheses. I'll leave my own proof in a spoiler below.

Assume $H_1 \equiv P\wedge Q$, $H_2\equiv P \to \overline{Q\wedge S}$, $H_3 \equiv R\to S$ and $H_4 \equiv T \wedge P \to R$ are true statements. Then, $P$ by simplification on $H_1$ and by $H_2$, $\overline{Q\wedge S}$. Then $\overline{Q}\vee \overline{S}$ by De Morgan's law, and $\overline S$ by simplification. Then, $\overline R$ by Modus Tollens/counterreciprocal on $H_3$ and thus $\overline{T \wedge P}$ by Modus Tollens on $H_4$. Then $\overline T \vee \overline P$ by De Morgan's law, and since $(\overline T \vee \overline P)\wedge P$ by conjunction (as $H_1$ implies $P$), we have $\overline T$.

Answer 2

Here I use the symbol " ~X " for " proposition X is false".

H1 : P & Q

H2 : P --> ~ (Q & S)

H3 : R --> S

H4 : (T&P) ---> R

To be proved : ~T

One method to discover the proof is analysis: you start with the goal you want to arrive at and ask yourself under which conditions that would be possible.

(1) I want ~T.

(2) The only way to get this is to have ~ R and P : indeed if I had ~R, I would have ~ (T&P) , by contraposition. And if I had also P, I would have ~T.

(3) Now, how to get P? P is actually given since I have (P&Q).

(4) How to get ~R? I need ~S , since by contraposition ~S --> ~R.

(5) How to get ~S? I need P ( and I already have it) in order to get ~ (Q&S) , equivalent to (~Q OR ~S); and I need the negation of "~Q" , i.e. Q , in order to get ~S ( by disjunctive syllogism). But I have Q ( by H1).

Now that I have all I need to do the proof, I go the opposite way ( using, at each step, the appropriate rule) :

• I prove I have Q
• I prove I have P
• I prove I have ~ (Q&S)
• I prove I have ~Q OR ~S
• I prove I have ~ ~ Q
• I prove I have ~S
• I prove I have ~R
• I prove I have ~ (T&P)
• I prove I have ~T OR ~P
• I prove I have ~ ~ Q
• I prove I have ~T.

The first way is called " analysis" ( discovering the conditions) the second one is called " synthesis" ( from the conditions to the consequences).