ex falso quodlibet in natural deduction (ND)?

Question

My understanding of inference rules is that they should be intuitively acceptable (like axioms). But my only intuition for ex falso quodlibet in natural deduction is not immediate but comes from its proof, which rests on disjunctive syllogism (which apparently holds in classical propositional logic and intuitionistic logic). I am probably missing something obvious. Is the principle of explosion really basic/intuitive and why?

The closest justification I see, based on the current answers, is that Ex Falso encodes a (metalogical) desire that the proof system be consistent. But also see the references to intuitionism in the answers.

Edit: as pointed out below, it may have been a distraction for me to raise "intuitionism" in the original question, although Ex Falso arises in classical and intuitionistic logic. This does not exclude that I would like axioms and rules to be intuitively acceptable (to me).

" There is surprisingly little agreement about the exact status of EFQ in the literature. Gentzen groups the ex falso rule alongside the other operational rules(1969, p. 77). Prawitz follows this practice. He treats ⊥ as a ‘0-place sentential operation’ (Prawitz 1978, p. 38), one ‘for which there is no canonical proof’ (Prawitz1977, p. 26). EFQ is then understood as the elimination rule for ⊥. Others, including Dummett, present ⊥ as an elimination rule for ¬ (e.g. Dummett 1991, p. 291). I contend that both of these approaches are mistaken...." ref.

Ex Falso has an ‘anomalous position [. . . ] inside the scheme of introduction/elimination rules’, tarnishing the neat symmetry of the intuitionistic system" (Weir 1986, p. 461).

According to here: "Kolmogorov’s criterion whether to keep an axiom was whether a proposition has an “intuitive foundation” or “possesses intuitive obviousness” (van Heijenoort 1967: 421, 422) ... Kolmogorov said that, just like PEM, Ex Falso “has no intuitive foundation” (van Heijenoort 1967: 419). In particular, he says that Ex Falso is unacceptable for the reason that it asserts something about the consequences of something impossible (van Heijenoort 1967: 421)."

... "I submit that these difficulties can be overcome simply by treating⊥as a punc-tuation marker, as Tennant suggests. Of course this means thatEFQis left withouta logical constant which it serves to eliminate. But on the view I am proposingno such constant is needed becauseEFQis not an operational rule at all. Its roleis best accounted for by according it the status of astructural rule. It simply isthe structural rule that tells us that any sentence whatsoever follows from a con-tradiction. Such a license isnotspecific to any logical constants, but amounts to ablanket policy. Our reclassification ofEFQthus fits neatly with our characterizationof structural rules as global rules that assign properties to our deducibility relation." ref.

Edit: I came across this reformulation of my initial uneasiness.

"... adopting disjunctive syllogism as an axiom or as a primitive rule of inference is not an option. Hence, it is ex falso that is naturally treated as primitive, not the principle of disjunctive syllogism. Therefore, rather than thinking of ex falso as the product of the mean-ings of disjunction and negation, we must treat disjunctive syllogism as a product of the ex falso rule. This suggests that the law of disjunctive syllogism is not a con-sequence only of the meanings of the constants involved, but also hinges on one’s stance concerning the general principle expressed by EFQ." ref.

Answer 1

Here is another possible idea behind the rule, although it is technically just presenting things Couchy said in a different way.

Because there is no specific way to prove $⊥$, for every proof we do come up with, we could have ostensibly just systematically replaced $⊥$ with $A$ to instead get a proof of $A$ (though there may be technicalities with respect to non-logical axioms). So, the elimination rule for $⊥$ is a local rule that internalizes this global property. The reason one introduces the local rule is to solidify the role of $⊥$ as classifying such degenerate proofs.

By contrast, minimal logic's $⊥$ is useful for other purposes. It can be useful to consider 'negating' with respect to an arbitrary unknown proposition that is not intended to be trivial in this way.

Answer 2

Ex Falso, in the form:

$$\dfrac {\bot}{\varphi} \ (\bot \text E)$$

is a rule of Natural Deduction, both classical and intuitionsitic.

In the Hilbert-style version it amounts to: $\lnot A \to (A \to B)$.

The formula is consistent with the operational interpretation of Intuitionistic Logic: there is no proof of $\bot$ (the absurdity).

See Arendt Heyting, Intuitionism. An Introduction (1st ed, 1956), page 106:

"Axiom X [the propositional form of EFQ] may not seem intuitively clear."

And see Andrei N. Kolmogorov, On the principle of excluded middle (1925):

"Axiom 5 [the propositional form of EFQ] is used only in symbolic presentation of the logic of judgments; therefore it is not affected by Brouwer's critique, especially since it has no intuitive foundation either. [...]

Hilbert's first axiom of negation, "Anything follows from the false", made its appearance only with the rise of symbolic logic. [...] the axiom does not have and cannot have any intuitive foundation since it asserts something about the consequence of something impossible: we have to accept $B$ if the true judgment $A$ is regarded as false."

For a system that rejects EFQ, see Minimal Logic.

Answer 3

The reason Brouwer used the term "intuitionist logic" is that a proof in intuitionistic logic necessarily corresponds to a construction. In particular, the rules of intuitionistic logic come in pairs: each symbol $\wedge,\vee,\to$ has an introduction rule and elimination rule (except for your observation that $\bot$ has no introduction rule), whereas excluded middle ($\vdash \phi\vee\neg\phi$) has no counterpart.

So the term "intuitionistic" is incidental.

The best explanation I can give for the elimination rule for $\bot$ (EFQ) is that its intended meaning is contradiction. Namely, the rule says "if a contradiction is derived, then anything is derivable." However, since the presence of a contradiction implies that the system is inconsistent, this can be reworded to mean "anything is derivable in an inconsistent system".

Being able to derive anything in a logical system would make the system useless (or trivial). So EFQ essentially guarantees that if a contradiction is present in the system, then the system is useless.

So to prove that a logical system is useful, or non-trivial (or consistent), we must show that no contradiction is derivable. Note that the absence of an introduction rule for $\bot$ does not guarantee that it cannot be derived. This is because of modus ponens, (or the cut rule in sequent calculus), which is why we need the cut elimination theorem to prove consistency results.

Mauro ALLEGRANZA pointed out that minimal logic (more generally paraconsistent logic) excludes EFQ. To quote wikipedia,

The characteristic or defining feature of a paraconsistent logic is that it rejects the principle of explosion. As a result, paraconsistent logics, unlike classical and other logics, can be used to formalize inconsistent but non-trivial theories.

Answer 4

Apparently a justification for Ex Falso in terms of disjunctive syllogism is due to Glivenko’s paper from 1929. Neither Heyting nor Kolmogorov ever justified Ex Falso by giving this traditional argument here.

"While [Ex Falso] can of course be justified byappeal to disjunctive syllogism, it should not be overlooked that disjunctive syllo-gism is itself only provable in the presence of EFQ" ref.

Regarding constructivists,

"Kolmogorov said that “As soon as ¬a is solved, then the solution of a is impossible and the problem a→b is without content” (Kolmogorov 1932: 62 [Mancosu 1998: 331]), and proposed that “The proof that a problem is without content [owing to an impossible assumption] will always be considered as its solution” (Kolmogorov 1932: 59: [Mancosu 1998: 329]). Taken together, this yields a justification of Ex Falso ." here.

Heyting gave the following justification of Ex Falso:

" Axiom X [¬p→(p→q)] may not seem intuitively clear. As a matter of fact, it adds to the precision of the definition of implication. You remember that p→q can be asserted if and only if we possess a construction which, joined to the construction p, would prove q. Now suppose that ⊢¬p, that is, we have deduced a contradiction from the supposition that p were carried out. Then, in a sense, this can be considered as a construction, which, joined to a proof of p (which cannot exist) leads to a proof of q. (Heyting 1956: 102)"here.

(It may be my ignorance, but I have no idea exactly how to "join" these proofs.)

Regarding the latter, as pointed out here.

"In its attempt to provide, “in a sense”, a construction, the explanation is clearly not of the same kind as Kolmogorov’s stipulation from 1932. But it does not fit into Heyting’s original interpretation of logic in terms of intentions directed at constructions and the fulfillment of such intentions either. For to fulfill an intention directed toward a particular construction we will have to exhibit that construction; we will have to exhibit a construction that transforms any proof of p into one of q. But how can a construction that from the assumption p arrives at a contradiction, and therefore generally speaking not at q, lead to q? It will not do to say that such a construction exists “in a sense”. A construction that is a construction “in a sense”, as Heyting helps himself to here, is no construction.""

Elsewhere, Glivenko stated that Ex Falso "is a consequence of the principle (p∨¬q)→(q→p), the admissibility of which he considers “quite evident” here.

"Undeniably, we take it that contradictionsoccur when we have derived two contradictories, a statement and its negation. Butthere is no reason in principle whyex falsocould not also play a role in systemsdevoid of a negation operator. A ‘contradiction’ in this sense is not necessarily theresult of asserting two contradictories. For example, we could conceive of systems inwhichA,B`⊥would hold, whereAandBare contraries. So long as our languagecontains antonyms like hot/cold, soft/hard, weak/strong, etc. or contraries likered/green, triangle/circle, there will be cases where various sentences fail to bejointly assertible. This we register with the sign ‘⊥’. In all such cases, we have achoice of whether or not to adoptEFQ. The fact that there is such a choice, thatthe decision for or againstEFQis not predetermined by the meanings of the logicalconstants in the system—for a system withEFQneed not contain any constants atall—shows thatEFQis indeed a structural rule. As such it expresses a general policyregarding our consequence relation, namely that we allow any statement whatsoeverto follow from a contradiction." ref.

Reasoning here "shows that the standard natural deduction formulations of intuitionistic logic are misleading. They present ex falso as an operational rule that appears to in-troduce an awkward asymmetry into the system. In fact, however, this is just asuperficial feature of the standard system. The move from minimal logic to intuitionistic logic, represented by the adoption of theex falsorule, turns on structural assumptions rather than on a shift in the meanings of the logical constants. Like-wise, relevant logic can tidily be characterized as the substructural logic obtainedby dropping the structural rule of weakening on the left from minimal logic" ...

"... natural deduction systems in sequent format most accurately capture the distinction between the structural rules expressing the structural properties of the system as a whole and operator-specific operational rules. In particular, I showed that, contrary to conventional wisdom, the rule of ex falso ought to be understood as a structural rule closely related to the rule of weakening on the right." (ibid).

Answer 5

In this answer, I'm going to try to convince you on very general grounds that the Ex Falso Quodlibet rule should be uncontroversial. The short version is that since $\bot\models \varphi$ for every sentence $\varphi$ (under any reasonable interpretation of these symbols), we also want $\bot\vdash \varphi$ for every sentence $\varphi$.

For me, a logic should always come with a notion of semantic entailment between sentences: $\varphi\models\psi$.

Depending on the logic, this notion could be defined in a variety of different ways, but the intuitive meaning should be the following.

(1) $\varphi\models \psi$ means: whenever $\varphi$ is true, $\psi$ is also true.

Let's suppose that the logic has a sentence $\bot$, called "false". Again, the intuitive semantics of $\bot$ should be the following.

(2) $\bot$ is never true.

Of course, we probably want to make the words "whenever" and "never" more precise. For this, we need a definition of "model" and what it means for a sentence to be "true in a model". But remember, I'm trying to be as flexible as possible here - I don't mean to suggest that a "model" needs to be a set-theoretic model like in first-order logic. If the word "model" makes you uncomfortable for some reason, feel free to substitute "world" or "situation". Anyway, we get the following more precise versions.

(1') $\varphi\models\psi$ means: For every model $M$, if $\varphi$ is true in $M$, then $\psi$ is true in $M$.

(2') For every model $M$, $\bot$ is not true in $M$.

Now it follows immediately from (1') and (2') that for any sentence $\psi$, $\bot\models \psi$. Indeed, for any model $M$, $\bot$ is not true in $M$ by (2'). So for any model $M$, if $\bot$ is true in $M$, then $\psi$ is true in $M$. Thus $\bot\models \psi$ by (1').

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We can also introduce a proof system for our logic, which allows us to define the derivability relation: $\varphi\vdash\psi$.

Now the primary criterion for a proof system is that it should be sound and complete for our intended semantics. This means that for any sentences $\varphi$ and $\psi$, $$\varphi\vdash \psi \text{ if and only if } \varphi\models \psi.$$

By the discussion above, this means that we must have $\bot\vdash \psi$ for every sentence $\psi$. This is exactly what the rule Ex Falso Quodlibet gives us.

Now the desiderata of soundness and completeness completely determine the derivability relation $\vdash$: it must be equal to the entailment relation $\models$. But our logic may admit many different proofs systems which are equivalent in the sense that they produce the same derivability relation. Different (but equivalent) proof systems may be preferred in different situations on aesthetic and practical grounds. This is a subjective matter, and no logic has a "correct" proof system - different people may disagree about whether natural deduction or sequent calculus is more aesthetically pleasing, a Hilbert system may be convenient for certain proof theoretic arguments, but less convenient for others, etc. The crucial thing at the end of the day is not the particular rules of the system, but the soundness and completeness of the derivability relation.

So I have no opinion about whether Ex Falso Quodlibet should be included in a proof system as a basic rule. Personally, I find it to be both aesthetically pleasing and intuitive, but you may disagree. But if it is not included as a basic rule, it must be an admissible rule! That is, under the basic assumptions (1) and (2) on our logic, for any satisfactory proof system, we must have $\bot\vdash \psi$ for any sentence $\psi$.