Logic: Same premises and different conclusion?

Question

Suppose that there are $n$ premises, say $p_1,\cdots, p_n$. Then, is it possible to obtain two different conclusion? That is, Can we obtain two conclusions $q_1$ and $q_2$ using the rule of inferences if we have the same premises $p_1,\cdots, p_n$?

For example, if $p_1 \rightarrow p_2$ and $p_2 \rightarrow p_3$, then the rule of inferences says that $p_1 \rightarrow p_3$ is valid. In this situation, can we get another valid argument?

Can we derive only one theorem from the axioms of group theory? — miracle173, Apr 09 '22 at 08:48
@miracle173 Sorry, what does it mean? You mean that we can get several valid argument. right? — filter hash, Apr 09 '22 at 08:50

score 2 · Accepted Answer · answered Apr 09 '22 at 11:42

2

For example, if $p_1 \rightarrow p_2$ and $p_2 \rightarrow p_3$, then the rule of inferences says that $p_1 \rightarrow p_3$ is valid. In this situation, can we get another valid argument?

Sure. Since $$(A→B) ∧ (B→C) \implies (A∧B → C) ∧ (A→C),$$ the arguments $$A→C\tag{A1}$$ and $$A∧B → C\tag{A2}$$ are both derivable from the premises $$A→B$$ and $$B→C.$$ Furthermore, since $$(A∧B→C) \kern.8em\not\kern-.8em\iff (A→C),$$ arguments $(A1)$ and $(A2)$ are not logically equivalent.

answered Apr 09 '22 at 11:42

ryang

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Thank you for your comments – filter hash Apr 09 '22 at 12:15
Can I ask one more question? If the conclusion is a conditional statement such as $p_1 \rightarrow p_3$ in the above example, then can we assume that $p_1$ is true to show that the argument is valid? – filter hash Apr 09 '22 at 15:17
@filterhash 1. Remember, it is the soundness—not the validity—of $A\to C$ that depends on $A$'s truth. $\quad$ 2. If this is just propositional logic, then this previous Answer that I wrote gives one method for verifying the validity of argument $A\to C.$ $\quad$ 3. If $A$ and $C$ are both atomic propositions, that $A\to C$ must be invalid. (Why?) – ryang Apr 09 '22 at 15:29
Thanks! Actually, based on my understanding, we do not assume that $p_1$ is true. If $p_1$ is true, then valid argument is $r$ in my example. Thus, I thought it implies the contradiction before the this question. – filter hash Apr 09 '22 at 15:32
Thanks. My last exercise is to show that $A\rightarrow C$ is invalid. – filter hash Apr 09 '22 at 17:09
Also: Valid argument? Sound argument?. – ryang Apr 10 '22 at 05:20

Logic: Same premises and different conclusion?

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