Many-valued logic and sequence arguments in

value theory

Simon Knutsson

October 1, 2020

I have submitted this manuscript to a journal. Commentsare still welcome to, e.g., simonknutsson@gmail.com

Abstract

Some find it plausible that a sufficiently long duration of torture isworse than any duration of mild headaches. Similarly, it has been claimedthat a million humans living great lives is better than any number of worm-like creatures feeling a few seconds of pleasure each. Some have relatedbad things to good things along the same line. For example, one mayhold that a future in which a sufficient number of beings experience a life-time of torture is bad, regardless of what else that future contains, whileminor bad things, such as slight unpleasantness, can always be counter-balanced by enough good things. Among the most common objections tosuch ideas are sequence arguments. But sequence arguments are usuallyformulated in classical logic. One might therefore wonder if they work ifwe instead adopt many-valued logic. I show that, in a common many-valued logical framework, we get valid sequence arguments if we grantany of several strong forms of transitivity of ‘is at least as bad as’ and anotion of completeness. But almost all of these forms of transitivity seemintuitively problematic. Other, weaker forms of transitivity seem accept-able as premises, but the resulting sequence arguments are invalid. All inall, the path to a convincing sequence argument in this logical frameworklooks narrow, but there are a few moderately strong forms of transitiv-ity that might be acceptable, and that are here shown to result in validsequence arguments.

1 Introduction

Some find it plausible that there are values that cannot be counterbalanced byother values; for example, that a sufficiently large amount of torture is worsethan any amount of mild headaches.1 An example concerning positive value isprovided by Lemos (1993, p. 487). Imagine ‘a world in which there are a million

1E.g., Carlson (2000, pp. 246–247). For discussion, see, e.g., Norcross (1997) and Schonherr(2018).

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people living pleasant and morally virtuous lives, engaged in aesthetically andintellectually excellent activities.’ In another world, the only beings are worm-like creatures who feel no pain but who each feel a few seconds of pleasure duringtheir lives. Lemos holds that the second world is good and that it would be evenbetter if it contained more such worm-like creatures, but he finds it plausiblethat the first world is better than the second, regardless of how many worm-likecreatures the second contains. Another example is that Ross (1930, p. 150)finds it likely that any amount of virtue, however small, is better than anyamount of pleasure, however large.2 One can relate bad things to good thingsalong the same line. For example, some authors seem sympathetic to followingidea: some horrible things such as a sufficiently large finite number of humansexperiencing a lifetime of torment cannot be counterbalanced by various goodthings, regardless of the amount of those good things, while trivially bad thingscan always be counterbalanced by sufficiently many good things.3

These ideas are important for the allocation of healthcare resources (Voorho-eve 2015). For example, should limited public funds be spent on treating manypeople with mild illnesses or a few with the worst health conditions? The ideasare also important for the impossibility theorems in population ethics (Carlson2015; Thomas 2018), and for what to prioritise when making policy or individ-ually trying to have an impact on the world.

I will deal with some of the most common objections to such ideas, namelya group of similar objections called sequence arguments (or spectrum or con-tinuum arguments). I will explain them in detail later, but the following is asketch of a sequence argument against the view that there is a finite number ofinstances of torture that together are worse than any number of mild headaches:There is a sequence of intermediate bads between torture and mild headachesuch as the following: torture, a terrible disease, a less serious disease, severeheadache, moderate headache, mild headache. Spelt-out sequence argumentsinclude more bads so that adjacent bads are more similar to each other. Ifthere is a finite number of instances of torture that are worse than any numberof mild headaches, there is a bad in the sequence such that this relation holdsbetween it and its successor; for example, there is a finite number of severeheadaches that together are worse than any number of moderate headaches. Itis implausible, the argument goes, that this holds between adjacent bads in thesequence, which are so similar. Hence, the plausibility of the original view oftorture versus mild headaches is undermined.

The main sequence arguments are formulated in classical logic, which as-sumes there are only two truth values, true and false, and that every declarativesentence is either true or false. I will investigate whether sequence arguments areconvincing if one instead uses many-valued logics; that is, logics with more thantwo truth values. The truth values in many-valued logic are sometimes calledtruth degrees, and I will assume, as is common, that they are numbers between0 and 1, where 0 is falsest and 1 is truest. For example, in some many-valued

2For more historical references, see Arrhenius (2005, p. 97).3Such authors include Mayerfeld (1999, pp. 176–180), Brulde (2010, p. 577), Hedenius

(1955, pp. 100–102), and Erik Carlson (e-mail to the author, Oct. 1, 2019).

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logics, a sentence can be true to degree 0.85.The idea of replying to sequence arguments based on many-valued logic has

been mentioned in the literature. More exactly, it has been suggested that onecan appeal to vagueness, and that one of the options is a theory of vaguenessinvolving degrees of truth (Qizilbash 2005) or many-valued logic (Knapp 2007).4

But there is more to be done. Although Qizilbash (2005) mentions degree theo-ries of vagueness, he focuses on another theory (supervaluationism), as work onvagueness and sequence arguments tends to do (e.g., Handfield and Rabinowicz2018). Knapp (2007) goes more into many-valued logic but the treatment of thetopic is still brief. Moreover, in contrast to the works just mentioned, I do notappeal to vagueness. I focus on the logic, and I leave it open whether vaguenesshas any role to play.

There are several reasons why it is worthwhile to investigate many-valuedlogic and sequence arguments.5 Broadly speaking, many-valued logic seems atleast as suitable for use in value theory as does two-valued (e.g., classical) logic,regardless of sequence arguments, but many-valued logic also has particularstrengths when it comes to such arguments. More specifically, many-valuedlogic allows for gradual changes in the phenomenon at hand to be mirrored bygradual changes in degrees of truth.6 For example, if someone who is goingbald loses one more hair, it can become slightly truer that the person is bald.Similarly, slight changes in evaluatively relevant features can be mirrored byslight changes in the truth degree of value statements about that phenomenon.A related advantage of using many-valued logic in value theory is that it allowsfor a nuanced, precise repertoire of positions. For example, one can assign atruth value such as 0.76 to a view in value theory.

A standard question is how to understand degrees of truth. The answerdoes not affect the results of this paper so I will leave the question open andmerely touch on the topic. I follow authors such as Hajek (1998, pp. 2, 4) andDubois and Prade (2001) in emphasising the distinction between truth degreesand probabilities, and I do not understand degrees of truth as degrees of beliefor probabilities. One can understand truth degrees objectively. For example,according to Smith (2008, p. 233), ‘For any proposition, its degree of truth is anobjective matter—determined by the world.’ An objective view can include theidea that possession of a property comes in degrees. As Smith (2008, p. 211) putsit, “if Bob’s degree of baldness is 0.3, then ‘Bob is bald’ is 0.3 true.” We woulddeal with betterness or worseness rather than baldness, but the story could besimilar: the holding of the relation of worseness between two items can come indegrees. Another option is to understand the truth degree an agent would giveto a sentence as the ease with which the agent can accept the sentence (Paris

4There is a relevant literature on many-valued logic and the sorites paradox (Paoli 2019),which has some resemblance to sequence arguments (Temkin 1996; Pummer 2018, §3; Asgeirs-son 2019).

5See Paoli (2003, forthcoming) for defences of many-valued logic, and Smith (2008) for adefence of degrees of truth. For writings favourable to many-valued logic, see, e.g., Behounek(2006), Hajek (2007), and Novak and Perfilieva (2011). For objections to many-valued logic,see Paoli (2003, pp. 367–368) and Smith (2008) and the sources cited there.

6A similar point is made by Paoli (2003, pp. 364–365) in relation to the sorites paradox.

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1997).Sequence arguments typically assume transitivity and sometimes complete-

ness of a relation such as ‘at least as good as.’7 The classical notion of tran-sitivity of ‘at least as bad as,’ which I denote 4, is that for any a, b and c,a 4 b and b 4 c together imply a 4 c. And a standard, classical statement ofcompleteness of 4 is that for any a and b, either a 4 b or b 4 a.

In many-valued logic, there are several notions of transitivity and complete-ness. And different many-valued logics have different sets of truth values, no-tions of logical consequence, and connectives for ‘and,’ ‘or,’ ‘implies,’ etc.

We can take at least two approaches to sequence arguments using many-valued logic. The first is to start with one or more many-valued logics includinglogical connectives. From the connectives in a logic, we can get versions oftransitivity and completeness. For example, in the family L of Lukasiewicz logicsI will work with, we can state transitivity of a relation R using the quantifier∀ (for all), the conjunction ∧ and the implication → as ∀a∀b∀c((aRb ∧ bRc)→aRc), and then consider sequence arguments with that formula as a premise.

The second approach is to consider which transitivity and completeness con-ditions are reasonable to place on many-valued (versions of) relations such as4, without first selecting many-valued logics such as those in L. For example, ifwe let J K denote the truth value, a reasonable completeness condition might bethat for all a and b, Ja 4 bK + Jb 4 aK ≥ 1, regardless of the definitions of theconnectives. We can treat such conditions as meta-level restrictions.

It is unobvious which approach is best, so I will use both approaches, one ata time, but with an emphasis on the second approach.

Along the lines of the second approach, I will state a few basic, commonproperties of a many-valued logic, and use the term ‘M’ to represent the familyof logics with those properties. I then consider ten versions of transitivity andseveral notions of completeness, regardless of whether and how they could beformulated as formulas in a specific logic such as one of the logics in L. In theend, I formulate and prove technical results about sequence arguments for alllogics in the family M.8

Regarding the first approach, I will use L in one technical result. L fallswithin the broader category M. Lukasiewicz logic is ‘the most intensely re-searched many-valued logic,’ according to Hahnle (2001, p. 323). It has beenused in many areas such as game theory and social choice which is closely relatedto value theory (e.g., Marchioni and Wooldridge 2015; Ovchinnikov 1991).

My key findings are that if we grant a notion of completeness, the strongestforms of transitivity can be used to formulate valid sequence arguments, butthese forms seem intuitively problematic. The weakest forms of transitivity seemacceptable as premises, but the resulting sequence arguments are invalid. Thereis a middle ground with a few forms of transitivity that might be acceptable as

7E.g., Norcross (1997), Arrhenius and Rabinowicz (2015), and Handfield and Rabinowicz(2018). Arguments without transitivity can be found in, e.g., Nebel (2018) and Pummer(2018, §3), and they are quite different from the one by Arrhenius and Rabinowicz I will focuson.

8I am grateful to a reviewer for suggesting essentially this approach.

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premises and that result in valid sequence arguments.In §2, I explain the views to which sequence arguments are objections, and in

§3, I describe previous sequence arguments. Then we turn to many-valued logicand sequence arguments. I describe my logical framework (§4), and considermany-valued relations, completeness (§5) and transitivity (§6). In §7, I presentmy results about sequence arguments, and §8 concludes.

2 The views targeted by sequence arguments

The ideas targeted by sequence arguments can and have been specified in dif-ferent ways. My focus will be on the view that there are bad things which areinferior to other bad things, where ‘inferior to’ is defined as follows:

Inferiority: An object b is inferior to another object b′ if and only if there isa number m such that m b-objects are worse than any number of b′-objects.9

There are different ways to specify what a bad b and m b-objects are, and what‘worse than’ refers to. I will give a few examples, but the following specificationsdo not matter for my results: An object b could be an experience with a givenunpleasantness that lasts for one second, and m b-objects could mean m suchexperiences. In general, I think of m b-objects as m objects of the same typeas b. ‘Intuitively, we might think of objects of the same type as being identicalin all value-relevant respects,’ as Arrhenius and Rabinowicz (2015, p. 232) say.The term ‘worse,’ could refer to the value of outcomes or something being worsefor an individual.

Although I will focus on inferiority between bads, my points in this paperare equally relevant to the analogous superiority relation between goods,10 andto the aforementioned views that relate bads to goods along the same line.

3 Previous sequence arguments

Norcross (1997) presented an early sequence argument against inferiority, andRachels (1998) and Temkin (1996) published similar arguments around thattime in the context of the transitivity of ‘better than.’ The perhaps clearestsequence arguments are the more recent ones by Arrhenius and Rabinowicz(2003, 2015) against various forms of superiority, and I will focus on what Iconsider their core sequence argument.11 All sequence arguments I will dealwith assume a finite sequence of goods g1, . . . , gn or bads b1, . . . , bn, where n isa positive integer. b1 could, for example, be torture and bn could be some minor

9I draw on the formulation of weak superiority by Arrhenius and Rabinowicz (2015, p. 232).10I define superiority as follows: A good g is superior to another good g′ if and only if there

is a number m such that m g-objects are better than any number of g′-objects (cf. Arrheniusand Rabinowicz 2015, p. 232).

11Another recent example of a sequence argument is the one by Ord (2013) against theview that ‘suffering and happiness both count, but there is some amount of suffering that noamount of happiness can outweigh.’

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bad such as mild discomfort. The argument by Arrhenius and Rabinowicz isabout goods, but I will rephrase it so that it is about bads because I focus onbads. Essentially, their argument is that in a finite sequence of bads b1, . . . , bn,if b1 is inferior to bn, then the sequence contains a bad bi that is inferior tothe bad bi+1 that immediately follows it. If the sequence is chosen such thateach item is only marginally better than the preceding item, it is implausibleor counterintuitive that bi would be inferior to the only marginally better bi+1.Since this is a consequence of that b1 is inferior to bn, the plausibility of that b1is inferior to bn is undermined.

Arrhenius and Rabinowicz assume classical logic throughout their argu-ment.12 For example, they assume a value statement such as ‘a is better thanb’ does not have an intermediate truth value such as 1

2 , while I will allow suchtruth values. They also assume that ‘at least as good as’ is transitive and com-plete in the classical sense. They prove the following theorem, which I haveadjusted so that it is about ‘at least as bad as’ instead of ‘at least as good as’and about inferiority instead of superiority:13

Theorem 1 (Arrhenius and Rabinowicz 2015, p. 241). Suppose that ‘is at leastas bad as’ is a complete and transitive relation on the domain. Then, in anyfinite sequence of objects in which the first element is inferior to the last element,there exists at least one element that is inferior to its immediate successor.

It is an open question whether it is a problem if there is inferiority or su-periority between adjacent items in a sequence. I will set the question asideand assume that it is desirable to avoid inferiority and superiority between ad-jacent items. Arrhenius (2005, p. 108) finds superiority between adjacent itemscounterintuitive when the difference in value between them is marginal. But Ra-binowicz does not find superiority between adjacent items worrying.14 He andothers have mentioned principles for valuing the objects in a sequence such thata bad could be inferior to the bad that immediately follows it. The principlesmentioned by Carlson (2000), Binmore and Voorhoeve (2003), and Rabinow-icz (2003) share the feature that the marginal value contributed by additionalobjects of the same type diminishes as there are more of them. For example,the marginal negative value of an additional hour of pain at a given intensitydiminishes as the total period becomes longer (Binmore and Voorhoeve 2003).One can, however, object that such principles of diminishing marginal value areimplausible at least in some cases such as when the number of lives lived intorment is increased (Norcross 2009, pp. 85–88). Another kind of argument forthe plausibility of inferiority between adjacent elements can be found in a paperby Klocksiem (2016). He argues that on the intensity scale for pain, there is apoint or threshold that separates mere discomfort from genuine pain, and thegenuine pains are inferior to the discomforts.

12Arrhenius has confirmed in conversation that they had classical logic in mind when theyformulated their argument.

13They call the theorem Observation 5.14Arrhenius (2005, p. 108), Arrhenius and Rabinowicz (2005, p. 138, 2015, p. 238), and

e-mail to the author from Rabinowicz in 2016.

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There are other sequence arguments against inferiority and superiority, butI will also set them aside because it would be separate projects to determinewhether and how they could be formulated as valid, persuasive arguments usingmany-valued logic. For example, the above sequence argument assumes that4 is complete, but Arrhenius and Rabinowicz (2015, p. 241) also present a se-quence argument without assuming completeness, which has a weaker conclusionthan their argument that uses completeness. Other examples are the sequencearguments by Handfield and Rabinowicz (2018) which allow indeterminacy orincommensurability.

4 Our logical framework

I use many-sorted many-valued first-order logics at the object level. At thislevel, we have, for example, many-valued predicates such as 4, connectives suchas ∧, and quantifiers such as ∀. I use sorted logics for convenience because weare dealing with three sorts of things: numbers, bads, and quantities of bads. Atthe meta level, I use classical logic and induction. For example, I use classicallogic when I use proof by contradiction, and when I assume that it is either trueto degree 1 that b is inferior to b′ or it is not true to degree 1 that b is inferiorto b′.

Our formal object-level language L is 3-sorted and contains the sorts σZ+ ,σB and σQ, which, intuitively, are about positive integers, bads and quantitiesof bads, respectively. Each sort has a set of variables: VZ+ = {k,m, n, k′,m′, n′,. . .}, VB = {b, b′, b′′, . . .} and VQ = {q, q′, q′′, . . .}. Similarly, the sorts have thesets of individual constants CZ+ , CB and CQ, respectively. L includes the binaryrelation symbols ≺, 4 and ∼ of type 〈σQ, σQ〉 (i.e., they concern quantities ofbads). The intended readings of ≺, 4 and ∼ are ‘is worse than,’ ‘is at least asbad as’ and ‘is equally bad as,’ respectively. L also contains the binary functionsymbol f of type 〈σQ, σZ+ , σB〉 (f will be associated with a function that takesa number and a bad as inputs and outputs a quantity of a bad).

The set of truth values will be either of the following: A finite set of equidis-tant rational numbers between 0 and 1, always including 0 and 1; that is,

Wp :=

{i

p− 1: 0 ≤ i ≤ p− 1

}for an integer p ≥ 2, where := is definitional equality. For example, W4 ={

0, 13 ,23 , 1}

. Or the infinite set of all real numbers between 0 and 1, including 0and 1; that is,

W∞ := [0, 1]

(Gottwald 2017). ‘W’ represents any of Wp or W∞.I will use the perhaps most basic notion of models and logical consequence

in many-valued logic. A conclusion is a logical consequence of the premises ifand only if (iff) the conclusion is true to degree 1 whenever all premises are

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true to degree 1. We can find this notion of consequence in several importantmany-valued logics (Gottwald 2001, pp. 180, 249, 267, 291, 313, 386). As usualin first-order logic, the truth value of a sentence depends on the interpretationof the language which involves a structure that corresponds to the language(Conradie and Goranko 2015, ch. 4). More exactly, in many-sorted many-valuedfirst-order logic, a structure S (containing domains, relations and functions) fora language J consists of the following:

• for each sort σ in J , a domain Dσ in S;

• for each constant symbol c in J of sort σ, an element cS in Dσ;

• for each predicate symbol P in J of type 〈σ1, . . . , σn〉, a relation PS onDσ1× . . . ×Dσn

(i.e., a mapping PS associating a truth value with eachtuple 〈d1, . . . , dn〉 where di ∈ Dσi

for i = 1, . . . , n);

• for each function symbol f in J of type 〈σ0, . . . , σn〉, a function fS :Dσ1 × . . .×Dσn → Dσ0

(cf. Hajek 1998, §5.5; Manzano 1993; Gottwald 2001, pp. 22, 27; Lucas 2019).The truth value of a sentence A in S is denoted JAKS . We say that S is a modelof A and write S � A iff JAKS = 1. For a set of sentences Σ, S is a modelof Σ and we write S � Σ iff JBKS = 1 for each B ∈ Σ. We say that A is alogical consequence of Σ and write Σ � A iff S � Σ implies S � A for all S (cf.Gottwald 2001, §3.3, 249). That is, Σ � A iff every model of Σ is a model of A.

I define the quantifiers ∀ and ∃ as is commonly done in many-valued logic(e.g., Malinowski 2007, pp. 49, 51; Gottwald 2001, pp. 26, 28, 250; Urquhart2001, p. 274) with the minor modification that the variable and domain are ofa sort. xσ is a variable of sort σ, and H is a well-formed formula with at mostone free variable xσ.

J∀xσHKS := inf{JH[xσ/d]KS : dS ∈ Dσ

};

J∃xσHKS := sup{JH[xσ/d]KS : dS ∈ Dσ

}.{

JH[xσ/d]KS : dS ∈ Dσ

}is the set of truth values of H gotten when, for every

dS in the domain Dσ, each free occurrence of xσ in H is replaced with theconstant d that names dS . Given a set S, inf {S} is the infimum (greatest lowerbound) of S. For example, let S be a subset of R. If inf{S} exists, it is thelargest r ∈ R such that for all s ∈ S, r ≤ s. Similarly, sup{S} is the supremum(least upper bound) of S.

I use the notation ‘M’ for the family of all logics with W, �, ∀ and ∃, asdefined above. ‘Mp’ and ‘M∞’ represent such families of logics with the sets oftruth values Wp and W∞, respectively.

To save on notation, I will omit S and S when it is clear from the contextwhat is meant and, for example, write J K instead of J KS . And I will typicallyuse the same notation for variables, constants, and objects in the domain; forexample, k, m and n for variables of sort σZ+ , constants in CZ+ , and objects inthe domain Z+.

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‘L’ denotes the family of Lukasiewicz logics I deal with. L has any of thesets of truth values W, and the notions of �, ∀ and ∃ are as in M. So L fallswithin M. But L has specific propositional connectives, while it is unspecifiedwhich connectives the logics in M have.

Lukasiewicz logic is often presented as having available two disjunction con-nectives ∨ and Y, and two conjunction connectives ∧ and &, (Hajek 1998, pp. 65,67; Gottwald 2001, pp. 179–181, 2017; Metcalfe, Olivetti, and Gabbay 2009,p. 146; Marra 2013). The connectives of L are listed in Table 1. I omit someparentheses when writing formulas. As usual, negation has preference overdisjunction and conjunction, which have preference over implication and bicon-ditional. For example, I write ((¬A) ∧B)→ (C ∨D) as ¬A ∧B → C ∨D.

Connective Definition Truth functionA→ B JA→ BK = min(1, 1− JAK + JBK)¬A J¬AK = 1− JAKA ∨B (A→ B)→ B JA ∨BK = max(JAK, JBK)A ∧B ¬(¬A ∨ ¬B) JA ∧BK = min(JAK, JBK)A YB ¬A→ B JA YBK = min (1, JAK + JBK)A&B ¬(A→ ¬B) JA&BK = max(0, JAK + JBK− 1)A↔ B (A→ B) ∧ (B → A) JA↔ BK = 1− |JAK− JBK|

Table 1: Propositional connectives of Lukasiewicz logic (L)

In the truth function for ↔, | | is absolute value.Let us briefly discuss the connectives of L, partly because it is also relevant

to other many-valued logics. A common objection is that ‘A and not A’ shouldget truth value 0, but JA∧¬AK = 0.5 if JAK = 0.5.15 Also, Y and & seem odd asdisjunction and conjunction connectives in various cases. For example, let P andQ be unrelated formulas such as P := a 4 b and Q := c 4 d. If JP K = JQK = 0.5,then JPYQK = 1, which sounds too high, and JP&QK = 0, which seems too low. Iwill mention two replies. First, regarding A∧¬A, there are other forms of the lawof contradiction which one can accept even if one rejects that JA∧¬AK is always0 (Rescher 1969, pp. 143–148). Second, one can argue that sometimes ∧ is asuitable formalisation of ‘and’ while in other cases & is appropriate; for example,that ‘A and not A’ should be formalised as A & ¬A, which always has truthvalue 0 (Fermuller 2011, pp. 200–201). An analogous claim can be made about∨ and Y as alternative formalisations of ‘or.’16 For example, Paoli (forthcoming)argues that classical logic is ambiguous and collapses a distinction between twotypes of connectives. Classical disjunction, conjunction and implication caneach be disambiguated in two kinds of ways; for example, classical disjunctioncan be disambiguated as ∨ or Y, and classical conjunction can be disambiguatedas ∧ or & (a formula may contain all of ∨, Y, ∧ and &).

15See Fermuller (2011, pp. 199–200) and Smith (2017) which contains further references.16Thanks to Erik Carlson for bringing up in an e-mail to the author that interpreting ‘and’

as ∧ is more plausible in some situations while interpreting ‘and’ as & seems more appropriatewhen ‘A and B’ is a contradiction. Carlson made a similar point about Y.

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I use classical logic and induction at the meta level for two reasons: First,it is common to do so (Williamson 1994, p. 130; Gottwald 2001, pp. 6–7;Chakraborty and Dutta 2010, p. 1889; Dutta and Chakraborty 2016, p. 238).Second, the object and meta levels are about different matters. It seems reason-able that value statements such as ‘a is worse than b’ can have more than twotruth values. But classical logic and induction may be suitable for whether asentence has a given truth value or not, which kinds of proofs to accept, etc. Inthe metalanguage, I use ‘⇒’ for implication in classical logic, and I have classicallogic in mind when I write ‘implies,’ ‘if . . ., then,’ ‘iff,’ ‘for all,’ ‘there is,’ etc.

5 Many-valued relations and completeness

We have available three many-valued binary relations: 4, ≺ and ∼. Which ofthem do we want to use? How are they related? And which completeness-likeconditions do we want to place on them?

I use 4 and ≺ in all of my results, and I sometimes use ∼. One might find ≺and ∼ conceptually clearer than 4, and therefore avoid 4 or define 4 in termsof ≺ and ∼.17 Or one might find it more parsimonious to take 4 as primitiveand define ≺ and ∼ in terms of 4 (Hansson 2001, p. 322).

I will use an idea about the link between weak and strict fuzzy preferencerelations (Banerjee 1994; Llamazares 2005, p. 480), which captures a notion ofcompleteness (Barrett and Pattanaik 1989, pp. 238–239), and which I call

F := for all a, b, Ja ≺ bK = 1− Jb 4 aK.

F is equivalent to the following formula in L having truth value 1:

F L := ∀a∀b(a ≺ b↔ ¬b 4 a).

For any relation R, ¬aRb means ¬(aRb).The most common definitions of completeness of the single relation 4 seem

to be

Completeness (C4) := for all a, b, Ja 4 bK + Jb 4 aK ≥ 1;

Strong completeness := for all a, b,max(Ja 4 bK, Jb 4 aK) = 1

(Barrett and Pattanaik 1989, pp. 238–239; Llamazares 2005, p. 479; Fono andAndjiga 2007, p. 668). I will use C4 but not strong completeness because it isimplausibly restrictive as it rules out both a 4 b and b 4 a having intermediatetruth values between 0 and 1.

One may want a notion of completeness for only ≺ and ∼, in which case thefollowing seems reasonable (Van de Walle, De Baets, and Kerre 1998, pp. 116–117):18

Trichotomy := for all a, b, Ja ≺ bK + Jb ≺ aK + Ja ∼ bK = 1.

17Thanks to a reviewer for bringing up this matter.18Thanks to a reviewer for suggesting the use of a trichotomy.

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Whether F , F L, C4 and trichotomy are ultimately plausible is beyond thescope of this paper. They seem reasonable if one wants to impose completeness-like conditions on the value relations.

6 Transitivity of many-valued relations

There are many versions of transitivity of many-valued relations. Ten of themare listed in Table 2 (I have shortened some of the names).19 There are morebut these ten cover a fair bit of the ground, and I have tried to include thosemost relevant to sequence arguments. In this section, R is a many-valued binaryrelation, the formulations of transitivity are for all a, b and c in the domain,and R is short for J R K.

Probabilistic-sum transitivity (T1) If 0 < aRb and 0 < bRc, thenaRb+ bRc− aRb · bRc ≤ aRc

Max-transitivity (T2) If 0 < aRb and 0 < bRc, thenmax(aRb, bRc) ≤ aRc

Weighted mean transitivity (T3) If 0 < aRb and 0 < bRc, then thereis λ ∈ (0, 1) such that λmax(aRb, bRc)+(1− λ) min(aRb, bRc) ≤ aRc

Min-transitivity (T4) min(aRb, bRc) ≤ aRcProduct transitivity (T5) aRb · bRc ≤ aRcSensitive transitivity (T6) If 0 < aRb and 0 < bRc, then 0 < aRc

Weak min-transitivity (T7) If bRa ≤ aRb and cRb ≤ bRc, thenmin(aRb, bRc) ≤ aRc

∆-transitivity (T8) aRb+ bRc− 1 ≤ aRcMultiplicative transitivity (T9) aRb · bRc · cRa = aRc · cRb · bRaAdditive transitivity (T10) aRb+ bRc− 1

2 = aRc

Table 2: Versions of transitivity from the literature on fuzzy preference relations

Observation 1. T1 ⇒ T2 ⇒ T3 ⇒ T4 ⇒ T5 ⇒ T6.

Dasgupta and Deb (2001, p. 493) mention this observation and refer tosources for proofs.

Eight of these forms of transitivity of 4 or ≺ seem problematic as premisesin a sequence argument in our framework (T1–T6, T9 and T10). T10 would beunsuitable so I will not consider it more, because if aRb + bRc > 1.5, thenaRc > 1, which is outside of our sets of truth values. T1–T6 and T9 would

19T1–T8 are listed by Dasgupta and Deb (2001, p. 493); T9 and T10 are from Tanino (1984,p. 119, 1990, p. 175) and Herrera-Viedma et al. (2004, p. 101).

11

seemingly be intuitively problematic premises because of the following case (cf.Barrett and Pattanaik 1985, p. 78): There are two bads b1 and b2. Hereafter,

I write m b-objects as mb; for example, 5b1 is 5 b1-objects. Let R represent 4or ≺. Suppose 100b1R100b2 and 100b2R101b1 are at least 1

4 , which could besensible if b1 and b2 are very different and neither appears clearly at least asbad as or worse than the other. Each of T1–T4 implies 100b1R101b1 is at least14 , T5 implies it is at least 1

16 , and T6 implies it is greater than 0. As long as100b1R100b2 > 0 and 100b2R101b1 > 0, each of T1–T6 implies 100b1R101b1 > 0.T9 has this implication if we plausibly assume 101b1R100b1 > 0 because the left-hand side of T9 becomes greater than 0 so all numbers on the right-hand sidemust be greater than 0. These implications are problematic. 100b1R101b1 isplausibly 0 (and even more plausibly less than 1

4 or 116 ) because, since b1 is

something bad, fewer b1-objects are not worse than or equally bad as moreb1-objects but clearly less bad.

Someone may object that these forms of transitivity are only problematicwhen bads of the same type are compared, while sequence arguments only needtransitivity for different types of bads such as b1, b2 and b3. One could weakenthe forms of transitivity as follows so that they only hold for different types ofbads (m, n and k are positive integers, and that b, b′ and b′′ are distinct meansthat b 6= b′, b′ 6= b′′ and b 6= b′′):

Restricted product transitivity (T r5 ) If b, b′ and b′′ are distinct, thenmbRnb′ · nb′Rkb′′ ≤ mbRkb′′

Restricted sensitive transitivity (T r6 ) If b, b′ and b′′ are distinct, thenif 0 < mbRnb′ and 0 < nb′Rkb′′,then 0 < mbRkb′′

For any form of transitivity, I write r when it is restricted to distinct b, b′ andb′′ as in T r5 and T r6 .

But the following case suggests that at least T r1 –T r4 are intuitively prob-lematic: Suppose mb1Rnb2 = nb2Rkb3 = w ∈ (0, 0.5). T r1 –T r4 each impliesmb1Rkb3 ≥ w, but it can plausibly be lower because ifmb1Rnb2 and nb2Rkb3 areequally close to false, it can reasonably be even closer to false that mb1Rkb3. T r5 ,T r6 and T r9 might be too restrictive because they rule out, for example, the com-bination mb1Rnb2 ∈ [0, 1], nb2Rkb3 ∈ [0, 1], mb1Rkb3 ∈ (0, 1], nb2Rmb1 ∈ (0, 1],kb3Rnb2 ∈ (0, 1], kb3Rmb1 = 0. And T r9 has the especially restrictive aspect ofbeing an equality, so I will set T r9 aside.

The seemingly most acceptable forms of transitivity we are left with are T7,T r7 , T8 and T r8 . T r5 and T r6 might be acceptable too. The others seem moreproblematic, and a few seem so unsuitable that I hereafter set them aside (T9,T r9 , T10 and T r10).

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7 Sequence arguments using many-valued logic

In this section, I consider sequence arguments assuming T1–T8 or T r1 –T r8 . I findthat either of T1–T5 or T r1 –T r5 results in a valid sequence argument. So doesT6 or T r6 when the number of truth values is finite, but not when it is infinite.T7,T r7 , T8 and T r8 generally do not result in a valid sequence argument, althoughT7 and T r7 may do so when there are only three truth values.

I assume M in all of my technical results below except one (Theorem 3)in which I assume L. When I speak of F , C4, trichotomy, T1–T8 or T r1 –T r8 , Iassume they are meta-level conditions on the structures (as above, a structureis denoted S). For example, if T4 is assumed, we are considering only the classof structures in which T4 holds; the structures that satisfy T4. In contrast, whenI treat such assumptions as object-level formulas, as in the proof of Theorem 3,I write L, for example, T L

4 .Recall that L is our formal language with three sorts and symbols ≺, f , etc.

as described in §4.I use � for the notion of ‘is inferior to’ I work with in this section. � is an

abbreviation defined as follows:

b� b′ := ∃m∀n(f(m, b) ≺ f(n, b′)).

Informally, I read b� b′ as ‘there is a positive integer m such that m b-objectsare worse than any number (in Z+) of b′-objects.’20 I abbreviate f(m, b) as mb,so we can write b� b′ as ∃m∀n(mb ≺ nb′). When I say ‘is inferior to’ withoutmentioning a truth degree, I mean that it is true to degree 1.

The first result is that, assuming M, F and that any of the transitivityconditions T1–T5 or T r1 –T r5 hold for 4, we get a valid sequence argument.

Theorem 2. In M, if F and any of T1–T5 or T r1 –T r5 of 4, then in any finitesequence of objects in which the first object is inferior to the last object, there isan object that is inferior to its successor.

Proof in appendix A. In other words, Theorem 2 says that, assuming M, inevery structure S for L in which F and any of T1–T5 or T r1 –T r5 of 4 hold, and inwhich there is a finite sequence b1, . . . , bn where S � b1 � bn, there is a bi withi ∈ {1, . . . , n − 1} such that S � bi � bi+1. Theorem 2 is phrased as it is forreadability, and the other theorems are phrased similarly for the same reason,but all could be stated in terms of S, �, �, etc. along the lines just indicatedfor Theorem 2.

Theorem 2 has the problem that at least T1–T5 and T r1 –T r4 seem problematic,or so I argued in §6. But this is a matter of intuition and debatable. Regardless,T r5 might be acceptable, so we have a valid sequence argument with potentiallyacceptable premises.

Someone sympathetic to inferiority can reply to Theorem 2 by saying thatit need not be true to degree 1 that b1 is inferior to bn. It may be true to degree

20Thanks to Graham Leigh regarding the formulation of �.

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0.99. But the next theorem (Theorem 3) shows that, given some assumptions, ifJb1 � bnK = 0.99, then there is a bi in the sequence such that Jbi � bi+1K ≥ 0.99.This holds not only for 0.99 but for any truth value w ∈ [0, 1]. So the upshotof the next theorem is that if one accepts the assumptions in it, one does notavoid sequence arguments by claiming that it is merely true to degree w ∈ [0, 1)that the first object is inferior to the last. The theorem assumes L insteadof M for the following reasons: One might find that assuming a more specificfamily of logics like L is a better approach than assuming M. It is also useful forillustration to use different approaches; for example, in the proof of Theorem 3,we formalise a lemma in L and reason using the semantics of the connectives ofL. Moreover, L is historically important and reasonable, and one can easily usethe connectives in L to state F , T4, which is perhaps the most widely used formof transitivity, and the idea of it being true to at least degree w that a bad isinferior to another.

Theorem 3. In L, if F and T4 of 4, then for any w ∈ [0, 1], and in any finitesequence of objects in which it is true to degree w that the first object is inferiorto the last object, there is an object such that it is true to at least degree w thatit is inferior to its successor.

Proof in appendix B.The forms of transitivity considered so far (T1–T5 and T r1 –T r5 ) are fairly

strong. The weaker T6 results in a valid sequence argument when the numberof truth values is finite, but not when it is infinite, as the next two theoremsshow.

Theorem 4. In Mp, if F and T6 or T r6 of 4, then in any finite sequence ofobjects in which the first object is inferior to the last object, there is an objectthat is inferior to its successor.

Proof in appendix C. The remaining theorems 5 and 6 deal only with unre-stricted forms of transitivity because if the unrestricted form holds, so does therestricted form (i.e., for all i ∈ {1, 2, . . . , 10}, Ti ⇒ T ri ).

Theorem 5. In M∞ there is a structure for L that satisfies F , C4, trichotomy,and T6 of 4, ≺ and ∼, with a finite sequence of objects in which the first objectis inferior to the last object, but in which no object is inferior to its successor.

Proof in appendix D. Theorem 5 shows that, assuming M∞, even if wegrant quite a large number of conditions such as trichotomy and T6 of all threevalue relations, we can still avoid the purportedly unappealing implications ofinferiority. Note that in theorems 2–4 we want to rely on few, weak premises,while in theorems 5 and 6 we want to allow many, strong conditions.

The next and final theorem shows that T7 and T8 are generally not enoughto get a sequence argument (so neither are T r7 and T r8 ). The theorem deals withT7 and T8 at the same time for brevity and because one might try to use severaltransitivity conditions as premises in one argument.

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Theorem 6. In M∞ and Mp≥4 there is a structure for L that satisfies F , C4,trichotomy, and T7 and T8 of 4, ≺ and ∼, with a finite sequence of objectsin which the first object is inferior to the last object, but in which no object isinferior to its successor.

Proof in appendix E. Theorem 6 is about when there are at least four truthvalues, which I find more interesting than the case of three truth values, butone can tell from the proof that an almost identical structure satisfies T8 in M3.We can thereby get a result like Theorem 6 in M3 about only T8 instead of bothT7 and T8. I leave it unanswered whether, assuming M3, T7 or T r7 results in avalid sequence argument.

One may respond to theorems 5 and 6, which show that one can avoid cer-tain sequence arguments, by saying that the truth values of 4, ≺ and ∼ inthose structures are counterintuitive from a value-theoretic perspective. A rea-son why one might find them counterintuitive is that the truth values of thevalue statements are independent of the number of each type of bad in mostcases. One may want to see a more reasonable way of making value compar-isons that avoids sequence arguments. That is a fair point. The structures intheorems 5 and 6 are very simple and merely meant to be sufficient for logicalpurposes. In appendix F, I present a more complex example structure with morereasonable value comparisons. In the end, one might very well want a differentand perhaps even more complex way of making value comparisons. My aimswith this example structure are merely to point out a direction towards makingreasonable value comparisons which avoid at least some type of sequence argu-ment and to illustrate how one can confirm that such a way of making valuecomparisons does not violate some reasonable conditions (I use F , C4 and T8of 4 as examples of such conditions).

In this example structure it is true to degree 0.7 that the first bad is inferiorto the last, but there is no bad such that it is true to at least degree 0.7 that itis inferior to its successor. I assume M∞, and my structure contains the threebads b1, b2 and b3. Jb1 � b3K = 0.7, but Jb1 � b2K = Jb2 � b3K = 0.5. Thetruth degrees of value comparisons depend on the quantities of the bads, whichis one respect in which this structure is more intuitive than those in the proofsof theorems 5 and 6. Value comparisons in terms of ≺ have the following truthvalues: If m ≥ n, Jmb1 ≺ nb3K = 1; that is, a given number of b1-objects aredefinitely worse than fewer or the same number of b3-objects. If m < n, then forany fixed m, Jmb1 ≺ nb3K decreases and approaches a limit, which we can callw, as n increases. This resembles existing ideas of diminishing marginal value,but, importantly, the intuition is not that additional b3-objects contribute lessand less disvalue to the whole. Rather, the intuition is that for a given numberm of b1-objects, it is true to some degree w that m b1-objects are worse thanany number of b3-objects. And while it should become less true that m b1-objects are worse than n b3-objects as n increases, it should always be true toat least degree w. For a higher fixed m, the limit, which we can call w′, ishigher (i.e., w < w′). The intuition is that for a higher m, it is truer that mis a sufficient number of b1-objects for this collection of b1-objects to be worse

15

than any number of b3-objects. As m and then n approach infinity, Jmb1 ≺ nb3Kapproaches 0.7; that is, Jb1 � b3K = 0.7. Value comparisons of b1-objects to b2-objects and of b2-objects to b3-objects work analogously, except that the truthvalue 0.5 instead of 0.7 is approached.

8 Concluding remarks

My findings are partly good news and partly bad news for inferiority and similarviews such as those in §1 and §2. My findings are good news in that one canreadily formulate arguments suggesting that most of the forms of transitivity Ihave considered are intuitively problematic. And the perhaps most acceptableforms of transitivity (T7,T r7 , T8 and T r8 ) are generally not enough to get a validsequence argument. The path to a convincing sequence argument in our logicalframework looks narrow. On the other hand, there are forms of transitivitythat might be acceptable, and that result in valid sequence arguments. Theseare T r5 and, when the number of truth values is finite, T r6 . If there are onlythree truth values, T7 and T r7 , which seem acceptable as premises, may resultin a valid sequence argument, although I would prefer to use more than threetruth values. We get the most convincing sequence arguments when we use themoderately strong forms of transitivity as premises. In particular, to make asequence argument in our framework convincing, a reasonable step would be toargue extensively for the plausibility of using T r5 or T r6 as premises.21

A Proof of Theorem 2

We can establish Theorem 2 using the following lemma and induction (cf.Arrhenius and Rabinowicz 2015, p. 241):

Lemma 1. In M, if F and any of T1–T5 or T r1 –T r5 of 4, then for any distinctobjects b, b′ and b′′, if b is inferior to b′′, then b is inferior to b′ or b′ is inferiorto b′′.

Proof. Suppose b, b′ and b′′ are distinct. Let w1 := Jb � b′K and w2 := Jb′ �b′′K. Suppose Jb � b′′K = 1 but w1, w2 ∈ [0, 1). Pick ε ∈ (0, 1) such thatw1 + ε < 1 and w2 + ε < 1. Let y := w1 + ε and z := w2 + ε. Pick m suchthat J∀k(mb ≺ kb′′)K > y + z − y · z. There is such an m because y + z − y ·

21I am grateful for comments on earlier versions by Roger Crisp, Kaj Borge Hansen,Francesco Paoli, Nils Sylvan and Alex Voorhoeve. I thank the ALOPHIS group at Universityof Cagliari, the LSE Choice Group, the PhD seminar in practical philosophy at StockholmUniversity, and Theron Pummer for helpful discussion. My supervisors Gustaf Arrhenius andKrister Bykvist have kindly contributed in many ways. The following people have been ex-ceptionally helpful: Erik Carlson, Valentin Goranko, Laurenz Hudetz, Graham Leigh, RupertMcCallum, Karl Nygren, Daniel Ramoller and Magnus Vinding. Anonymous reviewers gavevery useful comments. I am grateful for thoughts on an ancestor to this paper from Camp-bell Brown, Jens Johansson, Anna Mahtani and Wlodek Rabinowicz. Thanks to GunnarBjornsson and Mozaffar Qizilbash for answering questions related to my research.

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z < 1 and, by the assumption Jb � b′′K = 1 and the definitions of � and ∃,sup {J∀k(mb ≺ kb′′)K : m ∈ Z+} = 1. To see that y + z − y · z < 1, note that1− (y + z − y · z) = (1− y)(1− z) > 0, so y + z − y · z must be less than 1. Bythe definition of ∀, for all k,

(1) Jmb ≺ kb′′K > y + z − y · z.

Pick n such that

(2) Jmb ≺ nb′K < y.

There is such an n because Jb � b′K = w1 < y and, by the definitions of �, ∃and ∀, for all m there is an n such that Jmb ≺ nb′K < y. Analogously, pick ksuch that

(3) Jnb′ ≺ kb′′K < z.

By (1), (2), (3) and F ,

w3 := Jkb′′ 4 nb′K = 1− Jnb′ ≺ kb′′K > 1− z;w4 := Jnb′ 4 mbK = 1− Jmb ≺ nb′K > 1− y;

w5 := Jkb′′ 4 mbK = 1− Jmb ≺ kb′′K < 1− (y + z − y · z).

w3 · w4 > (1 − z)(1 − y) = 1 − (y + z − y · z) > w5, which contradicts T5 andT r5 of 4, which imply w3 · w4 ≤ w5. By Observation 1, T1–T4 and T r1 –T r4 arecontradicted too. Assuming classical logic at the meta level, we have a proof bycontradiction of Lemma 1.

We use Lemma 1 in the following induction on the length of the sequence toestablish Theorem 2: Base step: The sequence contains two objects. If the firstobject is inferior to the last object, the first object is inferior to its successor.Induction hypothesis: When the length of the sequence is n objects (n ≥ 2), ifthe first object is inferior to the last object, there is an object in the sequencethat is inferior to its successor. Induction step: The length is n + 1 objects.Suppose the first object is inferior to the last object (object n + 1). If objectn is inferior to object n + 1, an object is inferior to its successor. If object nis not inferior to object n+ 1, then, by Lemma 1, the first object is inferior toobject n. By the induction hypothesis, there is an object in the sequence thatis inferior to its successor.

B Proof of Theorem 3

We can establish Theorem 3 using the following lemma:

Lemma 2. In L, if F and T4 of 4, then for any w ∈ [0, 1] and any objects b,b′ and b′′, if it is true to degree w that b is inferior to b′′, then it is either trueto at least degree w that b is inferior to b′ or true to at least degree w that b′ isinferior to b′′.

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More exactly, we establish Theorem 3 by the following induction on thelength of the sequence,22 in which we use Lemma 2: Base step: The sequencecontains two objects. If it is true to degree w that the first object is inferiorto the second, it is true to degree w that the first is inferior to its successor.Induction hypothesis: When the length of the sequence is n objects, if it is trueto degree w that the first object is inferior to the last object, there is an objectin the sequence such that it is true to at least degree w that it is inferior toits successor. Induction step: The length is n + 1 objects. Suppose it is trueto degree w that the first object is inferior to the last object (object n + 1).If it is true to at least degree w that object n is inferior to object n + 1, thenthere is an object such that it is true to at least degree w that it is inferior toits successor. If it is not true to at least degree w that object n is inferior toobject n + 1, then, by Lemma 2, it is true to at least degree w that the firstobject is inferior to object n. By the induction hypothesis, there is an objectin the sequence such that it is true to at least degree w that it is inferior to itssuccessor.

I will prove a version of Lemma 2 called Lemma 2′. Lemma 2 has three partsbesides the assumption of L: assumptions of F and T4 of 4, and a statementabout inferiority. I formalise these parts in L as follows:

F L := ∀q∀q′(q ≺ q′ ↔ ¬q′ 4 q);T L4 := ∀q∀q′∀q′′(q 4 q′ ∧ q′ 4 q′′ → q 4 q′′);

I := ∀b∀b′∀b′′(b� b′′ → b� b′ ∨ b′ � b′′).

I is a formalisation of the part of Lemma 2 about inferiority because if JIK = 1,then for any b, b′, b′′, Jb� b′K or Jb′ � b′′K is at least as great as Jb� b′′K.

The version of Lemma 2 formalised in L becomes

Lemma 2′. F L, T L4 � I.

I do not see a direct proof of Lemma 2′, but the following is a proof bycontradiction:

Proof of Lemma 2′. Let S be an arbitrary structure for L. Suppose (1) JF LKS= 1; (2) JT L

4 KS = 1; (3) JIKS < 1. In the rest of the proof, JK is short for JKS .By the definition of ∀, (3) iff there are b, b′ and b′′, such that

(4) Jb� b′′ → b� b′ ∨ b′ � b′′K < 1.

By the semantics of →, (4) iff

(5) Jb� b′′K > Jb� b′ ∨ b′ � b′′K.

Let r := Jb � b′′K; s := Jb � b′K; t := Jb′ � b′′K. By the semantics of ∨, (5) iffr > max(s, t), so s < r and t < r.

22Thanks to Valentin Goranko for suggesting that one can do induction on the length ofthe sequence.

18

Claim. There are m, n and k, such that

(6) Jmb ≺ nb′K < Jmb ≺ kb′′K;

(7) Jnb′ ≺ kb′′K < Jmb ≺ kb′′K.

Proof of Claim. Pick a real number u such that s < u < r and t < u < r, andshow that there are m, n and k such that

(8) Jmb ≺ nb′K < u < Jmb ≺ kb′′K;

(9) Jnb′ ≺ kb′′K < u < Jmb ≺ kb′′K.

First we show there is an m such that

(10) u < J∀k(mb ≺ kb′′)K.

By the definitions of � and ∃, r = J∃m∀k(mb ≺ kb′′)K = sup{J∀k(mb ≺kb′′)K : m ∈ Z+}. This means that there is no r′′ < r such that for all m,J∀k(mb ≺ kb′′)K ≤ r′′. In particular, since u < r, it is not the case that for allm, J∀k(mb ≺ kb′′)K ≤ u. So there is an m such that u < J∀k(mb ≺ kb′′)K.

Next, we show that there is an n such that

(11) Jmb ≺ nb′K < u.

s = sup{J∀n(mb ≺ nb′)K : m ∈ Z+}, so for all m, J∀n(mb ≺ nb′)K ≤ s. Thus,J∀n(mb ≺ nb′)K ≤ s. By the definition of ∀, J∀n(mb ≺ nb′)K = inf{Jmb ≺ nb′K :n ∈ Z+}. So there is no s′′ > s such that for all n, Jmb ≺ nb′K ≥ s′′. Sinceu > s, there is an n such that Jmb ≺ nb′K < u.

Analogously, there is a k such that

(12) Jnb′ ≺ kb′′K < u.

t = sup{J∀k(nb′ ≺ kb′′)K : n ∈ Z+}, so J∀k(nb′ ≺ kb′′)K ≤ t. So by the definitionof ∀, there is no t′′ > t such that for all k, Jnb′ ≺ kb′′K ≥ t′′. Since u > t, thereis a k such that Jnb′ ≺ kb′′K < u.

By (10) and the definition of ∀,

(13) u < Jmb ≺ kb′′K.

(11), (12) and (13) imply (8) and (9), which imply (6) and (7). Claim isproved.

Let m, n and k represent three constants that satisfy both parts of Claim(i.e., (6) and (7)), and define truth values r′, s′ and t′ as follows:

(14) r′ := Jmb ≺ kb′′K;

(15) s′ := Jmb ≺ nb′K;

(16) t′ := Jnb′ ≺ kb′′K.

By (6), (7), (14), (15) and (16), (17) s′ < r′; (18) t′ < r′. By (1),

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(19) for any q and q′, Jq′ 4 qK = 1− Jq ≺ q′K.This equality in (19) holds if we replace q and q′ by, say, mb and kb′′ becausemb is shorthand for f(m, b) and the function fS named by the symbol f hascodomain Q, so each element in the range of fS is named by a constant in CQ.

By (14), (15), (16) and (19),

(20) Jkb′′ 4 mbK = 1− r′;

(21) Jnb′ 4 mbK = 1− s′;

(22) Jkb′′ 4 nb′K = 1− t′.By (2),

(23) Jkb′′ 4 nb′ ∧ nb′ 4 mb→ kb′′ 4 mbK = 1.

By (20), (21), (22) and the semantics of ∧ and →, (23) is equivalent to

(24) min(1− t′, 1− s′) ≤ 1− r′.By (17), 1− r′ < 1− s′, and by (18), 1− r′ < 1− t′, which contradicts (24).

We have seen that if JF LKS = JT L4 KS = 1, JIKS < 1 leads to a contradiction.

Assuming classical logic at the meta level, we have a proof by contradiction ofthat JF LKS = 1 and JT L

4 KS = 1 imply JIKS = 1. Because S is an arbitrarystructure, we can conclude that for any structure in which the truth value ofF L and T L

4 is 1, the truth value of I is 1.

C Proof of Theorem 4

We can establish the theorem by a lemma and induction. The induction is thesame as in appendix A except that Lemma 3 is used so I omit the induction.

Lemma 3. In Mp, if F and T6 or T r6 of 4, then for any distinct objects b, b′

and b′′, if b is inferior to b′′, then b is inferior to b′ or b′ is inferior to b′′.

Proof. Suppose b, b′ and b′′ are distinct and Jb � b′′K = 1. Jb � b′′K = 1 iffsup{J∀k(mb ≺ kb′′)K : m ∈ Z+} = 1 so because there are finitely many truthvalues, there is an m such that J∀k(mb ≺ kb′′)K = 1 and, by the definition of ∀,such that Jmb ≺ kb′′K = 1 for all k. By F ,

(1) there is an m such that Jkb′′ 4 mbK = 0 for all k.

Case 1. Jb′ � b′′K < 1. Thus, for all n, there is a k such that Jnb′ ≺ kb′′K < 1and, by F , such that Jkb′′ 4 nb′K > 0. So, by (1), there is an m such that forany choice of n, there is a k such that Jkb′′ 4 nb′K > 0 and Jkb′′ 4 mbK = 0. ByT6 or T r6 of 4, Jnb′ 4 mbK = 0 and, by F , Jmb ≺ nb′K = 1. So there is an msuch that for any n, Jmb ≺ nb′K = 1; that is, Jb� b′K = 1.

Case 2. Jb � b′K < 1. Hence, for all m, there is an n such that Jmb ≺nb′K < 1 and, by F, such that Jnb′ 4 mbK > 0. So, by (1), there is an m andan n such that Jnb′ 4 mbK > 0 and Jkb′′ 4 mbK = 0 for all k. By T6 or T r6of 4, Jkb′′ 4 nb′K = 0 for all k; hence, by F , Jnb′ ≺ kb′′K = 1 for all k. SoJb′ � b′′K = 1.

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D Proof of Theorem 5

Construct a simple structure S in which 4 and ≺ always get the same truthvalue in (0, 1] and ∼ always gets 0. This takes care of T6. Let S contain thedomains Z+ = {1, 2, 3, . . .}, B = {b1, b2, b3} and Q = Z+ ×B, and the functionf : Z+×B → Q, which is simply a bijection that maps each ordered pair 〈m, b〉in Z+ ×B to the same ordered pair 〈m, b〉 in Q. Let R represent ≺ and 4. Forall m,n ∈ Z+, let

Jmb1Rnb3K = 1− 1

2m;

Jnb3Rmb1K =1

2m;

Jmb1Rnb1K =1

2;

for b, b′ ∈ {b2, b3}, JmbRnb′K =1

2;

for b, b′ ∈ {b1, b2, b3}, Jmb ∼ nb′K = 0.

That was the description of S. In S, b1 is inferior to b3 (i.e., S � b1 � b3)because sup{J∀n(mb1 ≺ nb3)K : m ∈ Z+} = 1. It is easy to confirm that thereare no other inferiority relationships and that F , C4, trichotomy, and T6 of 4,≺ and ∼ hold in S.

E Proof of Theorem 6

Let the new structure S have the same domains and function as in appendix D,and let R represent ≺ and 4. I omit JK around relation expressions; for example,mbRnb′ is short for JmbRnb′K. For all m,n ∈ Z+ and b, b′ ∈ B let

mb1Rnb3 = 1;

nb3Rmb1 = 0;

mb1Rnb2 = mb2Rnb3 = w;

nb2Rmb1 = nb3Rmb2 = 1− w;

mbRnb =1

2;

mb ∼ nb′ = 0;

where w ∈ (0.5, 1).S � b1 � b3 because mb1 ≺ nb3 = 1 for all m,n ∈ Z+. No other bad is

inferior to any other. F and C4 hold in S because for m,n ∈ Z+ and b, b′ ∈ B,mbRnb′ = 1 − nb′Rmb. Trichotomy holds for this reason and because thetruth value of ∼ is always 0. T7 and T8 of ∼ hold because min(0, 0) ≤ 0 and0 + 0− 1 ≤ 0.

21

T7 and T8 of R hold when the form bR b is on the left-hand side of theinequality in the transitivity condition because to then get the form 〈aRb, bRc,aRc〉, we need 〈 bR b, bR b′, bR b′〉 or 〈 b′R b, bR b, b′R b〉. In either case, T7and T8 of R hold because the truth value of R is independent of m and n, andfor any w1, w2 ∈ [0, 1], min(w1, w2) ≤ w2 and w1 + w2 − 1 ≤ w2.

It remains to confirm T7 and T8 of R when nothing on the left-hand side ofT7 or T8 has the form bR b. In this case, to violate T7, both arguments of themin function in T7 need to be at least w for the antecedent (bRa ≤ aRb andcRb ≤ bRc) of T7 to hold. The only combination of arguments which are atleast w with the form 〈aRb, bRc〉 is 〈mb1Rnb2, nb2Rkb3〉, where m,n, k ∈ Z+.But then we get min(mb1Rnb2, nb2Rkb3) ≤ mb1Rkb3 = 1, which holds, so T7of R is confirmed.

To violate T8, the left-hand side of the inequality in T8 must be greater than0. There are three such cases in which the terms on the left-hand side have theform 〈aRb, bRc〉. In these cases, T8 implies the following for any m,n, k ∈ Z+:

mb1Rkb3 + kb3Rnb2 − 1 ≤ mb1Rnb2, i.e., 1 + 1− w − 1 ≤ w;

nb2Rmb1 +mb1Rkb3 − 1 ≤ nb2Rkb3, i.e., 1− w + 1− 1 ≤ w;

mb1Rnb2 + nb2Rkb3 − 1 ≤ mb1Rkb3, i.e., w + w − 1 ≤ 1.

These inequalities hold so T8 of R is confirmed. That concludes the proof.Note that T8 holds in a structure that is exactly like S except that w = 1

2 .In that case, we would only use the three truth values in W3 =

{0, 12 , 1

}, so we

would get a result like Theorem 6 in M3 about only T8.

F A more intuitive structure

I assume M∞ and present a structure S in which it is true to degree 0.7 thatthe first object is inferior to the last object, but in which there is no object suchthat it is true to at least degree 0.7 that it is inferior to its successor. I showthat F and C4 hold in S, and I present a partial demonstration of that T8 of4 holds.S has the same domains and function as in appendix D. Let R represent ≺

and 4. As in appendix E, I omit J K around relation expressions; for example,

22

mb1 4 nb2 is short for Jmb1 4 nb2K. For all m,n ∈ Z+ and i, j ∈ {1, 2, 3}, let

nb2Rmb1 = nb3Rmb2 =

0 if m ≥ n,

0.5(

1 + 1m+1

) √n−m√n

if m < n;

mb1Rnb2 = 1− nb2Rmb1;

mb2Rnb3 = 1− nb3Rmb2;

nb3Rmb1 =

0 if m ≥ n,

0.3(

1 + 1m+1

) √n−m√n

if m < n;

mb1Rnb3 = 1− nb3Rmb1;

mbi 4 nbi =

1 if m ≥ n,

0 if m < n;

mbi ≺ nbi =

1 if m > n,

0 if m ≤ n;

mbi ∼ nbj =

1 if m = n and i = j,

0 otherwise.

C4 holds in S because for all m,n ∈ Z+, mbi 4 nbi +nbi 4 mbi ≥ 1, and wheni 6= j, mbiRnbj = 1− nbjRmbi. Due to the latter, F holds when i 6= j. And Fholds when the same type of object is compared: If m > n, mbi ≺ nbi = 1 andnbi 4 mbi = 0. If m ≤ n, mbi ≺ nbi = 0 and nbi 4 mbi = 1.

I is true to degree 0.7 that b1 is inferior to b3, but there is no object suchthat it is true to at least degree 0.7 that it is inferior to an adjacent object inthe sequence:

Jb1 � b3K = limm→∞

limn→∞

(1− 0.3

(1 +

1

m+ 1

) √n−m√n

)= 0.7;

Jb1 � b2K = Jb2 � b3K =

limm→∞

limn→∞

(1− 0.5

(1 +

1

m+ 1

) √n−m√n

)= 0.5;

Jb3 � b1K = Jb3 � b2K = Jb2 � b1K = 0.

To confirm T8 of 4, we need to confirm that for all m,n, k ∈ Z+ and b, b′, b′′ ∈B,

mb 4 nb′ + nb′ 4 kb′′ − 1 ≤ mb 4 kb′′. (F1)

23

There are many cases such as when b = b′ 6= b′′ and m = k < n. I find that F1holds in all cases so that T8 of 4 holds in S. But it is a lengthy exercise to gothrough all cases so I will only confirm the six most difficult cases here:

Case 1 mb1 4 nb2 + nb2 4 kb3 − 1 ≤ mb1 4 kb3, when m < n < k;

Case 2 mb3 4 nb1 + nb1 4 kb2 − 1 ≤ mb3 4 kb2, when n < k < m;

Case 3 mb2 4 nb3 + nb3 4 kb1 − 1 ≤ mb2 4 kb1, when k < m < n;

Case 4 mb1 4 nb3 + nb3 4 kb2 − 1 ≤ mb1 4 kb2, when m < k < n;

Case 5 mb2 4 nb1 + nb1 4 kb3 − 1 ≤ mb2 4 kb3, when n < m < k;

Case 6 mb3 4 nb2 + nb2 4 kb1 − 1 ≤ mb3 4 kb1, when k < n < m.

We can deal with cases 1, 2 and 3 at the same time because they are equiv-alent. For example, case 1 becomes

0 ≤ 0.5

(1 +

1

m+ 1

) √n−m√n

+ 0.5

(1 +

1

n+ 1

) √k − n√k− 0.3

(1 +

1

m+ 1

) √k −m√k

, (F2)

where m < n < k. And case 2 becomes

0 ≤ 0.5

(1 +

1

k + 1

) √m− k√m

− 0.3

(1 +

1

n+ 1

) √m− n√m

+ 0.5

(1 +

1

n+ 1

) √k − n√k

, (F3)

where n < k < m. To notice that the two cases are equivalent, in F2 and itsm < n < k, rename m to n, n to k, and k to m to get F3 and its n < k < m.The way to get from case 2 to case 3 and from case 3 to case 1 is analogous. Sowe can confirm T8 of 4 for cases 1, 2 and 3 by confirming it for case 2, which Iwill do by checking that F3 holds for all m,n, k ∈ Z+ such that n < k < m.23

To minimise the right-hand side of F3 for any constant k ≥ 2, m should beas small as possible and n should be as large as possible; that is, m = k + 1and n = k − 1. The reason is that when 1 ≤ n < k < m, and n, k and mare real numbers, the first-order partial derivatives of the right-hand side of F3with respect to m and n are positive and negative, respectively.

The partial derivative of the right-hand side of F3 with respect to m is

0.25k(k + 2)

m√m(k + 1)

√m− k

− 0.15n(n+ 2)

m√m(n+ 1)

√m− n

. (F4)

23Thanks to Magnus Vinding for explaining how one can check F3 and several of the otherinequalities below.

24

To check that F4 is positive when 1 ≤ n < k < m, confirm the followinginequality for such n, k and m:

0.25k(k + 2)

m√m(k + 1)

√m− k

>0.15n(n+ 2)

m√m(n+ 1)

√m− n

. (F5)

On both sides of F5, multiply by m√m, k + 1 and n+ 1, and divide by 0.15 to

get

53k(k + 2)(n+ 1)√m− k

>n(n+ 2)(k + 1)√

m− n. (F6)

When 1 ≤ n < k < m, the following holds: The denominator on the left-hand side of F6 is less than the denominator on the right-hand side, and afterexpanding the products in the numerators, one can see that the numerator onthe left-hand side is greater than the numerator on the right-hand side. F5 isconfirmed, so the partial derivative of the right-hand side of F3 with respect tom is positive when 1 ≤ n < k < m.

The partial derivative of the right-hand side of F3 with respect to n is

0.15(n2 + n+ 2

)+ 0.3m

√m√m− n(1 + n)2

−0.25

(n2 + n+ 2

)+ 0.5k

√k√k − n(1 + n)2

. (F7)

To confirm that F7 is less than 0 when 1 ≤ n < k < m, note that for such n, kand m,

0.15(n2 + n+ 2

)√m√m− n(1 + n)2

<0.25

(n2 + n+ 2

)√k√k − n(1 + n)2

(F8)

because 0.15 < 0.25 and k < m, and then confirm

0.3m√m√m− n(1 + n)2

<0.5k√

k√k − n(1 + n)2

, (F9)

25

which we can do by simplifying and rearranging F9 to

m√m√m− n

<5

3

k√k√k − n

;

√m√

m− n<

5

3

√k√

k − n;

√m√k − n < 5

3

√k√m− n;

m (k − n) <

(5

3

)2

k (m− n) ;

mk −mn < 25

9km− 25

9kn;

25

9kn <

16

9km+mn;

16

9kn+ kn <

16

9km+mn.

This holds when 1 ≤ n < k < m because 169 kn <

169 km and kn < mn. So the

partial derivative of the right-hand side of F3 with respect to n is negative when1 ≤ n < k < m.

In F3, replace m by k + 1 and n by k − 1 to get

0 ≤ 0.5

(1 +

1

k + 1

)1√k + 1

− 0.3

(1 +

1

k

) √2√

k + 1+ 0.5

(1 +

1

k

)1√k, (F10)

where k ≥ 2. F10 holds for all k ≥ 2 because 0.3√2√

k+1< 0.5√

kfor all k ≥ 2. Thus,

F3 holds for all m,n, k ∈ Z+ such that n < k < m. T8 of 4 for cases 1, 2 and 3is confirmed.

Cases 4, 5 and 6 can be treated all at once for the same reason as cases 1,2 and 3. I will confirm T8 of 4 for cases 4, 5 and 6 by confirming the followinginequality based on case 5:

0 ≤ 1− 0.5

(1 +

1

m+ 1

) √k −m√k

− 0.5

(1 +

1

n+ 1

) √m− n√m

+ 0.3

(1 +

1

n+ 1

) √k − n√k

, (F11)

where n, k,m ∈ Z+ and n < m < k. The partial derivative of the right-handside of F11 with respect to k is negative when n, m and k are real numbers, and1 ≤ n < m < k. So k should be as large as possible to minimise the right-hand

26

side of F11. The limit of the right-hand side of F11 as k goes to infinity is theright-hand side of

0 ≤ 1− 0.5

(1 +

1

m+ 1

)− 0.5

(1 +

1

n+ 1

) √m− n√m

+ 0.3

(1 +

1

n+ 1

). (F12)

F12 holds for all n,m ∈ Z+ such that n < m, as one can see by expanding thebrackets in F12 and simplifying. So T8 of 4 is confirmed for case 5 and thusalso for cases 4 and 6.

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