Why is any sentence a logical consequence of a set of inconsistent premises?

Question

If a set of premises is inconsistent, there is no situation that makes all the premises true simultaneously.

Given a sentence S, there is no situation in which a conjunction of a set of inconsistent premises is true and S is false, simply because there is not situation that makes the conjunction of the set of premises true.

I heard the following quote in an online lecture on inconsistent premises: "This is the same as saying that the sentence 'S is true in every situation that makes the premises true'". When a sentence is true in every situation that makes the premises true, we say that S is a logical consequence of the premises.

I don't understand how the phrase in bold above follows from the fact that there is no situation that makes a set of premises all true simultaneously.

Why couldn't we also say that "S is false in every situation that makes the premises true"?

Answer 1

In propositional logic, logical consequence is just defined in terms of applying rules of inference to the premises, regardless of whether the premises are true or even whether they could possibly be true. In this case, to show that any arbitrary proposition Q follows from a contradiction like "P AND ~P", you only need three basic rules of inference (a minimal set of 8 rules for propositional logic is listed on page 296 here, with other useful rules deducible from those 8 given on pages 298, 300, 302, and 304):

--The "simplification" or "conjunction elimination" rule: if you have a proposition of the form "P AND Q", you are allowed to then infer P, or then infer Q.

--The "addition" or "disjunction introduction" rule: if you have some proposition P, you can infer the proposition "P OR Q" for any other proposition Q.

--The "disjunctive syllogism" rule: if you have a proposition of the form "P OR Q", and you also have the proposition ~P (i.e. 'P is false'), then you can infer the proposition Q.

So, say you start with the premise "P AND ~P" (line 1). Using the simplification rule, you can then infer ~P (line 2). Using the simplification rule again, you can also infer P (line 3). Then using the "addition" rule with proposition P that you had on line 3, you can infer the proposition "P OR Q" (line 4), where Q is any arbitrary proposition. Now you have both "P OR Q" (line 4) and ~P (line 2), so using the "disjunctive syllogism" rule, you can then infer Q (line 5). Since Q was a completely arbitrary proposition, this shows that you can infer any arbitrary proposition starting from a contradiction of the form "P AND ~P" as a premise.

line 1: P AND ~P (premise)

line 2: ~P (simplification applied to line 1)

line 3: P (simplification applied to line 1)

line 4: P OR Q (addition applied to line 3)

line 5: Q (disjunctive syllogism applied to line 2 and line 4)

Alternately, you could just take P (line 2) and ~P (line 3) as independent premises, and in this way show that any conclusion follows from a set of inconsistent premises without needing to use the simplification rule.

Answer 2

The explanation that you quoted in your question is correct. This is the notion of vacuous truth.

Something is a logical consequence of something else if the former is true in every interpretation in which the latter is true. If there are no interpretations at all in which the latter holds, then the logical consequence relation is vacuously true.

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Let's define what it means to be a logical consequence.

I'll discuss classical propositional logic.

Let M (for "model") be an intepretation. We can make M just a set of variables that are true, e.g. {A, B, C} or similar.

M ⊨ V holds iff V is in M, where V is a propositional variable.
M ⊨ a∧b holds if and only if M⊨a holds and M⊨b holds.
M ⊨ a∨b holds if and only if M⊨a holds or M⊨b holds.
M ⊨ ¬a holds if and only if M⊨a fails.

Let Γ be a set of well-formed formulas and let φ be a well-formed formula.

Γ ⊨ φ means "φ is a semantic consequence of Γ".

Here is its definition.

Γ ⊨ φ holds if and only if for all M, if M ⊨ Γ holds then M ⊨ φ holds.

If Γ contains a formula and its negation, then M ⊨ Γ is false.

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As an addendum, there are some logical systems that do not use a standard logical consequence relation.

This is a specialized topic within logic, but people do study systems of logic that are specifically designed to avoid certain paradoxes or counterintuitive behavior like consequences following from contradictory premises.

Here's a link to a discussion of one such system https://www.youtube.com/watch?v=QcdcKb8nH5Y&t=1249s.

Answer 3

Why is any sentence a logical consequence of a set of inconsistent premises?

It is not true that any sentence is a logical consequence of a set of inconsistent premises. This is not how humans reason logically and not how human logic works. This is only what mathematical logic says.

This is what mathematical logic says essentially because the philosophers and mathematicians who developed mathematical logic in the 19th and 20th centuries adopted the logical operation ¬φ ∨ ψ, called "material implication" for the occasion, as their mathematical model of a logical implication φ → ψ. The whole of "classical logic", a misleading term, follows logically from this choice.

They made this choice essentially because after years of efforts, they could not find a better model.

The consequence of this choice has been that some mathematicians, unhappy with it, developed their own alternative formal models. Different formal models then led mathematicians to the idea that each formal system was "a logic", not just a formal system proposed as a model of human logic, but "a logic", which led to the fallacious narrative that there are different logics when there are only different models, and then all wrong. This narrative also contradicted 2,500 years of formal logic, including Aristotle's own perspective.

The narrative has also evolved from Boole to Quine to refer to "logic" only as a mathematical discipline or a mathematical system, not as a characteristic of human reasoning, as most people outside mathematical logic still keep believing that it is.