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Can two formal logical deductions A and B, starting from different sets of assumptions H1 and H2 (if it helps, consensual facts about reality), each of them independent (that is, they don't imply any assumption of the other set of assumptions, nor any negation of one of them), arrive in different conclusions C1 and C2 such that C1 and C2 considered together will create a contradiction in both systems A and B?

PS: logical framework and semantics used in both systems are the same. Let's say the classical logical framework is used (if any other, that would make a difference here?).

Thanks and sorry if I'm using terms non-technically.

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    The answer is trivially no. If they did then by definition H1 and H2 will not be logically independent. Take the conjunction of all H1 assumptions and the conjunction of all H2 ones, if a contradiction follows from them together then one of them implies the negation of the other. – Conifold Dec 8 '17 at 0:52
  • Conifold is correct, the fact that they are independent assures that this is not possible. – Not_Here Dec 8 '17 at 1:15
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    @Conifold This should really be expanded into an answer --it's not appropriately a comment, since it provides a definitive answer to the question as asked. – Chris Sunami Dec 8 '17 at 17:47
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    @ChrisSunami It seemed too trivial, so at the time I expected that the OP meant something else and would refine the question. I am not even sure that this answered it since there was no feedback. – Conifold Dec 8 '17 at 21:14
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    @Conifold It's not trivial to a non-expert. You're allowing your familiarity with the topic to skew your perception of what is obvious and what is not. Whether or not it's what the OP meant, it addresses the question in a way that could be useful to other SE readers. – Chris Sunami Dec 8 '17 at 21:22
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The answer to your question is "It depends". Different axiomatic systems can lead to different conclusions. The truth of a statement S can only be said to be true or false within a given model.

This issue is not really a problem if you are talking about reaching a conclusion in two different axiomatic systems. We usually only work in one axiomatic system at a time and there is usually some kind of justification for using that system. And in this case, H1 and H2, in the OP's question would not be independent, as Conifold points out.

However, it is also possible for a single axiomatic system to lead to contradicting results. Such a system is said to be "inconsistent". We generally do not work with such systems and it is usually assumed that reality itself is logically consistent, but we do not have to make that assumption. There is a whole field of study involving inconsistent logical systems. They do have their benefits. For instance, in a consistent logical system, there are theorems which can never be proven true or false, while in an inconsistent logical system, we can always determine whether a theorem is true or false.

Additionally, Conifold's argument does not hold in a logically inconsistent system. Reductio ad absurdum, or proof by contradiction, is a form of valid proof, specifically because we have assumed that our logical system results in only true or false statements and that a statement cannot be both true and false.

The link I provided discusses this concept in more detail and also explains some reasons why we might not be certain that our reality is logically consistent and also how we can take a current axiomatic system and add to it in such a way as to make it useful but not consistent.

  • This seems intuitively correct, but how would you respond to Conifold's point in the comments on the original post? – Chris Sunami Dec 8 '17 at 17:46
  • "If a contradiction follows from them together then one of them implies the negation of the other." This result is true, only if the axiomatic system is consistent. – Daniel Goldman Dec 8 '17 at 18:12
  • I've updated my answer. Does that help? – Daniel Goldman Dec 8 '17 at 18:14
  • Great. Let me know if you have any other questions or want me to add more detail. – Daniel Goldman Dec 8 '17 at 20:05
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Can 2 logical deductions starting from 2 different and independent sets of assumptions lead to 2 contradictory conclusions?

Yes. If assumptions H1 and H2 are randomly related, they are independent of each other in the sense that one assumption implies nothing about the other. But in a random relation, there can be correlation without causation, and two factors can still have a correlation of minus one (-1).

That is, in this random relation, the two factors being measured have never appeared together. In all cases, when one factor was present, the other was not. So if the correlation of H1 and H2 is (-1), the two assumptions can be independent and still yield contradictory results.

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I think the answer to your question is yes.

Take the following example: H1={A} and H2={A -> B, not B}.

Now H1 and H2 are independent but their union is contradictory.

edit: Hm I just re-read your question and I think I misunderstood it.

edit #2: Thank you for Conifold for pointing out my silly mistake. So if the question is "is it possible that the union of C1 and C2 is contradictory" then the answer is yes in my opinion. (Hope this time I am not wrong).

S is a constant symbol, F,G,H are unary relations.

H1={F(S), F(x)&G(x)->J(x)}

H2={G(S), not J(S)}

C1=H1, C2=H2

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Consider a scientific theory and an experiment. The theory includes a mathematical model (an axiom system that we think useful). In the setup of the experiment, we make certain assumptions and deduce a numerical result. After executing the experiment, we use the result of the experiment as an axiom, along with axioms about interpreting such experiments, and deduce a different result. (This is likely to be a relatively simple deduction.) The two deductions don't necessarily have to share axioms, in that it's quite possible that none of the assumptions of the model are necessary to figure out the result.

For example, say we were sending up GPS satellites but had not yet developed general relativity. We would take a set of assumptions about the satellite's passage of time, and formally treat them as axioms running through special relativity, and arrive at an adjustment to the atomic clock to keep proper time. Now, we launch the satellite into a circular orbit and observe the time signals being sent from it. We don't bother with orbital mechanics, but merely monitor the time signals every time it's at a particular point in its orbit relative to the monitoring station.

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