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In forall x: Calgary, by P. D. Magnus, appears this reasoning:

$\neg(A \land B), A \lor B, A \leftrightarrow B \therefore C$

Examining its truth table, I see there are no rows where all premises happen to be true.

Is this reasoning still valid ?

F. Zer
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2 Answers2

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Yes.

The requirement

In every row where all premises are true, the conclusion must be true as well

can be equivalently reformulated as

There is no row where all premises are true and the conclusion is false

In classical logic, the only way for an inference to become invalid is if there is at least one counter valuation which makes all the premises true but the conclusion false.

If there is no valuation under which the premises are true in the first place, then there can be no such counter example, and the conclusion follows vacuously.

If there is no row that satisfies all of the premises, then this means that the premises are contradictory. In classical logic, according to the principle ex falso quodlibet, from a contradiction anything may be concluded, precisely for the reason that there can be no counter valuation.

Natalie Clarius
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  • I suppose a more modal view leads the following statement: No true, nor false proposition necessarily follows from a false premise, but every true or false proposition may follow possibly from a false premise. – Doug Spoonwood Mar 02 '20 at 14:56
  • Thank you @lemontree ! So, soundness is not directly related to validity. To classify an argument as valid, I am not interested if that argument would ever have all its premises true at once. Is that right ? – F. Zer Mar 02 '20 at 16:00
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    Ignore my previous, now deleted comment. As forallx defines it, soundness is related to validity in the sense that a sound argument must also be valid, plus in a sound argument the premises must be true (under the current valuation in consideration), but that's not a requirement in validity. – Natalie Clarius Mar 02 '20 at 16:09
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    As for what you say in your second sentence: You are interested, when checking for validity, in whether you ever have all the premises true at once, because when it does happen, then you also need to check the conclusion in these cases. But if it never happens, then the argument is valid for sure, but unsound, because then the premises can not possibly be true. – Natalie Clarius Mar 02 '20 at 16:11
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First, consider a pair simpler examples:

Example 1

$a\land b \implies b\land a$

This is a valid deduction since it is true regardless of the truth values of $a$ an $b$.

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Example 2

$a\lor b\implies a\land b$

This is not a valid deduction since it is sometimes false (on lines 2 and 3 of the truth table).

enter image description here

In the case of your example, it is a valid deduction since it is true regardless of the truth values of $A, B$ and $C$.

enter image description here

Dan Christensen
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