The inapplicability of (selected) paraconsistent logicsDraft

Rafal Urbaniak1

Pawe l Sini lo2

Abstract

In some cases one is provided with inconsistent information and has toreason about various consistent scenarios contained in that information.Our goal is to argue that filtered paraconsistent logics are not the righttool to handle such cases and that the problems generalize to a large classof paraconsistent logics.

A wide class of paraconsistent (inconsistency-tolerant) logics is ob-tained by filtration: adding conditions on the classical consequence oper-ation (one example is weak Rescher-Manor consequence — which bears Γto φ just in case φ follows classicaly from at least one maximally consis-tent subset of Γ). We start with surveying the most promising candidatesand comparing their strength. Then we discuss the mainstream views onhow non-classical logics should be chosen for an application and arguethat none of these allows us to chose any of the filtered logics for action-guiding reasoning with inconsistent information, roughly because such areasoning has to start with selecting possible scenarios and such a pro-cess does not correspond to any of the mathematical models offered byfiltered paraconsistent logics. Finally, we criticize a recent attempt to de-fend explorative hypothetical reasoning by means of weak Rescher-Manorconsequence operation by Meheus et al. (2014).3

1 Motivations

One fairly standard view among researchers in non-classical logics is that suchlogics are often constructed as mathematical and normative theories of validityof (certain) natural language arguments (Shapiro, 2006; Cook, 2010). Naturallanguage arguments as employed and assessed by competent language users arethought of as the (fallible) data, and a non-classical logic — a purely math-ematical theory — is assessed against such data in order to choose the rightpure logic for a given natural language phenomenon. Finding the right set ofnatural language expressions with respect to the meaning of which (perhaps ina specific type of contexts) the assessed arguments are to be valid and identify-ing the formal system best modeling such arguments results in an applied (andperhaps non-classical) logic. For the task of selecting a logic, standard criteriaof rational theory choice are conjured, such as adequacy to the data, simplicity,non-ad-hocness, unifying power etc. (Priest, 2013).

1Centre for Logic and Philosophy of Science, Ghent University, Belgium. Institute ofPhilosophy, Sociology and Journalism, Gdansk University, Poland

2Authored sections 1, 2 and 4. Bristol University, UK.3We’d like to thank Peter Verdee, Frederik Van De Putte, Christian Straßer, Joke Meheus,

Nicole Wyatt and the audience members at the 5th World Congress of Paraconsistency inKolkata for their comments on earlier versions of this paper.

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While this sounds nice in theory, it is not clear whether such considerationsand natural language practice are unambiguous and fine-grained enough to de-termine a single particular formal system, even if we restrict our attention toa fairly narrow kind of phenomena in a rather narrow context. When it comesto actually choosing a logic, the abundance of mathematical systems seems tosurpass the richness of the data. One of the goals of this paper is to illustratethis point using the case of filtered paraconsistent (henceforth FP) consequencerelations and to suggest a diagnosis of the phenomenon different from the stan-dard pluralist one. According to this diagnosis, the logics under considerationare bound to fail because of the unrealistic assumption that the notion of relia-bility of parts of information can be captured (in extension) by merely syntacticcriteria.

We’ll start with a description of a class of paraconsistent consequence oper-ations that we’ll be looking at. Then we’ll compare them in terms of strength.Then we’ll discuss the undetermination of the choice of logic by the data, andexplain why the pluralist reaction to it is not satisfactory. Next, we’ll sketch ourdiagnosis of the problem. Finally, we’ll describe and criticize a recent attemptof a defense of some filtered paraconsistent relations by Meheus et al. (2014).

2 Some filtered paraconsistent consequence relations

To get things started, we need to look at some options that one might initiallythink are interesting candidates for the formal tools for handling inconsistentinformation. The nice thing about this group of consequence operations is thatthey’re constructed in a similar manner. Roughly speaking, a paraconsistentfiltered consequence operation is constructed by means of applying classicallogic to some parts of the premise set and then applying some operation to theresult. Various choices of methods of choosing parts and an operation result indifferent consequence relation. Quite a few different paraconsistent filters havebeen suggested so far, so the first step will be to present the main candidateson the market and give some examples of how they’re supposed to work, for thesake of clarity. Next, we’ll look at how they’re related to each other in terms ofstrength.

We’ll work with a standard propositional language. Sets of formulas willbe represented by capital Greek letters. Lower case Greek letters are meta-variables standing for formulas. Sometimes we’ll say things like Γ ∪ α insteadof Γ ∪ {α} and α, β |= γ instead of {α, β} |= γ. Unless indicated otherwise, |=will represent the standard classical consequence relation. A set of formulas Γis (classically) consistent iff there exists at least one formula α such that Γ 6|= α.A maximal consistent subset (mcs) Γj of Γ is any consistent subset of Γ suchthat for any α, if α ∈ Γ \ Γj , then Γj ∪ α |= ⊥.

We’ll start with Rescher-Manor consequence operations (Rescher and Manor,1970).

Definition 1. Weak Rescher–Manor consequence. A formula α is anRMW -consequence of Γ iff it is a classical consequence of at least one mcs of Γ.

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In other words: Γ |=RMWα iff there exists a set Γj ⊆ Γ such that Γj is an mcs

of Γ and Γj |= α.4 �

Example 1. Let Γ = {¬p, q, p, p ∨ ¬r, r}. Then we can list all mcss of Γ:

Γ1 = {¬p, q, p ∨ ¬r}Γ2 = {¬p, q, r}Γ3 = {p, q, p ∨ ¬r, r}

Γ1 classically entails ¬r (because ¬p, p ∨ ¬r |= ¬r), so Γ |=RMW¬r. 2

Definition 2. Strong Rescher–Manor consequence relation. A formulaα is an RMS-consequence of Γ provided that it is a classical consequence of allthe mcss of Γ. In other words, Γ |=RMS

α iff for all Γj ⊆ Γ, if Γj is an mcs ofΓ, then Γj |=CL α.5 �

Example 2. Say Γ is the same as in Example 1. Then q will be an RMS-consequence of Γ, because is entailed by all the mcs. 2

Example 3. Now assume Γ = {p,¬p}. Γ will have two mcss: Γ1 = {p} andΓ2 = {¬p}. The only RMS-consequences of Γ are classical tautologies. 2

Definition 3. Free consequence relation. A formula α is anF -consequence of Γ (Benferhat et al., 1997, 24)6 iff it is a classical consequenceof the intersection of all mcss of Γ. In other words:

Γ |=F α iff⋂Γi

{Γi is an mcs of Γ} |= α

�

Example 4. Now let Γ = {p ∧ ¬q,¬p ∧ q, p → r, q → r}. There are two mcssof Σ: Γ1 = {p ∧ ¬q, p → r, q → r} and Γ2 = {¬p ∧ q, p → r, q → r}. ThenΓ1 ∩ Γ2 = {p → r, q → r}. Hence Γ |=F p → r and Γ |=F q → r. Observe alsothat Γ 6|=F r because Γ1 ∩ Γ2 does not classically entail r. 2

Definition 4. Argued consequence relation. A formula α is said to bean A-consequence of Γ (Benferhat et al., 1997; Simari and Loui, 1992)7 iff thereis an mcs of Γ, Γj , such that Γj |= α and there is no mcs Γk of Γ such thatΓk |= ¬α. �

Example 5. Let Γ = {p,¬p,¬q, q → r, p ∧ q}. Then Γ has the following mcss:Γ1 = {p, p∧ q, q → r}, Γ2 = {¬p,¬q, q → r}, Γ3 = {p,¬q, q → r}. Γ1 classicallyentails r while there is no mcs that classically entails ¬r. Hence, Γ |=A r. 2

4This consequence relation is also called existential consequence relation in(Benferhat et al., 1997, 28).

5RMS is also called universal consequence relation in (Benferhat et al., 1997, 23).6The notion is originally formulated in terms of minimally inconsistent sets. For the sake

of uniformity we chose to use a different but equivalent formulation.7The original formulation proceeds through consistent subsets. We use a definition shown

to be equivalent by Benferhat et al. (1997).

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Definition 5. Cardinality based consequence relation. A formula p is saidto be a CB-consequence of Γ (Reggia et al., 1985)8 iff p is a classical consequenceof every mcs Γj of Γ with highest cardinality. �

Example 6. Let Γ = {p∧q,¬p∧q,¬q → r}. It has two mcss: Γ1 = {p∧q,¬q → r}and Γ2 = {¬q → r,¬p ∧ q}. |Γ1| (the cardinality of Γ1) and |Γ2| is 2 andthere is no mcs with higher cardinality (in fact, there is no other mcs). SoΓ |=CB ¬q → r. 2

Example 7. Now let Γ = {p, q,¬q, p → q}, Then the following are mcss of Γ:Γ1 = {p, q, p → q}, Γ2 = {p,¬q} and Γ3 = {¬q, p → q}. Since |Γ1| = 3,while |Γ2| and |Γ3| is 2, the CB-consequences of Γ are exactly the classicalconsequences of Γ1. 2

To introduce our next candidate, we need a bit of preliminary explanations.

Definition 6. Logical cover and its width. =(∆) is a logical cover of a setof formulas ∆ iff it is an indexed set such that ∆1 = ∅, every element ∆i of =(∆)is consistent and ∆ is a subset of the union of (classical) deductive closures ofall elements of the cover: ∆ ⊆

⋃i∈I CCL(∆i). If =(∆) is of cardinality n, then

n− 1 is its width w(=(∆)).9 �

Often, sets have more than one logical cover. In such a case, we’ll use = withsubscripts. Not all sets have logical covers. If ∆ contains an absurd formulap ∧ ¬p = φ, at least one of the elements of ∆ (namely φ) cannot be an elementof a union of consequence sets of consistent sets.

Example 8. Let ∆ = {p, q,¬p ∨ ¬q}. Then:

=1(∆) = {∅, {p,¬p ∨ ¬q}, {q}}=2(∆) = {∅, {p}, {q}, {¬p ∨ ¬q}}=3(∆) = {∅, {p, q}, {¬p ∨ ¬q}}=4(∆) = {∅, {q,¬p ∨ ¬q}, {p}}

w(=1(∆)) = w(=3(∆)) = w(=4(∆)) = 2, w(=2(∆)) = 3. 2

Definition 7. Level of a set of formulas. The level of ∆ is the minimalwidth of a logical cover of ∆.

`(∆) = min{w(I(∆)) | I(∆) is a logical cover of ∆}

where c is the number of logical covers of ∆. If ∆ has no logical cover, `(∆) =∞.If w(=(∆)) = `(∆) we’ll say that =(∆) is a minimal logical cover of ∆. �

8The notion originally was used for multisets, which might contain repeated occurrencesof formulas. We’ll ignore this detail as irrelevant for the discussion which will follow.

9Our definitions related to the forcing consequence relation are equivalent modifications ofdefinitions occurring in (Payette and Schotch, 2009).

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While there can be many different logical covers of a set, every set has aunique level (if the set contains an absurd formula, its level is ∞).

Example 9. Let ∆ be as in Example 8. Then =2(∆) is not of minimal widthand all the other covers are. 2

Observe that definition 6 allows for “wild” elements in logical covers, evenwithout any change to their width. Thus, Γ = {p,¬p} will not only have theobvious cover {∅, {p}, {¬p}}, but also such covers as {∅, {p, q}, {¬p, q}}.

Definition 8. Level–forcing consequence relation. A formula α is anLF -consequence of Γ iff every minimal logical cover of Γ contains at least oneset which classically entails α. If `(Γ) = ∞, then the only LF -consequences ofΓ are the consequences of the empty set, that is, classical tautologies.10

�

However, LF -consequence is rather unmanageable, because any set has aninfinite number of covers. So for the sake of simplicity and our convenience, wewill replace it in our discussion with the following cousin.

Definition 9. Normality. A logical cover =(Γ) is normal iff (1) no elementof =(Γ) contains a formula which does not belong to Γ, (2) each formula fromΓ is in exactly one element of =(Γ).

A formula α is a normal forcing consequence of Γ (Γ |=NF α) iff every normalminimal logical cover of Γ contains at least one set which classically entails α.If `(Γ) =∞, then the only normal LF -consequences of Γ are the consequencesof the empty set. �

An alternative way of formulating the normality requirement is to say alogical cover {I}(−) is normal iff both

I(Γ) ⊆ ℘(Γ)

I(Γ) \ ∅ is a partition of Γ

Example 10. Let Γ = {¬p,¬q, p∨ q, p∧ q}. Γ will have only one minimal logicalcover devoid of “wild elements”: =1(Γ) = {∅, {¬p,¬q}, {p∨ q, p∧ q}}.11 Amongthe NF -consequences of Γ are p, q,¬p,¬q. 2

Example 11. Take Γ = {p ∧ q,¬p ∧ q, q → r}. There are two minimal (normal)logical covers:

=1(Γ) = {∅, {p ∧ q, q → r}, {¬p ∧ q}}=2(Γ) = {∅, {¬p ∧ q, q → r}, {p ∧ q}}

Each of these covers contains an element which classically entails r, so Γ |=NF r.2

10This is a slightly modified definition of X-level forcing (Payette and Schotch, 2009, 96).11Try to make a cover starting with a different consistent subset. Say you start with

{¬p, p∨ q}. Then you can’t put the remaining formulas ¬q and p∧ q together, so you have to“cut” your set into three. The same applies to other attempts.

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3 Comparison

Now, let’s see how the consequence operations we introduced compare in termsof strength. Most of the results are available in (Benferhat et al., 1997), butbefore we draw a picture we first prove the bits that cannot be found in thatpaper.

Fact 1. NF is contained in RMW .

Proof. Suppose Σ |=NF α. This means that any normal cover of Σ contains anelement Σi which classically entails α. By definition of a cover, Σi is consistentand by the normality condition Σi ⊆ Σ. That being the case, Σi can be extendedto an mcs of Σ, call it Σ′

i. By monotonicity of classical logic, Σ′i |= α so

Σ |=RMWα.

Fact 2. RMW goes beyond NF , and therefore by fact 1 is properly strongerthan NF .

The proof employs the following example:

Example 12. Let Σ = {p, q,¬p∨¬q}. The following are all the minimal normalcovers of Σ:

=1(Σ) = {∅, {p,¬p ∨ ¬q}, {q}}=2(Σ) = {∅, {p, q}, {¬p ∨ ¬q}}=3(Σ) = {∅, {q,¬p ∨ ¬q}, {p}}

Clearly, Σ 6|=NF ¬p and Σ 6|=NF ¬q, because no element of =2(Σ) classicallyentails any of these.

Now let’s turn to RMW . Σ will have the following mcss:

Σ1 = {p, q}Σ2 = {p,¬p ∨ ¬q}Σ3 = {q,¬p ∨ ¬q}

Σ |=RMW¬p (because Σ3 |= ¬p) and Σ |=RMW

¬q (because Σ2 |= ¬q).The following example will show that NF and RMS are not comparable:

Example 13. Suppose that Σ = {p, p → q, r,¬r}. Then Σ has the followingmcss:

Σ1 = {p, p→ q, r}Σ2 = {p, p→ q,¬r}

and the following minimal normal covers:

=1(Σ) = {∅, {p, p→ q, r}, {¬r}}=2(Σ) = {∅, {p, p→ q,¬r}, {r}}=3(Σ) = {∅, {p, r}, {p,→ q,¬r}}=4(Σ) = {∅, {p,¬r}, {p→ q, r}} 2

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Σ |=RMSq, while Σ 6|=NF q, since there is no element Σi of =3(Σ) and =4(Σ),

such that Σi |= q. Moreover, Σ |=NF r and Σ |=NF ¬r, while neither r or ¬rare RMS-consequences of Σ. Hence:

Fact 3. RMS and NF are not comparable.

To show that F and NF are not comparable, we will again refer to example13. It’s clear that Σ |=F q, while Σ 6|=NF q. Moreover, neither r or ¬r areF -consequences of Σ, while Σ |=NF r and Σ |=NF ¬r.

Fact 4. NF and F are not comparable.

The next example will show that NF and A are not comparable:12

Example 14. Let Σ = {p,¬p, p→ q}, Then there are two mcss of Σ:

Σ1 = {p, p→ q}Σ2 = {¬p, p→ q}

and the following normal minimal covers of Σ:

=1(Σ) = {∅, {p, p→ q}, {¬p}}=2(Σ) = {∅, {¬p, p→ q}, {p}} 2

So Σ |=A q, but there is no element Σi of =2(Σ) that classically entails q,hence Σ 6|=NF q. On the other hand Σ |=NF p, Σ |=NF ¬p, while neither ofthem is an A-consequence of Σ.

Fact 5. NF and A are not comparable.

Now we are left with the case for NF and CB:

Example 15. Let Σ = {p,¬p, p ∧ q, p→ r}. Then there are two mcss of Σ:

Σ1 = {p, p ∧ q, p→ r}Σ2 = {¬p, p→ r}

and two normal minimal logical covers of Σ:

=1(Σ) = {∅, {p, p ∧ q, p→ r}, {¬p}}=2(Σ) = {∅, {¬p, p→ r}, {p, p ∧ q}} 2

It’s clear that Σ |=CB r, since Σ1 is the only mcs with highest cardinalityand it classically entails r, while Σ 6|=NF r since there is no element of =2(Σ)that classically entails r. On the other hand, Σ |=NF ¬p, while Σ 6|=CB ¬p.

Fact 6. NF and CB are not comparable.

12Example from (Benferhat et al., 1997, 27).

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The following diagram summarizes the results of comparison. Double arrowsindicate proper inclusion while the lack of arrows indicates incomparability.

CLKS

RMW8@ _gKS

A ^f NF CB7?

RMSKS

F

4 Choosing logic: a theoretical perspective

Now, one striking thing to observe is that even within a fairly narrow group ofFP-logics there are quite a few non-equivalent options. Let’s think about oneof the standard views on how one is to go about choosing a non-classical logic(Priest, 2013).

First, let’s follow Priest in the distinction between logica docens, logica utensand logica ens. Logica docens is what logicians claim to be the right logic and canbe found in logic textbooks, logica utens captures how people actually reasonand Logica ens13 determines what actually is valid: what really follows fromwhat.

The general picture of the task of a non-classical logician, as far as whatthey want to do is to develop a logic for natural language reasoning, seems tobe the following. Such a logician, let’s call him Griest, first looks at a particularphenomenon in natural language reasoning — this might be a particular typeof reasoning in some particular context, perhaps with an eye on validity due tothe meaning of certain selected expressions.

For instance, one might be interested in how negation, disjunction, conjunc-tion and implication behave in the context of, say, nurses and doctors collectingdata about patients from various sources, and one might want to know what inthis context follows from what in virtue of the meaning of those connectives,especially when a contradiction in the data arises.

Then, Griest has to figure out what the logica ens for that particular phe-nomenon is, playing around with logica utens for this phenomenon. So, Griestlooks at various arguments people use and consider valid and various ways peo-ple recognize their mistakes in the ways they actually reason to figure out thecorrect forms of reasoning.

13We’d rather say logica entis, but we’re following Priest here.

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So, on this model, Griest has to go and look at what arguments nurses anddoctors use when their data turns out to be inconsistent and what argumentpatterns they recognize as valid upon reflection. This context has the followingimportant characteristics: (1) contradictions being true is out of the question,(2) epistemic decisions as to what to accept and what to infer are to guideaction, and (3) whether the right action is taken is of crucial importance.

Finally, Griest has to either construct or pick a formal, mathematical logicthat captures the correct pattern of natural language reasoning for the phe-nomenon he’s interested in.

Thus, from the abundance of formal systems, Griest chooses the one thatmodels the phenomenon the best, and for this purpose he treats pure formallogics as various competing theories of the phenomenon in question, and decidesbetween them employing rational theory choice criteria for picking the right purelogic to capture logica ens. Such criteria of course include adequacy to the datafor which the theory is meant to account (logica utens), but not only. The datamight be fallible — after all, logica utens doesn’t have to be exactly the same aslogica ens and people might simply tend to make mistakes of particular kinds.Other important factors might be simplicity, non-(ad hocness), unifying power,fruitfulness, etc.

5 Choosing logic: top-down

All this sounds quite nice in theory, but it is very far from what people develop-ing FP-logics actually do (it is also quite far from what non-classical logiciansusually do, but in this paper we will focus on FP-logics only). Instead of theprocedure described above, authors usually employ the following strategy. Theygive a general description of what they think the goal of the logic they wantto develop is, and then they try to explicate this notion (in Carnap’s sense)by developing a formal system.14 If all goes well, they will sprinkle their workwith a few funny and somewhat artificial examples of unusual or paradoxicalreasonings that the system they developed is advertised as being able to handle,but we are aware of no major attempt to take the data (for instance, informa-tion about how inconsistency in data, say in hospital data, is actually treated)seriously. The contrast between these two approaches we’ll call the contrastbetween the top-down and the bottom-up approach.

So, the filtered operations mentioned in the technical sections have somecommon general goal: to model sensible reasoning with inconsistency and tolimit the damage to the data caused by its presence. All of the systems can bethought as various ways of explicating this idea mathematically:

• The focus on consistent subsets reflects the desire to avoid contradictions.

• The focus on maximal consistent subsets reflects the desire to preserveinformation as much as possible.

14Come to think of it, this is also what Carnap was doing with induction, ending up withan embarrassment of the riches (see Urbaniak and Batens, 2014).

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• Looking at the intersection of maximal consistent subsets (free conse-quence operation) mirrors the wish to isolate problematic formulas andto rely only on the unproblematic ones.

• Relying on unopposed arguments (argued consequence operation) seemsto be a different way of capturing the desire to stay consistent and to relyon safe parts of the data.

• Looking at largest maximal consistent subsets is another way to try tomake the desire to preserve as much information as possible mathemati-cally precise.

• Preservationist forcing consequence operation is motivated by the ideathat inference should not increase the level of inconsistency of a base ofpremises. (The idea is that the lower the level of the minimal cover of aset, the less inconsistent it is.)

Now, this is all good fun and tough mathematics. Wondering what math-ematical forms the generic hand-wavy slogans might take, investigating whatrelations between them there are, looking into the properties of the constructedsystems and so on. In many respects it is a challenging and fascinating enter-prise. But we would like to submit that as long as we stay at the level of generalconsiderations of the sort mentioned above and stick to the top-down strategy,they won’t be specific enough to determine which of particular non-equivalentmathematical explications we should choose (if any), when it comes to FP-logicsin the context that we are currently concerned with (a nurse handling incon-sistent information). Are there any other considerations that could be used toimprove on the situation?15

6 Reacting to the top-down strategy failure

One move available to the defendant of FP-logics is perhaps exploring the optionof being a pluralist about logic. While pluralism is a philosophical perspectivenot developed for this purpose, it will be worthwhile to see whether this generalstance can help one to face the issues we raised. In the context of our currentdiscussion we think that pluralism comes in two interesting varieties.

A pluralist of the first sort will concede that there might be many interestingFP-logics, but they will insist that particular contexts and uses will determine

15One could complain that what we’re describing is just a general issue with armchairmathematical philosophy and the way it handles empirical data. But there is more to ourpoint. The case at hand is a clear example where a general philosophical motivation resulted ina multiplicity of mathematical options with no reasons to choose between them. This is quiteunlike the case of computability (where mathematical explications are equivalent) or even thecase of many “armchair” philosophical considerations which result in a multitude of options,as long as they provide a clear and interesting account of the pros and cons of choosing any ofthem. The inability of a priori considerations to result in a sensible selection of a mathematicaloption is one problem. The inability of FP-logics to handle more practice-oriented cases isanother one.

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which of the logics to use. So, perhaps, in certain contexts the cardinality-basedrelation will be appropriate and in some other context the argued consequenceoperation will be more appropriate and so on.

While initially promising, this strategy claims more than it actually achieves.For to make it work, one has to not only hand-wavily express the general con-tent of the previous passage, but rather do more. One has to be more specificabout what one means by a context and what possible relevantly different typesof contexts there might be. One should also explain in a principled and com-pelling manner which context goes with which consequence relation and why.We haven’t seen any extensive work on this being done and our conjecture isthat no compelling work on this will be done.

To illustrate this point, imagine you’re completely ignorant as to know how touse tools and someone gives you a very elaborate toolbox, saying: use whicheverworks, depending on the context. This is quite unhelpful. They should rathershow you which tools are used in which context and for what purpose. Onlythen you would believe that they understand how to use tools.

Another pluralist approach might be inspired by the considerations of thesort employed by Beall and Restall (2006). Roughly speaking, one of the mainreasons they think the equi-adequacy of non-equivalent logics might arise is thevagueness of the notion of validity. A standard definition of the notion is thata valid argument is one whose conclusion is true in every case in which all thepremises are true. Restall and Beall insist that there might be many equallywell specifications of what those cases are and what the range of quantificationin the definition is, thus resulting in many equally plausible candidates for TheLogic That Is.

General debate about logical pluralism aside, one could try to say somethinganalogous about our case of choosing between FP-logics. Namely, one couldargue that similarly, the general descriptions of the goals that FP-logics aresupposed to achieve are too vague to determine a logic and that there are manyequally good but non-equivalent explications of these goals.

The problem with this approach is that it doesn’t help much when it comesto guiding action in situations in which various epistemic decisions would resultin various actions with clearly different consequences and in which there areimportant risks related to various options. The nurse whose reasoning Griestinvestigates wants to know which inferential moves he can rely on in his decisionmaking when his data about a patient is inconsistent, and telling him to dowhatever because anything goes (within a certain array of options the pluralistgives us no criterion to choose from) might not be the best advice.

7 Bottom-up strategy and its failure

Perhaps it will be easier to decide between the options, if we look at the exampleswe used to show that one logic goes beyond another and ask ourselves: if wewere a nurse receiving information of the form of the premise set, would we bewilling to accept all the conclusions that follow from it in a given logic and act

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upon them?Let’s start with an easy case. The premise set {p,¬p} RMW -entails both p

and its negation. Now, if a nurse received information of this form, would hebe happy and willing to conclude simply that both p and ¬p and rely on bothin her action? Very unlikely. So, RMW is off the table (see, however, a moredetailed discussion in section 9).

Similarly, NF-consequence admits quite simple situations in which the con-sequence set still contains a contradiction, and for this reason it doesn’t seemlike a good tool for this job.

The strategy exemplified in the last three paragraphs, perhaps, looks promis-ing. We could now start going over various examples, hoping that those on whichthe FP-logics differ would provide us with intuitions clear enough to choose theright one for the context under consideration. Alas, the way this would developisn’t that satisfactory. To cut this short, let’s look at the weakest logic in thegang (we can safely ignore NF given that we’ve already excluded it). If thereis an inference that is validated by this logic and yet we can easily imagine acase in which the nurse would refuse to make this inference, this logic (and alllogics that contain it) is not up to the task of correctly modeling the inferenceshe makes.

Consider Γ = {p ∧ ¬q,¬p ∧ q, p→ r, q → r} and imagine the nurse receivedan information sheet containing data of this format. The set that he shouldclassically rely on, if F is his logic, is {p → r, q → r}. So, in such a case,whenever he obtains information of the format mentioned above, he should bealways happy and willing to infer p→ r and q → r.

Alas, we submit, a nurse which would bet a patient’s life on this would bepatently insane. After all, he doesn’t know if the source of information thatprovided him with inconsistent data isn’t also responsible for the rest of theinformation. Something went wrong with information gathering, and the factthat p → r is not syntactically involved in a contradiction doesn’t mean thenurse should bet a patient’s life on its truth.

Now, the pluralist might be worried that the example is so coarse-grainedthat in the absence of a more elaborate epistemic context it is simply impossibleto decide which logic should be used. This response seems to us unsatisfactoryon two counts. For one thing, it is void without any further specification ofwhat impact various possible choices as to the epistemic context would have onthe choice of logic. For another, while it might seem that a further specificationof the epistemic context might support one logic rather than another (or atleast help disqualify some of the logics), it will still fail to determine the choiceof logic, because even within that context we still will be able to imagine anddescribe in more detail a situation in which some of the inferences validated bythe suggested logic will seem ridiculous.

One might think that our worries about choosing the right logic to usewhen faced with inconsistent information don’t have much with the generalmethodology described in section 4, which seems to be devised for identifyingthe true truth-preserving logic, not with looking for the right non-monotonicstrategy to use in a difficult epistemic situation. Yet, our consideration still

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apply. It still would be good if for a given application some non-monotoniclogics were better or worse than others, and it would be good to have someprincipled explanation as to why we should use any of them.16 Perhaps theremight be a few equally useful ones — but it rather seems that FP-logics satisfythis condition only in a degenerate way by being equally useless.

A more detailed discussion of a recent attempt to defend the applicability ofsome filtered paraconsistent consequence operations is the subject of section 9.First, however, a rough picture of the direction in which the epistemic behaviorof the nurse should be expected to proceed.

8 What to do then?

One sensible thing for the nurse to do, it seems, is to take a step back: to suspendhis belief in any of the informations coming from an unreliable source, andeither to investigate the source of the error in detail, to figure out which piecesof information are still reliable, or to gather the information anew, dependingon what’s more convenient, or if these options are not available or impractical,to weigh the possible consequences of acting on each of the currently availableoptions and to evaluate the risks and benefits of doing so.

If the nurse decides to investigate the source of the contradiction, whichpart of the information he will accept after the process is a matter sensitivethe details of his investigation. This by no means is a phenomenon that canbe easily captured by a mere formal description (after all, the answers to thenurse’s questions will not be determined by the mere form of the information)and definitely it cannot be captured by FP-logics which are simply too crude tomodel considerations of this sort.

Possible epistemic decisions are pretty much as many as possible scenariosof what went wrong. Perhaps a stoned lab assistant mistyped everything andno part of the information is reliable. Perhaps there were two lab assistantsand only one of them responsible for only some items was stoned, in which caseonly the part of information he was responsible for is unreliable. Perhaps oneof the lab processes failed, in which case the information obtained on its basisis unreliable, quite independently of its syntactic form, etc. In each of thesescenarios, deciding which part of the information is reliable doesn’t have muchto do with its syntactic form, with the question whether the relevant sentenceshave some constants in common, and in general with the methods employed byvarious FP-logics.

An extremely optimistic pluralist might insist that each such scenario tellsus which logic to use in a given instance. On this approach, one is equippedwith a bunch of logics to choose from and with instructions telling him which

16This point has been emphasized by Wyatt (2004, 418), who insisted that agreement withreasoning practice might provide a good criterion for choosing a logic as truth-preservingvalidity: “if logics have to meet the additional requirement of modeling some aspect of ouractual reasoning practice, debate between relevant logicians and classical logicians [. . . ] will bejust as viable as the debate between the modal logicians over the right account of metaphysicalnecessity.”

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one to use depending on the particularities of the investigation of the source ofthe inconsistency. This however would make the logics rather redundant andwould move all the heavy lifting to the process of deciding which of the initialpremises to rely on depending on what exactly led to the situation in which wereceived inconsistent information.

To put it differently, FP-logics are advertised as helping us to identify thereliable part of the information and to reason with it. If to choose the rightFP-logic we have to do the actual work of identifying the reliable part of theinformation beforehand, we can equally well dispose of the FP-logic and simplyapply classical logic to the part of information we discovered to be reliable duringour investigation.

Moreover, for each consistent selection of some of the initial premises thereis a possible context in which exactly those sentences which belong to it are reli-able, and the other ones are not because they were produced by the same processwhich was responsible for the contradiction, because in general contradiction-generating epistemic processes are not neat enough to obey any enlighteningsyntactic regulations that we might come up with.17 Now, we could speak ofit in terms of for each such a selection there being a “logic” which tells us torely exactly on the elements of this selection, but given that choosing the rightselection is an extra-syntactic matter of substance, this would be an abuse ofthe word “logic.”

Perhaps, one could argue that the goal of FP-logics is not to model the waywe think a person should proceed in the situation of, say, a nurse obtaininginconsistent information, but rather should be used as a formal tool to programan expert system, and so the way humans would proceed can be safely ignored orperceived only as a rough guide to developing such formal tools. But then, again,lots of extra homework lies ahead. For one needs to provide independent andprincipled standards of assessment of such logics when they’re supposed to beused for such a purpose, clearly connected to the goal they are supposed to fulfill,and to assess various formal systems against those standards. Before we finishthis paper, however, let’s take a closer look at some more elaborate attempts atproviding selection criteria for strong/weak/free consequence relations.

9 Is paraconsistent existentialism defensible?

The problem of choosing between the free and the two Rescher-Manor conse-quence operations has been discussed by Meheus et al. (2014). One of the maingoals of the authors is arguing for the usefulness of the existential Rescher-Manorconsequence relation (and of credulous consequence in default logics, but thisis inessential for our current purpose) – the view that we’ll call paraconsistentexistentialism. The philosophical core of the argument starts with a discussionof an example put forward by Pollock (1995, 63):

17Think of a stoned lab assistant. Would he say: oh, well, I won’t screw up this part ofinformation, because it has no expressions in common with this other part of informationthat I have already mishandled?

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If the agent is performing medical diagnosis and the evidence favor-ing two diseases is tied, we do not want the agent to decide randomlyto treat the patient for one disease rather than the other. It couldhappen that the diseases are not serious if left untreated, but if thepatient is treated for the wrong disease, that treatment will gravelyexacerbate his condition.

Pollock claims that it would be a bad decision to choose any of the treatmentsat random. This case is contrasted with a situation in which we’re picking apicnic site and lack reasons to choose between two locations and in which itmakes sense to choose at random. The difference, as he points out, lies in theresults of the calculation of risks and profits. As Meheus et al. (2014) correctlypoint out, Pollock rather than discussing the credulous consequence on its own(where all options are in some sense accepted and treated as equal), brings upthe additional issue of choosing between various options available according tothe credulous consequence. While the latter issue is very important, it’s quiteseparate from our current question of whether credulous existential consequenceoperations can be useful for mapping all the options.

The intuitive reason why we should “keep an open mind for both options”and use the existential consequence is that “treating the patient in a justifiedway also means that one should at least try to obtain the best possible insightin the underlying causes. This in turn may require that one investigates thedifferent alternatives and their different outcomes. In our view, this purpose isbest served by the weak/credulous option.” (Meheus et al., 2014, 5) A similarintuitive defense of weak/credulous reasoning can be found in (Prakken, 2006),who insists that “a rational agent should map out all credulously acceptableaction alternatives that have sceptically acceptable epistemic support and thenmake a choice between them” (Prakken, 2006, 2) and that “a logic for practicalreasoning should simply present all credulously acceptable arguments to a userof the system and let the user decide.”18

A similar attempt at an intuitive defense of the weak/credulous consequenceoccurs also one page further (Meheus et al., 2014, 6):

The most rational way to proceed in cases like this is to weigh thedifferent pros and cons and to try to design some priority ordering.However, before one is able to do this, one needs to know whatthe different options and their expected outcomes are. In order tofind that out, one will have to reason from the information one has.Explorative reasoning seems to be best suited here.

Examples used by Meheus et al. (2014) seem to support this picture. Oneis that of an investigation of a plane crash, during which “it is harmless thatpeople start exploring different paths” [p. 7] Another is of scientists trying toexplain a new phenomenon, where finding the explanation “is best served by

18Prakken also wants to connect the skeptical/credulous distinction to the epis-temic/practical distinction. Meheus et al. (2014) succeed at showing this account to be toosimplistic.

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a period of exploratory reasoning.” Yet another [p. 9] is that of you beingon a committee picking candidates for open positions and trying to “figure outin which direction the discussion is going, before you start entering the debateyourself. . . [when] it might be wise to try to figure out first which participantshave compatible views and which have not . . . But, in order to figure this out,you will have to reason from their statements. What this comes to is that youwill have to make inferences from the premises, in such a way that you not onlyavoid the derivation of trivial conclusions, but also that you are able to detect,for instance which statements are jointly compatible.”

To be fair, Meheus et al. (2014) never say one should act upon the explo-rative consequences of contradictory premises. Rather, they insist that once oneconstructs a dynamic proof theory for a given system, it is useful to reason dy-namically using the moves captured by this proof theory. While to large extentthe dynamic proofs are revealing, their results are ultimately at best as reliableguide to action as the consequence operation they capture, and so we need toask about the status of the latter.

Although at the informal level of description explorative reasoning soundsboth tempting and useful, it is only fair to ask to what extent explorative rea-soning is adequately modeled by the mathematical apparatus of the existentialRescher-Manor consequence. Let’s recall what “steps” are involved in derivingRMW -consequences from a set Γ:

1. List all maximally consistent subsets of Γ.

2. Derive the classical consequences of those subsets separately.

3. Join them together into one large set of consequences.

Now consider a nurse who obtained inconsistent information and doesn’tknow how to proceed. The first problem is that the above recipe doesn’t tell himto investigate further and to identify the source of the problem. But let’s evenignore this and unrealistically assume that the nurse has no way of investigatingthis and has to take actions based on the currently possessed information.

The nurse might want to isolate separate scenarios and separately considertheir consequences. This might help him to evaluate the potential risks andprofits from acting upon any of them. And this is the exploratory reasoningwhich the examples described above make intuitively compelling, at least tosome extent.

Whether he will obey point 2 is unclear, because following the consequencesof a particular scenario might require something different than a classical logic(say, because counterfactuals, or caeteris paribus conditions, or causal infer-ences, or probabilistic reasoning also may play an important role). But let’signore this point, assuming that whatever consequence operation is sensible toapply within each scenario can be safely plugged in instead of classical logic inpoint 2.

A more pressing question is whether separating scenarios to investigate cor-responds to listing maximal consistent subsets of the original premise set. We

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would like to submit that it doesn’t. In most real-life cases, there will be defi-nitely fewer scenarios to be investigated than maximal consistent subsets, giventhat content-wise many of these maximal consistent sets will be very artificialcombinations that no one would ever hypothesize about. This indicates anotherdrawback of using the existential consequence and following steps 1 and 2: itwould be computationally much more demanding to actually work on each ofthe maximal consistent subsets.

But it’s not just an issue of computational complexity. As we already pointedout, mere consistency will not guarantee reliability and so some consistent sce-narios (and therefore some things we may infer from them) might be completelyunreliable. This means that we will end up putting in one bag of paraconsistentconsequences things that will still quite clearly not be something to act upon.

This should be quite clear even for the case of syntactically speaking thesafest consequences of the premise set, its free consequences. Meheus et al.(2014, 3) call them “innocent bystanders” suggesting that “they are in no wayrelated to the inconsistencies, and hence, even in situations in which the incon-sistencies have not yet been resolved, they can safely be accepted.” And again,on p. 7: “In the case of the RM-consequence relations, the safe consequencesare the free consequences. As they are not related to the inconsistencies, theycan safely be relied upon.”

Our discussion so far should make it clear, however, that just because thefree consequences don’t have to be involved in the derivation of an inconsistencyfrom the premise set, it doesn’t mean that they can be safely relied on. Whilethey might satisfy certain syntactic precautionary requirements, they might stillbe as unreliable as the claims directly involved in the inconsistency, becausethey might come from the same unreliable source and there is no way to knowwhether this is the case if we rely only on syntactic considerations.

Finally, once the nurse has separated scenarios to be investigated, separatelyconsidered and evaluated their consequences to choose one (or some) to actupon, there is no further need or use in putting these consequences togetherinto one big bag of paraconsistent consequences of the original premise set.Separate consideration of the scenarios either led him to a decision as to whichscenario to act on or not, but putting all such consequences on a par is not goingto help anything. Quite the opposite, it will just muddle the waters by losingtrack of which consequences belong to which scenario.19

10 Syntactic sensitivity to the rescue?

Perhaps the following approach might be closer to what we should be gunningfor. First of all, premises should come equipped with information as to whichsource they come from. This can be modeled by taking the data to be a set ofsets of premises, so that each set of premises corresponds to a source. So let’ssay that a sourced premise set (SPS) is a set of sets of formulas.

19Having said that, there are some virtues to the dynamic proof theory which (Meheus et al.,2014) provide, but our current discussion is about the consequence relations.

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Next, we want to say that once a source is (partially responsible) for acontradiction, it should not be relied on. So let’s say that an element of an SPSΓ, Γ1 is guilty iff Γ1 is a member of a minimal subset of Γ whose union entailsa contradiction. An element of an SPS is innocent just in case it is not guilty.That is, a source is guilty if a contradiction can be derived from it (perhaps,together with other sources), and this source is necessary for this contradiction.Finally, we say that φ is entailed by the data iff it classically follows from theunion of innocent members of the SPS.

At this point one might object:

But this is exactly what we can do using the free consequence rela-tion, provided that we formalize things properly, by putting thingsfrom one source in one big conjunction and things from differentsources in separate formulas.

In a sense, this is true. Once we use conjunction to keep track of sources inthe way described above, the free consequence operation will give us the sameresult. However, this doesn’t mean that this is the way to go.

Once you encode coming from the same source by being a member of thesame conjunction, your consequence relation becomes syntax-sensitive. Thismeans that the same basic data, formulated in a syntactically slightly differentmanner, might give you difference consequences. What you end up with is non-adjunctive conjunction: one whose introduction is not always legitimate (thatis, you won’t be always able to derive p ∧ q from p and q).

To be fair, this is mathematically manageable: there are logics which providesome ways of handling syntax sensitivity and non-adjunctive conjunction. Yet,we think, there is something unsatisfactory about going this route.

For one thing, the formal approaches to non-adjunctive conjunction are cum-bersome. The mathematics works out, but it is very difficult to connect the be-havior of conjunction in these systems with the way people handle conjunctionin natural language.

For another, the whole strategy seems a bit ad hoc and it results in requir-ing the conjunction connective to perform two different task: to express thestandard meaning of conjunction and to keep track of sources of information.One of the main nice things about logical devices is that they help us to distin-guish between importantly different functions of linguistic items and to identifythe principles governing them. Conflating two importantly different functionsis from this perspective a step back. So, we are inclined to think, a separatemathematical device handling the source-tracking is needed. But the issue goesbeyond the scope of this paper.

11 Final remarks

We have presented a variety of options when it comes to filtered paraconsistentoperations and compared them in terms of strength. This opened the field for aphilosophical discussion of the problem of choosing any of them for a particular

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epistemic context of a person who needs to act upon receiving inconsistentinformation. We argued that none of those logics is adequate when it comesto explaining what such a person should rationally do. The main reason ofthis fact is that following merely syntactic criteria will not result in rationaldecisions regarding the reliability of parts of the information. We surmise manyother paraconsistent logics face the same problem.

Perhaps the reader might pessimistically conclude that FP-logics try to pro-vide answers to the question “what should I infer from this set of (inconsistent)information,” but the real question asked in practical cases of dealing with in-consistent information is “which subset of this inconsistent information should Iuse as an inferential base”, which on the face of isn’t a question falling into thedomain of logic at all, so perhaps its not surprising that the logics don’t work.

This pessimistic conclusion, however, is a bit too hasty. Just because theright subset cannot be sensibly chosen on purely formal grounds, it doesn’t meanthat there is no part of the procedure principled enough to be captured by aformal system (for instance, in the vein of the consequence operation discussedin section 10). There is definitely something systematic in how inconsistentinformation should be handled, and it is possible that certain important as-pects of this phenomenon can be studied by mathematical methods familiarto logicians. However, the attempts we have so far, while definitely being astep forward from the stage where only classical logic was on the market, arestill far from the right way(s) to handle inconsistent information, at least in animportant class of cases.

References

Beall, J. and Restall, G. (2006). Logical Pluralism, volume 78. Oxford UniversityPress.

Benferhat, S., Dubois, D., and Prade, H. (1997). Some syntactic approaches tothe handling of inconsistent knowladge bases: A comparative study. StudiaLogica, 58:17–45.

Cook, R. T. (2010). Let a thousand flowers bloom: A tour of logical pluralism.Philosophy Compass, 5(6):492–504.

Meheus, J., Straßer, C., and Verdee, P. (2014). Which style of reasoning tochoose in the face of conflicting information? Journal of Logic and Compu-tation. (in print).

Payette, G. and Schotch, P. (2009). Preseving what? In On Preserving: Essayson paraconsistent logic, pages 81–99. University of Toronto Press.

Pollock, J. (1995). Cognitive Carpentry. MIT Press.

Prakken, H. (2006). Combining sceptical epistemic reasoning with credulouspractical reasoning. In Dunne, P. and Bench-Capon, T., editors, Computa-tionalModels of Argument. Proceedings of COMMA-06., pages 311–322.

19

Priest, G. (2013). Revising logic. [unpublished paper].

Reggia, J. A., Nau, D. S., and Wang, P. Y. (1985). A formal model of diagnosticinference. i. problem formulation and decomposition. Information Sciences,37(1):227–256.

Rescher, N. and Manor, R. (1970). On inferences from inconsistent premises.Theory and Decision, 1(2):179–217.

Shapiro, S. (2006). Effectiveness. in. In The Age of Alternative Logics, pages37–49. Springer.

Simari, G. R. and Loui, R. P. (1992). A mathematical treatment of defeasiblereasoning and its implementation. Artificial intelligence, 53(2):125–157.

Urbaniak, R. and Batens, D. (2014). Induction. In Hansson, S. O. and Hen-dricks, V. F., editors, Handbook of Formal Philosophy. Springer. (commis-sioned and forthcoming).

Wyatt, N. (2004). What are beall and restall pluralists about? AustralasianJournal of Philosophy, 82(3):409–420.

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