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I know that an argument is invalid when its premises are true while it's conclusions are false. But what if the premises are inconsistent?

!(a=>b) , b | !a |- c

$$\lnot (a\to b), b\lor\lnot a~\vdash~c$$

I read from (https://philosophy.stackexchange.com/questions/23148/suppose-you-know-the-premises-of-an-argument-are-inconsistent-do-you-have-to-do) that it is valid if the premises are inconsistent, but how can you prove that following the rules of natural deduction?

In the case I gave, [a]=T, [b]=F is the only case the first premise is true, but that would result in the second premise being false or false.

I did read the second answer in my link about the principle of explosion, but doesn't that mean I just have to ignore blatant contradictions during my proof?

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  • $\begingroup$ One premise may contradict another. A premise may even be self-contradictory, e.g. $P \land \neg P$. Your system of logic has to be able to handle contradictions (unless you are an intuitionist?). What you cannot have is a contradiction after all premises are discharged. Otherwise your system of logic would be inconsistent. $\endgroup$ – Dan Christensen Oct 1 '18 at 12:31
  • $\begingroup$ (1) Here is a formal proof of $\neg A \to [A \to B]:$ dcproof.com/FromFalsehood.htm (2) Here is a formal proof of $P \land \neg P \to Q:$ dcproof.com/MIParadox1.htm $\endgroup$ – Dan Christensen Oct 1 '18 at 14:08
  • $\begingroup$ @DanChristensen $\lnot(\varphi\land\lnot \varphi)$ is an intuitionistically accepted theorem. Intuitionism disagrees with the law of excluded middle ($\varphi\lor\lnot\varphi$).. $\endgroup$ – Graham Kemp Oct 12 '18 at 1:19
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If the premises are inconsistent then they cannot all be true. So the statement "if all the premises are true, then the consequence is true" holds, since the premises are never all true.

On the syntactic side, the details depend on what deductive system we use, and there are many different possibilities even if we confine ourselves to natural deduction-style systems. In the kind I prefer, there is a primitive falsity sentence $\bot$ and negation is not primitive, rather $\lnot A$ abbreviates $A\to \bot.$ So clearly from $A$ and $\lnot A,$ you can use implication elimination, aka modus ponens to infer $\bot.$ The principle of explosion is encapsulated in an elimination rule for $\bot$ that says you can any sentence from $\bot$.

It might seem unsatisfying that explosion seems to be assumed, not proven, in this framework but this is actually a major advantage, especially if you’re particularly interested in the rule. This way it is simple to track what proofs “use explosion” and which don’t, and it’s easy to come up with a systems (e.g. minimal logic) that don’t have it: just consider subsystems that don’t have this rule. The question of “why” it’s true classically, is better answered by a semantic argument, anyway, e.g. the one I gave above or observing $A\land\lnot A\to B$ is a tautology.

Regarding your last paragraph, it doesn't matter that you can ignore the contradictions in your premises. If you can find a consistent subset $\Sigma'$ of your assumptions $\Sigma$ such that $\Sigma' \models \phi,$ that's great, and much more interesting that the fact that $\Sigma\models \phi.$ But that just means you should have been working with $\Sigma'$ all along rather than $\Sigma.$

We know that since $\Sigma$ is inconsistent, $\Sigma\models \psi$ for any sentence $\psi,$ so the fact that $\Sigma\models \phi$ is not very informative. Thus we only usually care about reasoning from a consistent set of premises. (Unless, say, we’re using a logic that doesn’t have explosion, where the consequences of an inconsistent set of premises can be nontrivial.)

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    $\begingroup$ I feel like this answer is still missing something: addressing the question about how ex falso quodlibet can be done within natural deduction. $\endgroup$ – Mees de Vries Oct 1 '18 at 9:08
  • $\begingroup$ @MeesdeVries (1) Here is a formal proof of $A \to [\neg A \to B]:$ dcproof.com/FromFalsehood.htm (2) Here is a formal proof of $P \land \neg P \to Q:$ dcproof.com/MIParadox1.htm $\endgroup$ – Dan Christensen Oct 1 '18 at 12:42
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    $\begingroup$ @MeesdeVries I agree, thank you. Added something on that $\endgroup$ – spaceisdarkgreen Oct 1 '18 at 17:26
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I did read the second answer in my link about the principle of explosion, but doesn't that mean I just have to ignore blatant contradictions during my proof?

No, it means that you may use any contradictions that do arise.   Ex falso quodlibet is one method for handling contradictions.

$$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}\fitch{\neg(a\to b)\\b\lor \neg a}{\fitch b{\fitch a{b}\\a\to b\\\bot}\\b\to\bot\\\fitch{\neg a}{\fitch{a}{\bot\\b\qquad\text{via ex faslo quodliet}}\\a\to b\\\bot}\\\neg a\to \bot\\\bot\\c \qquad\text{via ex faslo quodliet}}$$

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