Prove tautological consequence from a set of premises question

Question

For this problem, I am asked to provide an informal proof to show that whether if $S$ is a tautological consequence of $P_1, . . . , P_n$, then the set of sentences {$S, P_1, . . . , P_n$} is consistent. I will appreciate it very much if anyone can verify my solution.

Here is my attempt:

A sentence $S$ is a tautological consequence of some premises $P_1, . . . , P_n$ if $S$ follows from the premises based on the meanings of the truth-functional connectives on a truth table. So, $S$ is a tautological consequence of the premises if and only if every row of their joint truth table that assigns true to each of the premises also assigns true to $S$. Thus, the premises must be consistent with each other to have a true value assigned to each of the premises. If the premises are inconsistent, for example, $P$ and $\neg P$, then there will be a contradiction of the premises and it is also impossible to assign a true value to each of the premises regardless of the truth value of the conclusion. So, in the case of inconsistent premises, $S$ will never be a tautological consequence of $P_1, . . . , P_n$. Therefore, if $S$ is a tautological consequence of $P_1, . . . , P_n$, then the set of sentences

{$S, P_1, . . . , P_n$} must be consistent.

My Doubts:

I understand what is required for $S$ to be tautological consequence of the premises, however, what happens if we have a truth table that has inconsistent premises, thus not a single row of the table has a True value assigned to each of the premises and the conclusion. What do we call this type of truth table? Is it a trivial logic?

Thanks

LZ

Answer 1

Your concern is sound: there is a tricky point here.

We say that a formula $S$ is a tautological consequence of a set $\{ P_1,\ldots,P_n \}$ of premises if:

in all cases when all the propositions $P_{1},\ldots, P_{n}$ are true, also the proposition $S$ is true.

This amounts to saying that:

$S$ is a tautological consequence of $\{ P_1,\ldots,P_n \}$ iff $\{ \lnot S,P_1,\ldots,P_n \}$ is unsatisfiable (or inconsistent).

But if the set $\{ \lnot S,P_1,\ldots,P_n \}$ is unsatisfiable, i.e. no truth assignment evaluates all the formulas to true, we have that every truth assignment taht evaluates all $P_1,\ldots,P_n$ to true must evaluate $\lnot S$ to false.

Thus, such an assignment will evaluate $S$ to true.

Is this enough to conclude that there must be a truth assignment that satisfies $\{ S,P_1,\ldots,P_n \}$ ?

Not necessarily: we may have no assignment evaluating to true all of $P_{1},\ldots, P_{n}$.

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What happens if we have a truth table that has inconsistent premises ?

If the set of premises $\{ P_1,\ldots,P_n \}$ is inconsistent (or unsatisfiable), we have that there is no assignment of truth values that can simultaneously satisfy them all.

But then, we can build a truth table with premises $P_1,\ldots,P_n$ and conclusion $S$ whatever and we have that in every row where all the premises are true (there is none) also the conclusion $S$ is.

This means that a formula $S$ whatever is tautological consequence of an inconsistent set of premises.

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See also Ex falso quodlibet.