De Finettian Logics of Indicative Conditionals

This paper explores trivalent truth conditions for indicative conditionals, examining the"defective"table put forward by de Finetti 1936, as well as Reichenbach 1944, first sketched in Reichenbach 1935. On their approach, a conditional takes the value of its consequent whenever its antecedent is True, and the value Indeterminate otherwise. Here we deal with the problem of choosing an adequate notion of validity for this conditional. We show that all standard trivalent schemes are problematic, and highlight two ways out of the predicament: one pairs de Finetti's conditional (DF) with validity as the preservation of non-False values (TT-validity), but at the expense of Modus Ponens; the other modifies de Finetti's table to restore Modus Ponens, but fails to preserve intersubstitutivity under negation. In Part I of this paper, we present both alternatives, with specific attention to a variant of de Finetti's table (CC) proposed by Cooper 1968 and Cantwell 2008. In Part II, we give an in-depth treatment of the proof theory of the resulting logics, DF/TT and CC/TT: both are connexive logics, but with significantly different algebraic properties.


Introduction
Bivalent logic struggles to specify adequate truth conditions for the indicative conditional "if A, then C" (henceforth, A → C) of natural language. For instance, if the indicative conditional is said to have the same truth conditions as the material conditional ¬A ∨ C, then it is automatically declared true whenever the antecedent is false or when the consequent is true. This is notoriously problematic because it severs the link between antecedent and consequent: suppose John was not in Paris yesterday, then "if John was in Paris yesterday, then he will be in Turin tomorrow" is true regardless of John's travels plans. The inferential dimension of conditionals is completely lost in this picture. Beside, and perhaps more fundamentally, this view cannot reconstruct the intuition voiced by de Finetti (1936), Reichenbach (1944, 168) and Quine (1950) (crediting Ph. Rhinelander for the idea), that uttering a conditional amounts to making a conditional assertion: the speaker is committed to the truth of the consequent when the antecedent is true, but committed to neither truth nor falsity of the consequent when the antecedent is false.
Several strategies out of this predicament have been suggested. One is to strengthen the truth conditions of the two-valued conditional. In particular, Stalnaker (1968Stalnaker ( , 1975 proposed that a conditional A → C with false antecedent is true if and only if C is true in the closest possible A-world-i.e. the closest world in which the antecedent is true. This proposal has many virtues but also some limitations, on which we say more in the next section. A second strategy admits that the truth conditions of the indicative conditional may not be truth-functional, or perhaps agree with those of the material conditional (e.g., Jackson 1987), but in any case they are a matter of secondary importance. What matters, ultimately, is the assertability or "reasonableness" of a conditional A → C, a topic that can be analyzed in probabilistic terms, e.g., focusing on the probability of C given A, in symbols p(C|A). Bracketing the topic of truth conditions is a popular strategy among cognitive scientists (e.g., Evans et al. 2007;Over et al. 2007), and among philosophers who focus on the evidential and inferential dimension of a conditional (e.g., Adams 1965Adams , 1975Edgington 1995;Krzyzanowska 2015;Douven 2016). To our mind, however, it would be preferable to have a theory that explains how assertability conditions are related to, and can be motivated from, the truth conditions of a conditional.
The idea that a conditional with a false antecedent is indeterminate in truth value is sometimes summarized in what Kneale and Kneale (1962) have named the "defective" truth table, in which the symbol '#' marks a truth value gap (Figure 1). When the gap is handled as a value of its own (we represent it by 1 /2, for "indeterminate"), and so as a possible input for semantic evaluation, then the "defective" two-valued conditional naturally leads to truth conditions within a trivalent (=three-valued) logic. For de Finetti, asserting a conditional of the form "if A then C" is a conditional assertion: an assertion that is retracted, or void, if the antecedent turns out to be false. In this respect, it is akin to making a bet that if A then C. When A is realized and C is false, the bet is 1 0 1 1 0 0 # # 1 # 0 1 1 · 0 # · · · 0 # · # Figure 1: "Defective" bivalent table (left) and trivalent incomplete expansion (right) lost; when A is realized and C is true, the bet is won; when A is not realized, however, the bet is simply called off.
f → DF 1 1 /2 0 1 1 1 /2 0 1 /2 1 /2 1 /2 1 /2 0 1 /2 1 /2 1 /2 Figure 2: The truth table for de Finetti's trivalent conditional. Reichenbach (1944) introduces the same table as de Finetti's, which he calls quasi-implication. His motivations are related but partly distinct, for they rely on measurement-theoretical considerations in quantum physics. Closer to the interpretation of the third truth value that features in Bochvar (1937), Reichenbach considers that some conditionals are meaningless when the antecedent concerns an event whose precise measurement is impossible (for instance, we cannot in general simultaneously measure position and momentum of a particle with arbitrary degree of precision). Reichenbach treats the third truth value as objectively indeterminate rather than as expressing a notion of subjective ignorance, as de Finetti does. In motivating this interpretation, Reichenbach refers explicitly to the Bohr-Heisenberg interpretation of quantum mechanics. 1 Setting aside those interpretive differences, de Finetti's and Reichenbach's trivalent truth table for the conditional functor f → DF mirrors an explication of the indicative conditional whereby the conditional is void if the antecedent is not true (Figure 2). 1 See Reichenbach (1935, 381) for what is likely the first occurrence of the left table in Figure 1, and Over andBaratgin 2017 on the history of the defective table. De Finetti presented his paper in Paris in the same year 1935, with explicit reference to Reichenbach 1935, but criticizing the latter's objective interpretation of probability. To the best of our knowledge, Reichenbach's 1935 book does not quite present De Finetti's three-valued table, but some variants instead. However Reichenbach (1944, 168, fn. 2) rightly traces quasi-implication back to his previous opus. In our view, the de Finetti conditional may therefore be called the de Finetti-Reichenbach conditional, but for simplicity and partly for established usage, we stick to calling it the DF conditional. We note that the DF table was reintroduced several times over the past decades, very often without prior notice of either de Finetti or Reichenbach, and sometimes with separate motivations in mind, viz. Blamey (1986), who calls it transplication, to highlight its hybrid character between a conjunction and an implication, or recently Kapsner (2018), who came up with the scheme specifically to deal with connexiveness. More on this will be said below.
Which trivalent table is the most adequate? This question was investigated experimentally by (Baratgin et al. 2013), who asked participants to evaluate various indicative conditional sentences as "true", "false" and "neither", by manipulating the truth value of the antecedent and consequent (making them clearly true, false, or uncertain). From their analysis, Baratgin and colleagues conclude that the original de Finetti table is better-supported than its competitors. Moreover, their findings indicate that participants's judgments are well-correlated with the de Finettian bet interpretation of conditionals. From a logical point of view, however, we note that the choice of a truth table cannot be made in isolation, only by looking at intuitions about the composition of truth values. A valuation scheme for a connective does not determine its inferential properties, unless it is supplemented by a definition of validity. Hence, the question about the appropriate truth table can only be answered in conjunction with an analysis of the appropriate notion of validity.
In trivalent logic, several notions of validity can be considered, and they yield significantly distinct predictions (Égré and Cozic 2016). Consider validity as preservation of truth (i.e., the value 1) from premises to conclusion in an argument. Following the terminology of Cobreros et al. (2012), we call this strict-to-strict validity, or SS-validity. An alternative is to define validity as the preservation of non-falsity ({1, 1 /2}), also known as tolerant-to-tolerant or TT-validity. Other schemes considered in the literature are the intersection of SS and TT (see McDermott 1996), as well as so-called mixed (strict-to-tolerant, tolerant-to-strict) consequence relations (ST, TS). All schemes have advantages and drawbacks, but some combinations of a conditional operator with a validity scheme appear better than others.
In this paper, we bring together the research strands on validity in trivalent logic and trivalent semantics for indicative conditionals. More precisely, we conduct a systematic investigation of the main trivalent semantics for defective conditionals, and isolate the most promising combinations of truth tables and validity relations. To the best of our knowledge, no such systematic comparison has been conducted so far. In particular, apart from Cooper (1968), we are not aware of an axiomatization of the logics based on a trivalent semantics for the indicative conditional.
We fill this gap in our paper and proceed in two main parts. Part I of this paper focuses on semantics: it reviews the main motivations for the de Finetti conditional (Section 2) and expounds the problems it faces when selecting an adequate trivalent consequence relation. This is what we call the "validity trilemma" for the de Finetti conditional (Section 3): the de Finetti conditional must either fail to support any sentential validity, support unacceptable arguments, or fail Modus Ponens. We present two ways out of this predicament: the first bites the bullet and associates de Finetti's conditional with a notion of tolerant-to-tolerant validity that fails Modus Ponens (Section 4). The other consists in modifying de Finetti's table so as to restore Modus Ponens for the same notion of validity. We specify the class of trivalent conditionals that support Modus Ponens and are adequate for TT-validity ("Jeffrey conditionals"), and we distinguish, among those, the conditional introduced independently by Cooper and Cantwell (Section 5). We end part I of this paper with a comparison between the two logics that ensue from those considerations, DF/TT (de Finetti-TT) and CC/TT (Cooper-Cantwell TT), with an indication of their commonalities (in particular both are connexive logics, Section 5) and limitations (in particular both retain the Linearity principle of two-valued logic, see Section 6). In part II, we further this comparison with an in-depth investigation of the proof theory and algebraic properties of those two logics.

Philosophical Motivation
Frank P. Ramsey (1926) was likely the first philosopher to connect an assertion of a proposition A with an implicit disposition to bet on A, and to interpret an indicative conditional A → C as a conditional assertion where we suppose the antecedent, and reason on that basis about the consequent. His views strongly influenced Bruno de Finetti, who combined both ideas of Ramsey's by postulating an isomorphism between the conditions that settle the truth of a (conditional) proposition, and the conditions that settle the winner of a (conditional) bet. Evaluating the truth or falsity of a conditional proposition, assertion or event requires supposing the antecedent in the same way that a conditional bet on C given A can only be won or lost if A is true; if A is false, the bet will be called off.
Hence, while the truth value of an ordinary, non-conditional proposition A is settled by either A or ¬A, the truth value of a conditional proposition or assertion-de Finetti uses the notation C/A-is settled by the corresponding pair A ∧ C and A ∧ ¬C (de Finetti 1936, 568, emphasis in original): 3 "C'est ici qu'il paraît indiqué d'introduire une logique spéciale à trois valeurs, comme nous l'avions déjà annoncé : C et A étant deux événements (propositions) quelconques, nous dirons triévénement C/A (C subordonné à A), l'entité logique qui est considérée This approach explains the intuition that upon observing A ∧ C, we feel compelled to say that the (previously made) conditional assertion C/A was right, that it has been verified. 4 Similarly, the conditional assertion C/A is falsified by the observation of A ∧ ¬C: we have been proven wrong by the facts. The indicative conditional A → C shall, in the rest of this paper, be understood as a conditional assertion C/A whose truth conditions correspond to the conditions that determine the result of a conditional bet. We now define a corresponding class of conditional operators: Definition 2.1 (de Finettian operators). A trivalent binary operator is called de Finettian if it agrees with de Finetti's truth conditions when the antecedent is determinate, that is, when the antecedent takes the value 1 or the value 0.
Equivalently, an operator is de Finettian if it agrees on the first and third row of the table in Figure 2. From the class of de Finettian operators, de Finetti selects the truth conditions that assign value 1 /2 to the conditional whenever the antecedent is itself indeterminate. Note that this grouping of indeterminate with false antecedents is not covered by the above epistemological motivation; in fact, this choice is a classical point of contention between trivalent logics of conditionals. De Finetti's choice resembles Bochvar's scheme for trivalent operators (a.k.a. the Weak Kleene scheme), where the value 1 /2 is carried over from any part of a sentence to the whole sentence (Bochvar 1937). Similarly, he assumes that a conditional is undefined as soon as antecedent or consequent are undefined. As we know from the theory of presupposition projection (Beaver and Krahmer 2001), however, Bochvar's choice is not the most adequate to account for the transmission of indeterminate values from smaller to larger constituents, (propositions) whatever, we will speak of the tri-event C/A (C given A), the logical entity which is considered: 3. null if A is false (one does not distinguish between "not A and C" and "not A and not C", the tri-event being only a function of A and A ∧ C)." 4 See also Cantwell 2008, andthe "hindsight problem" in Khoo 2015. and therefore it should not be viewed as mandated by the rest of de Finetti's motivations for the conditional. In fact, de Finetti himself does not handle conjunction and disjunction à la Bochvar/Weak Kleene, but in line with the Strong Kleene scheme (see below).

Main benefits of the approach
De Finetti's trivalent approach has the potential to avoid the paradoxes of material implication and yields a variety of benefits. 5 First of all, it is very simple and has a clear motivation: asserting a conditional amounts to making a conditional assertion; conditionals express dispositions to bet just as ordinary assertions do. The trivalent approach treats conditionals as expressing propositions, in agreement with their linguistic form and assertive usage; only their truth conditions cannot be expressed in bivalent logic. This is a substantial advantage over non-propositional views that have to explain the gap between linguistic form and philosophical theorizing. Second, de Finettian conditionals keep the epistemic notion of assertability and the semantic notion of truth separate, while allowing for a fruitful interaction: degrees of assertability can be defined directly in terms of the truth conditions. For a probability function p on a propositional language, we can define degree of assertability as Ast(X) = p(X is true|X has a classical truth value) (see also McDermott 1996;Cantwell 2006;Rothschild 2014). Trivalent semantics replaces the familiar norm of asserting what is probably true by the equally plausible norm of asserting what is (much) more likely to be true than to be false. This collapses to the classical picture Ast(X) = p(X is true) for bivalent propositions, while for X = A → C, we obtain We obtain Adams' Thesis (sometimes also called "The Equation", and read as a thesis about the probability of A → C), a plausible principle for the assertability of conditionals supported by patterns observed in natural language (Stalnaker 1968;Adams 1975;Dubois and Prade 1994;Evans et al. 2007;Over et al. 2007;Égré and Cozic 2011;Over 2016). 6 Similarly, the suppositional reading of conditionals as expressing conditional degrees of belief (e.g., Ramsey 1926;Edgington 1995) can be naturally grounded in trivalent semantics.
5 In particular, paired with SS-validity, the de Finetti conditional supports neither the entailment from ¬A to (A → C), nor the entailment from C to(A → C). For TT-validity, only the former scheme is blocked.
6 For recent criticisms of Adams' Thesis, see Douven andVerbrugge 2010 andSkovgaard-Olsen, Singmann, andKlauer 2016. The close relationship between truth and assertability allows us to explain intuitions which conflict at first with the trivalent view. For example, a sentence such as: (1) If Paul is in Paris, then Paul is in France.
would typically be judged as true, whereas trivalent semantics regard this as an empirical question: when Paul is in Berlin, the sentence has indeterminate truth value. However, the trivalent view can offer an error theory since (1) is maximally assertable regardless of Paul's whereabouts (p(C|A) = 1). When we call sentences such as (1) "true", what we really mean is that they command consent, that they are "maximally assertable" (see also Adams 1975). Since assertability conditions are fully defined in terms of truth conditions, this defense is arguably not ad hoc. In sum, on this view, indicative conditionals are factual-their truth and falsity is a matter of correspondence with the world-, like for predictions about future events, while their assertability is epistemic and is represented probabilistically. Thirdly, the de Finetti conditional satisfies the following identity: Here, "≡" means that the truth values of A → (B → C) and (A ∧ B) → C coincide according to the de Finetti tables. Import-Export expresses the idea that right-nesting a conditional is just the same as adding a further supposition. Gibbard (1980) proved that there is no truth-conditional operator → that (i) satisfies Import-Export; (ii) validates A → C whenever A classically entails C; (iii) is strictly stronger than the material conditional. In Stalnaker's and Lewis's possible world semantics, Import-Export thus fails. McGee (1989) proposed a modification of Stalnaker's semantics that restores Import-Export and is stronger than the material conditional, giving up (ii). 7 However, it involves syntactic restrictions on the sentences appearing as antecedents. De Finetti's conditional too will fail (ii), but on the other hand it can satisfy Import-Export without any syntactic restriction, and within a truth-functional framework. In that regard it appears prima facie more general and simpler.

Comparing Schemes for Validity
We now introduce and compare the main notions of validity that can be used in relation to de Finetti's conditional. By so doing, we expose a problem for the de Finetti conditional: all of the basic schemes available for validity in trivalent logic appear to overgenerate or to undergenerate relative to general principles of conditional reasoning.

Evaluations and Validity
Throughout the paper, we let L be a propositional language featuring denumerably many propositional variables (indicated as p 0 , p 1 , . . .), whose logical connectives include ¬ and ∧ (the others, ∨ and ⊃, are defined as usual). We call L → the language obtained from L by adding a new conditional connective, in symbols →, to the primitive stock of logical constants of L. We use uppercase Latin letters (A, B, C, . . .) as meta-variables for L-and L → -sentences, and For to denote the set of formulae of the language L → . For all trivalent semantics of the conditional that we consider, negation and conjunction are interpreted via the familiar Strong Kleene truth tables (introduced by Łukasiewicz 1920, also featuring in de Finetti 1936): f ∧ 1 1 /2 0 1 1 1 /2 0 1 /2 1 /2 1 /2 0 0 0 0 0 We can now proceed to define evaluations and consequence relations for the de Finetti conditional.
-A classical evaluation is a function from L-sentences to {1, 0} that interprets ¬ and ∧ by the functors f ¬ and f ∧ restricted to the values 1 and 0.
-A Strong Kleene evaluation (or SK-evaluation) is a function from L-sentences to {1, 1 /2, 0} that interprets ¬ and ∧ by the functors f ¬ and f ∧ .
Given an evaluation, we can distinguish two levels of truth for a sentence, namely T-truth (for tolerant truth) and S-truth (for strict truth), following Cobreros et al. 2012 andCobreros et al. 2015. 8 Identifying the value 1 with the True, the value 1 /2 with the Indeterminate, and the value 0 with the False, then S-truth is for a sentence to be True, whereas T-truth is for a sentence to be non-False. The two notions obviously coincide relative to classical evaluations, but they come apart relative to trivalent evaluations.
Following Chemla, Égré, and Spector (2017) and Chemla and Égré (2018), we single out five notions of validity in a trivalent setting, depending on whether validity is defined as the preservation of truth, non-falsity, or as some combination of those. Those five notions of validity are not the only conceivable ones in trivalent logic, but there is a sense in which they form a natural class. 9 In particular, the five schemata under discussion are all monotonic, and they are all the monotonic trivalent schemata (see Chemla and Égré 2018 for a proof), meaning that an inference remains valid by the inclusion of more premises. We leave open whether a nonmonotonic scheme for validity might offer a good fit for the original de Finetti table. 10 -Γ |= SS A, provided every evaluation that makes all sentences of Γ S-true also makes A S-true.
-Γ |= TT A, provided every evaluation that makes all sentences of Γ T-true also makes A T-true.
-Γ |= (SS∩TT) A, provided every evaluation that makes all sentences of Γ S-true also makes A S-true, and every evaluation that makes all sentences of Γ T-true also makes A T-true.
-Γ |= ST A, provided every evaluation that makes all sentences of Γ S-true also makes A T-true.
-Γ |= TS A, provided every evaluation that makes all sentences of Γ T-true also makes A S-true.
Relative to L and to SK-evaluations, SS-validity determines the so-called Strong Kleene logic, whereas TT-validity determines Priest's logic LP. SS ∩ TT corresponds to the so-called Symmetric Kleene logic, whereas TS and ST correspond to the so-called 9 See Chemla et al. (2017) for general arguments regarding the oddness of SS ∪ TT in particular. In the present case, taking the union of SS and TT would obviously not solve the overgeneration problem raised in the next section, in particular regarding the entailment to the converse conditional. Cooper (1968) restricts TT to bivalent atomic valuations (what Humberstone (2011, §7.19, 1044 and following) calls 'atom-classical' valuations): we set aside that restriction, which makes no essential difference to our discussion here. Farrell (1979) sketches another variant, which we can set aside on the same grounds (see next footnote). 10 Farrell (1979) introduces a notion of sentential validity that may be generalized into a nonmonotonic notion of argument-validity. On his definition, A is valid provided it is TT-valid, and there is a valuation that gives A the value 1. We may generalize this to: Γ |= A provided Γ TT-entails A and there is at least one valuation that gives the formulae in Γ and A the value 1. On that definition, A |= A, but A, ¬A |= A (we are indebted to a remark by T. Ferguson in relation to that fact). We note that like standard TTvalidity, this nonmonotonic restriction still fails Modus Ponens. As such, it would not add a separate route from the one described with standard TT-validity.
Tolerant-Strict and Strict-Tolerant Logics (also called the logics of q-consequence and p-consequence : Malinowski 1990;Cobreros et al. 2012;Frankowski 2004). In general, our definitions of validity are relative to the choice of a type of evaluation function (e.g., classical, SK, DF); however, in the rest of this section, we always refer to DFevaluations, in line with our focus on the de Finetti conditional. A remark on our notation: we will indicate a logic with an acronym of the form 'AB/XY', where 'AB' is the label of the evaluation function, and 'XY' is the acronym of the notion of validity. For example, DF/TT is the logic defined by de Finetti evaluations with tolerant-tolerant validity.
An interesting feature of the DF/TT-logic is that it implies mutual entailment between its conditional and the material conditional. The following inferences are DF/TT-valid: Moreover, we also have: That said, although ⊃ and → are equivalent in DF/TT-logic, they don't obey the same principles. For instance:

A trilemma for de Finetti's conditional
Among the previous schemes, which one is the most adequate relative to de Finetti's conditional? We begin with applying the SS-validity scheme over DF-evaluations, and similarly, mutatis mutandis, for the other schemes. It is easy to see that: That is, the conditional entails conjunction. This property is not intuitive, but perhaps less bad than it seems since the trivalent approach is based on de Finetti's idea of identifying the truth conditions for conditionals with the conditions for winning a conditional bet. Worse is that the de Finetti conditional entails its converse on a SSvalidity scheme: 11 The SS-scheme is thus very distant from an intuitive notion of reasonable inference with conditionals since supposing A and asserting B is very different from supposing B and asserting A. The TT-scheme avoids this problem since McDermott (1996) therefore proposes the SS ∩ TT-scheme to preserve the idea that validity is preservation of the value 1, but to weed out the implication from a conditional to the conjunction and to its converse. The SS ∩ TT consequence relation suffers, however, from the drawbacks of both of its constituents, as evidenced by the following observations: DF/(SS ∩ TT) fails both the Identity Law (A → A) and Modus Ponens: the first because DF/SS has no sentential validities (as is the case in the Strong Kleene logic SK/SS), the second because Modus Ponens is not valid in DF/TT (as is the case for the material conditional in Priest's LP = SK/TT). As a result, the logic DF/(SS ∩ TT) ends up being very weak. Consider now the so-called "mixed consequence" schemes, namely TS and ST, in which the level of truth varies from premises to conclusion (Cobreros et al. 2012). DF/TS squares well with the degrees of assertability defined in Section 2 since Ast(A) ≤ Ast(B) for all underlying probability functions if and only if either A and B are logically equivalent, or A |= TS B (Cantwell 2006, 166). Hence, the logic connects well to epistemology, and it also eschews the conjunction-and converse-conditional fallacies. Unfortunately, Modus Ponens and the Identity Law fail (like other sentential validities), not to mention other oddities of the logic, in which A |= TS A. In DF/ST, on the other hand, Modus Ponens and the Identity Law are retained, but also the entailment of the conditional to conjunction and to its converse remain.
We may summarize these observations in the form of a trilemma: Fact 3.4. Irrespective of whether SS, TT, ST, TS, SS ∩ TT is chosen for validity, a logic on (L → , f → DF ) must either (1) fail Modus Ponens; or (2) fail the Identity Law (and other sentential validities); or (3) validate the inference from a conditional to its converse.
The trilemma at a glance: The interest of this trilemma is that it involves schemata that depend on no other connective than the conditional. In what follows, we explore two main ways out of the trilemma: both select TT validity as comparatively the best choice for validity, but the second moreover involves a modification of the de Finetti table so as to restore Modus Ponens.

Giving up Modus Ponens: DF/TT
Given that no validity scheme satisfies the three desiderata of making the DF conditional validate Modus Ponens, avoid the entailment to its converse, and validity the Identity Law, one way out of the trilemma is to follow Quine 1970's maxim of "minimum mutilation", and to elect as optimal the scheme or schemes that violate the fewer of those constraints. 12 Three of the schemes violate two constraints, but DF/TT and DF/ST violate only one. However, DF/ST badly overgenerates (by validating the entailment to the converse), whereas DF/TT mildly undergenerates (by failing Modus Ponens, but still satisfying Conditional Introduction, see below). Arguably therefore, DF/TT appears to be the less inadequate of all options: it retains the Identity Law and avoids the entailment to the converse conditional, only at the expense of losing Modus Ponensa principle that is given up in other logics such as Priest's LP (i.e., SK/TT) for the material conditional. 13 Two more facts are worth highlighting about DF/TT. Firstly, despite the failure of Modus Ponens, the conditional supports Conditional Introduction, namely Γ, A |= B implies Γ |= A → B. In DF/SS, the situation is reversed, since Conditional Introduction fails despite Modus Ponens holding. Secondly, DF/TT supports full commutation of the conditional with negation, a schema widely regarded as plausible in natural language (see Cooper 1968;Cantwell 2008, and Section 4.1 below).
12 As in Optimality Theory (see Prince and Smolensky 2008), we also assume that constraints can be rank-ordered in terms of how their comparative importance. We don't state the ordering explicitly here, the discussion makes it clear enough.
13 Note that unlike McGee's logic (McGee 1989), which fails Modus Ponens for complex conditionals, DF/TT can fail Modus Ponens for simple conditionals, composed of atomic sentences.
Despite blocking the entailment to the converse conditional, DF/TT validates several sentential schemata that are intuitively controversial. Farrell (1979) for example points out that it validates the problematic schema (B ∧ (A → B)) → A, a sentential version of the fallacy of affirming the antecedent. More generally, we have: Given the conditions the de Finetti conditional puts on TT-validity, however, this schema does not necessarily constitute an unwelcome prediction. Firstly, it does not hold in argument form (that is, A → B |= TT A), consistently with the fact that TTvalidity does not satisfy Modus Ponens. Secondly, consider the left-nested conditional sentence: (2) If Peter visits if Mary visits, then Mary will visit [indeed].
This seems intuitively acceptable, in line with the suppositional reading of the conditional. The upshot is that DF/TT loses some classical inferences based on the conditional (like Modus Ponens), and introduces some conditional sentences as validities that are not classical (viz. Fact 4.2), though not necessarily problematic under a suppositional reading.
If, on the other hand, we wish to retain Modus Ponens as a central property of the conditional along with the Identity Law, then the trilemma presented in Fact 3.4 implies that either some further notion of validity must be sought for the de Finetti conditional, or the de Finetti conditional itself is not adequate. However, we have already argued that the notion of validities considered in this section exhaust the most natural and well-motivated class of monotonic notions of consequence defined over trivalent evaluations. For this reason, in the next section we explore that second option and explore alternatives to the de Finetti conditional.

Jeffrey conditionals
In a short and underappreciated note, Jeffrey (1963) highlighted the following condition for a trivalent operator to satisfy Modus Ponens when TT is used for validity: We may therefore call a conditional operator Jeffrey if it extends the bivalent "gappy" conditional as follows (Jeffrey 1963): Definition 5.2. A Jeffrey conditional is any binary trivalent operator of the form: An operator can therefore satisfy Jeffrey's constraint and be de Finettian at the same time, namely comply with the truth conditions of de Finetti's conditional when the antecedent has a classical truth value (see Definition 2.1). We thus say that: Fact 5.3. A Jeffrey conditional is de Finettian provided it is of the form: Clearly, there exist four de Finettian Jeffrey conditionals (see Figure 5). Two of them are the Cooper-Cantwell (CC) and the Farrell conditional (F). We call the other two J1 and J2. For each such table, we modify the notion of DF-evaluation accordingly (call it a CC-, F-, J1-, and J2-evaluation respectively). It is straightforward to see that Jeffrey conditionals (whether de Finettian or not) eschew the trilemma faced by de Finetti's: -invalidates the entailment of the conditional to its converse. Proof.
-Identity: All values on the diagonal of any Jeffrey conditional differ from 0.
-Avoiding the entailment to the converse: Like de Finetti's conditional, all Jeffrey conditionals TT-validate Conditional Introduction, but unlike the de Finetti conditional they satisfy the converse, namely the full Deduction Theorem. In fact, there is a precise sense in which TT-validity and Jeffrey conditionals fit each other: 15 Proof.
Deduction Theorem for TT-validity: This result is important since our consequence relation is meant to capture a suitable logic of suppositional reasoning, in line with de Finetti's original motivation. Just as the truth table for the trivalent conditional is motivated by the idea of evaluating the consequent under the supposition of the antecedent, the consequence relation should describe the inferences that are licensed by supposing the antecedent. Therefore, a deduction theorem is an important adequacy condition for a logic of trivalent conditionals, making a strong case for TT-validity in combination with Jeffrey conditionals. Relatedly, it can be seen that no Jeffrey conditional supports (A → B) → A as a valid schema relative to TT-validity (to see this, let v(A) = 0, v(B) = 1 /2), unlike de Finetti's conditional (see Fact 4.2 and compare Farrell 1979, whose motivation for → F lies precisely here).

Negation and CC/TT
To choose between the various Jeffrey conditionals, we suggest to look at the interplay of the conditional with the other logical connectives. The interplay between conditional and negation is especially relevant, since several of the most debated principles involving indicative conditionals concern negation as well. One common fact about Jeffrey conditionals is that they fail contraposition relative to Strong Kleene negation: Proposition 5.6. For any Jeffrey conditional, A → B |= TT ¬B → ¬A.
The failure of Contraposition may be seen as a welcome prediction. First of all, supposing A and supposing ¬B are just two different things. For example, when v(A) = v(B) = 1, then A → B is obviously true, whereas ¬B → ¬A is now "void"-the conditions for evaluating its truth or falsity are not satisfied. Therefore v(¬B → ¬A) = 1 /2. Second, contraposition does not always preserve meaning. The contrapositive of a sentence like "if Sappho did not die in 570 BC, then she is dead by now" would be "if Sappho is not dead by now, then she died in 570 BC". The latter obviously conveys a different thought. Hence the inference to the contrapositive is not warranted in all situations. 16 On the other hand, as noted by Cooper (1968) and Cantwell (2008), the Cooper-Cantwell conditional supports the full commutation of Strong Kleene negation with the conditional, namely the logical equivalence between ¬(A → B) and (A → ¬B) . In fact, it is the only Jeffrey conditional that does so: Proposition 5.7. Among all Jeffrey conditionals, only the Cooper-Cantwell conditional validates the full commutation schema for negation. For de Finettian Jeffrey conditionals, in particular, SK-negation is a separating connective: The truth table of a Jeffrey conditional is given by This implies that the truth tables for ¬(A → B) and A → ¬B look like this: ¬(A → B) 1 1 /2 0 1 0 ¬d 1 1 1 /2 ¬d 2 ¬d 3 1 0 1 /2 ¬d 4 1 /2 For TT-entailment to go in both directions, necessarily, ¬d 2 = 0, hence d 2 = 1, and d 1 , d 3 , d 4 must all equal 1 /2, which yields the table for the Cooper-Cantwell conditional. For the other de Finettian Jeffrey cases: let v be an F-evaluation, or a J1-evaluation: In classical logic, only the commutation from outer to inner negation is valid. On the other hand, inferences in natural language appear to support both directions in many contexts. Ramsey (1929), Adams (1965), Cooper (1968), Cantwell (2008) and Francez (2016) give a theoretically motivated defense of the commutation scheme, while the studies by Handley, Evans, and Thompson (2006) and Politzer (2009) provide some empirical support. See, however, Égré and Politzer (2013), Olivier (2018) and Skovgaard-Olsen, Collins, Krzyżanowska, Hahn, and Klauer (2019) for a more complex picture.

Connexivity
We conclude this section by briefly relating our discussion of the TT-logics of de Finettian and Jeffrey conditionals to a slightly wider logical context. A conditional logic is called connexive if it validates the two following schemata: On the other hand, systems of connexive logic lack some classical principle, lest they are trivial (of course, DF/TT and CC/TT are no exception). Informally construed, Aristotle's Thesis requires that it is never the case that a formula is implied by its own negation, while Boethius' Thesis requires that if a conditional A → C holds, then it is not the case that the conditional that results from the former by negating the consequent, i.e. A → ¬C (which is equivalent to the negated conditional ¬(A → C) in both DF/TT and CC/TT) hold. Now, since both DF/TT and CC/TT employ a tolerant-tolerant notion of validity, the fact that they satisfy Boethius' Thesis can hardly be interpreted as saying that they show that a conditional is 'incompatible' with its negation (and similarly for Aristotle's Thesis). Nevertheless, in requiring such a strict, extra-classical connection between antecedent and consequent of a conditional, connexive logics-including DF/TT and CC/TT-arguably ensure that the conditional interacts reasonably well with negation. Nevertheless, the interaction of conditional and negation displayed by connexive logics of De Finettian and Jeffrey conditionals, DF/TT and CC/TT in particular, is not entirely free from worries. For one thing, connexivity comes at a price when it comes to reductio proofs (see Cooper 1968 for discussion). For another, like de Finetti's conditional, the Cooper-Cantwell conditional also validates the following equivalence, where ≡ m is the material biconditional: As a consequence, both conditionals validate Conditional Excluded Middle is a moot principle, but it is a natural one to have if negation is to commute with the conditional. 19 Moreover, since every de Finettian Jeffrey conditional validates Conditional Excluded Middle, this does not tell against the Cooper-Cantwell variant. Thanks to the fact that it is the only one, within the de Finettian Jeffrey conditionals, to support the full commutation with negation, the Cooper-Cantwell conditional stands out as the closest to de Finetti's original connective.

Comparisons and Limits
We have distinguished two trivalents logics of indicative conditionals, namely DF/TT and CC/TT, whose proof theory and algebraic semantics we will explore in Part II of this paper. Before doing so, let us summarize the commonalities between the two logics, their principal differences, and draw comparisons with other logics of conditionals.
Four main features are common to DF/TT and CC/TT: they are truth-functional logics, they share the same de Finettian semantic core, they are connexive, and both support the law of Import-Export without restriction. The main difference between DF/TT and CC/TT is that the former fails Modus Ponens, whereas the latter preserves it, so that only CC/TT supports the full Deduction Theorem. This property is in line with the fact that for TT-validity, the designated values are 1 and 1 /2, and the Cooper-Cantwell conditional is only evaluated as false when the antecedent is designated and the consequent undesignated. Conversely, relative to Strong Kleene negation the Cooper-Cantwell conditional fails Contraposition, whereas de Finetti's conditional supports Contraposition.
The preservation of Modus Ponens may be seen as virtue of CC/TT compared to DF/TT. However, one common fact about both logics, given our assumption that they share the same Strong Kleene disjunction, is that they fail the rule of Disjunctive Syllogism (¬A, A ∨ B |= B). Clearly, this concerns the table for disjunction for a TTconsequence relation (see Priest 1979;Cantwell 2008), independently of the particular truth conditions for the conditional, and Cooper (1968) actually selects a different table for disjunction (and conjunction). On the other hand, the choice of conjunction and disjunction needs care, if their interdefinability via negation is to hold, and if the Law of Import-Export is to hold.
Because the Law of Import-Export is validated, in both CC/TT and DF/TT only one of the paradoxes of material implication is blocked, namely the schema A → (¬A → B). On the other hand, A → (B → A) holds in both logics, consistent with the fact that A ∧ B → A is valid. As discussed in Section 4, this property squares well with the proposed suppositional interpretation of the conditional. On the other hand, both CC/TT and DF/TT validate the so-called Linearity principle (A → B) ∨ (B → A). This schema was famously criticized by MacColl (1908), who pointed out that neither of "if John is red-haired, then John is a doctor" and "if John is a doctor, then he is red-haired" seems acceptable in ordinary reasoning. 20 Given the way conjunction and disjunction are handled in DF/TT and CC/TT, we can therefore conclude that whereas both logics are connexive, neither is relevantist, except in a weak sense (by failing one of the paradoxes of material implication).
Relatedly, there is a certain tension between our extensional semantics of conditionals and the intensional use to which they are often put. Suppose Mary believes the following conditional: If the Church is East of the City Hall, then the City Hall is West of the Church Intuitively the proposition that Mary believes appears analytically true. Nonetheless, on the de Finettian analysis its truth value depends on the position of the City Hall with respect to the Church: the conditional may be evaluated either as true or as indeterminate. The apparent analyticity of (3) has to be explained by reference to it being maximally assertable, regardless of its actual truth value. In fact, also Lewis (1986, 315) observes that "there is a discrepancy between truth-and assertability-preserving inference involving indicative conditionals; and that our intuitions about valid reasoning with conditionals are apt to concern the latter, and so to be poor evidence about the former." In other words, while DF/TT and CC/TT aim at describing a logic of suppositional reasoning and their analysis of (3) should be evaluated by these criteria, reasonable inferences with conditionals, including "apparent analytic truths", may need to be analyzed in terms of a (probabilistic) theory of assertability. This theory can again be anchored in, and motivated by, trivalent truth conditions for conditionalssee Section 2. Detailing the division of labor between semantics (truth conditions, validity) and epistemology (degrees of assertability) is, however, a project for future work. All these features of a trivalent logic of conditionals are important to bear in mind. Importantly, some of them depend more on the treatment of other connectives than on the particular logic we advocate, but others may indicate fundamental limits of a truth-functional approach.

Summary and Perspectives
De Finetti's trivalent conditional was put forward by de Finetti to qualitatively model the way in which conditional statements are probabilistically represented. Since its discovery, the DF table has received a fair amount of attention from mathematicians as well as psychologists, but there have been surprisingly few investigations of the trivalent logics supported by the conditional as well as the variants in its vicinity. Our main motivation for this paper has been to fill this gap.
We started with the observation that de Finetti's truth table faces a trilemma when confronted with the choice of a trivalent validity relation: give up the Identity Law and other sentential validities, support the entailment from a conditional to its converse, or give up Modus Ponens. We have argued that the latter option is the less costly in relation to its alternatives, if the DF conditional is paired with a notion of TT-validity. On the other hand, Trivalent Jeffrey conditionals, which have the property f → ( 1 /2, 0) = 0, avoid this trilemma when endowed with the same TT-consequence relation: they block the entailment to the converse conditionals, they support the Identity Law, and moreover they support the full Deduction Theorem (Modus Ponens and Conditional Introduction), in line with the fact that the values 1 and 1 /2 are designated for consequence, and pattern in the same way for those conditionals.
Zooming in on Jeffrey conditionals, we see that the Cooper-Cantwell conditional stands out in that it satisfies the full commutation schema for negation, a schema widely regarded as plausible in natural language, also supported by the de Finetti conditional. Prima facie therefore, the Cooper-Cantwell conditional appears to strike the best balance between logical and epistemological properties: like Farrell's conditional, but unlike de Finetti's, it satisfies Modus Ponens. Its motivation for the middle line of its truth table-to treat an indeterminate antecedent like a true one-is more stringent than Farrell's, and well-aligned with the TT-consequence relation.
As pointed out in the previous section, both CC/TT and DF/TT share features which may be seen as problematic, and which are given up in intensional logics of conditionals (based on possible worlds). One immediate example is the Linearity principle, while connexivity can also be considered problematic (although judgements diverge significantly on the plausibility of connexive principles for indicative conditionals). From a methodological point of view, however, we think it matters to any further work on conditionals to locate exactly the (actual and alleged) limits of the trivalent approach, in particular because they should be carefully compared to some of the benefits we highlighted. In Part II of this paper, we therefore propose a more elaborate treatment of the proof theory and algebraic semantics of both CC/TTand DF/TT, in order to give a more informed assessment of both logics.

De Finettian Logics of Indicative Conditionals
Paul Égré, Lorenzo Rossi, and Jan Sprenger Part II: Proof Theory and Algebraic Semantics

Abstract
In Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti (1936) and Reichenbach (1944) on the one hand, and by Cooper (1968) and Cantwell (2008) on the other. Here we provide the proof theory for the resulting logics DF/TT and CC/TT, using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: DF/TT allows for algebraic completeness, but not for the construction of a canonical model, while CC/TT fails the intersubstitutivity of equivalents and the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects.
In Part I of this paper, we have reviewed the motivations for a trivalent semantic treatment of indicative conditionals, centered on the proposal made by de Finetti (1936), and Reichenbach (1944), to treat indicative conditionals as conditional assertions akin to conditional bets. We have singled out two de Finettian logics of the indicative conditional, the first based on de Finetti's table, paired with a notion of logical consequence as preservation of non-Falsity (TT-validity), the other based on a close kin of De Finetti's table, the Cooper-Cantwell table, paired with the same notion of validity. These logics are called DF/TT and CC/TT, respectively. We repeat the truth tables of the conditional operator in Figure 6 and the definition of TT-validity below. In both logics, the other connectives '¬', '∧' and '∨' are interpreted via the Strong Kleene truth tables.
We have seen that both logics share some common features, in particular both satisfy Conditional Introduction and the law of Import-Export, but they differ foremost on the law of Modus Ponens, which is preserved in CC/TT but given up in DF/TT. In this second part of our inquiry, we turn to an investigation of the proof theory of these logics. We proceed in three main steps: in Section 1, we give sound and complete tableaux calculi for either logic; in Section 2, we give sound and complete sequent calculi for either logic; in Section 3, finally, we examine the prospect for an algebraic semantics for both DF/TT and CC/TT. As we shall see, neither logic admits a 'nice' algebraic semantics, but there is a sense in which CC/TT, despite satisfying Modus Ponens, falls even shorter than DF/TT in that regard. We give a discussion of that result and further the comparison between both logics in Section 4 and in the Appendix A.

Tableau Calculi
In this section, we introduce sound and complete tableau calculi for CC/TT and DF/TT. Tableau calculi are a proof-theoretical formalism that is very close to the semantics. To prove a sentence, tableaux employ trees that can be conceptualized as reverse truth tables. In building a tableau, one starts from the assumption that certain sentences A 0 , . . . , A n have certain semantic values, and iteratively works out all the value assignments to the sub-sentences of A 0 , . . . , A n that result from the initial assignment. In the propositional case, this process always terminates after a finite number of steps, resulting in either an open or a closed tableau: in the former case, the initial assignment is possible according to the chosen semantics, whereas in the latter it is not. Therefore, in order to prove that A follows from a (finite) set of sentences Γ in a tableaux system, one shows that all the tableaux resulting from the initial assignments in which all the sentences in Γ have a designated value but A does not are closed.

Tableau calculus for CC/TT
The CC/TT tableau calculus, in symbols CC/TTt, is given by the following tableau construction rules: We now give a precise characterization of the tableaux generated according to the above rules, and of CC/TTt-derivability.
-For every formula A, the CC/TTt-n-tableau of A (for n = 0, 1 /2, or 1) is the tree whose root is A : n, and that is obtained by applying the rules of CC/TTt.
-For every finite set of formulae Γ = {B 0 , . . . , B k }, the CC/TTt-n 0 ; . . . ; n k -tableau of A (for n i = 0, 1 /2, or 1, and i ∈ {0, . . . , k}) is the tree whose root is B 0 : n 0 ; . . . ; B k : n k , obtained by applying the rules of CC/TTt. 1 Since we are only concerned with the tableau calculus for CC/TT in this subsection, we suppress the label 'CC/TTt' whenever possible, to improve readability. Before proving soundness and completeness for CC/TTt, we give a sample of how to reason in this calculus. In particular, we prove the commutation with negation in CC/TTt. The following two closed tableaux establish that A → ¬B follows from ¬(A → B) in CC/TTt. The first tableau shows that ¬(A → B) cannot have value 1 while A → ¬B has value 0. We now prove that CC/TTt is sound and with respect to CC/TT-validity.
Definition 1.4. A quasi-CC-evaluation is a non-total function from the formulae of L → to {0, 1 /2, 1} that is compatible with the CC truth tables.
More compactly, a quasi-CC-evaluation is a proper subset of a CC-evaluation. For example, the function that sends p and (p ∧ q) to 1 is a quasi-CC-evaluation.
Lemma 1.5. For every finite set {B 1 , . . . , B k } of formulae and every CC-evaluation v, the completed CC/TTt-tableau whose root is is open, and all partial functions from sentence to {1, 1 /2, 0} induced by its open branches are quasi-CC-evaluations.
Proof. By induction on the height of the tree.
-The tableau consisting only of the root B 1 : v(B 1 ) ; . . . ; For suppose it is closed. Then, there are at least two sentences B i and B j s.t.
, against the hypothesis that v is a CC-evaluation: no CC-evaluation assigns two different values to the same sentence, because CCevaluations are functions.
-Assume by the inductive hypothesis (IH) that the (incomplete) tableau T n whose root is B 1 : v(B 1 ) ; . . . ; B k : v(B k ) and that has height n is open, and that its open branches induce quasi-CC-evaluations. Suppose also (in contradiction with the lemma to be shown) that the tableau T n+1 of height n + 1 resulting by applying one tableau rule to the terminal nodes of T n is closed. We reason by cases, according to the last rule applied to the nodes in a branch of T n (we only do two cases): (∧) Suppose a conjunction rule is applied to a node v occurring in an open branch B n of height n in T n , and all the branches of height n + 1 resulting from this application are closed. There are three possibilities: v has in its label A ∧ B : 1, or A ∧ B : 0, or A ∧ B : 1 /2.
-If v has A ∧ B : 1 in its label, then there is exactly one successor node v 1 in the resulting branch B n+1 of height n, and v 1 has A : 1 ; B : 1 in its label. If B n+1 is closed as a result of the addition of v 1 , this means that there is at least one node w, a predecessor of v, such that: w has A : 0 in its label, or w has A : 1 /2 in its label, or w has B : 0 in its label, or w has B : 1 /2 in its label Since we assumed that B n+1 is closed, B n has a node (namely v) that has A ∧ B : 1 in its label, and a node (namely w) whose label is as in one of the cases just listed. By IH, B n induces a quasi-CC-evaluation. But no quasi-CC-evaluation assigns value 1 to a conjunction and a value different from 1 to both conjuncts. Contradiction. -If v has A ∧ B : 0 or A ∧ B : 1 /2 in its label, the reasoning is exactly analogous to the previous case.
(→) Suppose a conditional rule is applied to a node v occurring in an open branch B n of height n in T n , and all the branches of height n + 1 resulting from this application are closed. There are three possibilities: v has in its label A → B : 1, or A → B : 0, or A → B : 1 /2.
-If v has A → B : 1 /2 in its label, then there are exactly two branches B 1 n+1 and B 2 n+1 of height n + 1 extending B n with three successor nodes of v, call them v 1 and v 2 such that: v 1 has A : 0 in its label v 2 has B : 1 /2 in its label Since we assumed that B 1 n+1 and B 2 n+1 are both closed, then B n has two nodes w 1 and w 2 , predecessors of v, such that: w 1 has A : 1 or A : 1 /2 in its label w 2 has B : 0 or B : 1 in its label By IH, B n induces a quasi-CC-evaluation. But no quasi-CC-evaluation assigns value 1 /2 to a conditional while assigning any of the following pairs of values to its antecedent and consequent respectively: 1, 0 , 1, 1 , 1 /2, 0 , and 1 /2, 1 . Contradiction. -If v 0 has A → B : 1 or A → B : 0 in its label, the reasoning is exactly analogous to the previous case.
Proposition 1.6 (Soundness). For every finite set Γ of formulae and every formula A: if Γ ⊢ CC/TTt A, then Γ |= CC/TT A Proof. We prove the contrapositive. Suppose that Γ |= A, for Γ = {B 1 , . . . , B k }. Then there is at least one CC-evaluation v such that v(B 1 ) ∈ {1, 1 /2}, . . . , v(B k ) ∈ {1, 1 /2} but v(A) = 0. Then, by Lemma 1.5, the tree whose root is labeled as is open. Therefore, not all the trees whose root is labeled as We finally show that CC/TTt is complete and with respect to CC/TT-validity (for inferences with finite sets of premises).

Lemma 1.7. Every open branch of a completed CC/TTt-tableau induces a quasi-CCevaluation.
Proof (Sketch). Let T be a completed CC/TTt-tableau with B an open branch. The branch is finite and it has a unique terminal node v of the form p i : k. Consider now the partial function that only sends p i to k (i.e., that is constituted by the single pair p i , k ). This is clearly a quasi-CC-evaluation. Call this function v B 0 . Then construct a new function v B 1 that simply adds to v B 0 every pair A, k , where A : k is in the label of the predecessor of v in B. Proceed in this fashion until the root of T is reached. It is easy to show that the resulting function v B n (for n + 1 the length of B) is a quasi-CCevaluation. 2 Proposition 1.8 (Completeness). For every finite set Γ of formulae and every formula A: if Γ |= CC/TT A, then Γ ⊢ CC/TTt A Proof. We prove the contrapositive. Assume Γ CC/TTt A. By definition this means that not all the CC/TTt-tableaux whose root is labeled as Then v B can be extended to at least one CC-evaluation using Zorn's Lemma. Call one such evaluation v. v and v B agree on Γ and A, and therefore v(B 1 ) ∈ {1, 1 /2}, . . . , v(B k ) ∈ {1, 1 /2}, and v(A) = 0.
But this means that Γ |= CC/TT A.

Tableau calculus for DF/TT
The tableau calculus for DF/TT, in symbols DF/TTt, is given by the rules of CC/TTt, with the conditional rule replaced by the following one: The proof is entirely similar to the proof of Propositions 1.6 and 1.8. As mentioned at the beginning of §1, tableau calculi are very close to truth table semantics. They are also quite informative: their construction determines all the possible truth value assignments that follow from the hypothesis that a given inference is valid. However, tableau calculi are not a particularly convenient formalism to work with. In particular, since tableau calculi are refutation calculi, in order to show that A follows from Γ in a tableau system, one has to show that the hypothesis that Γ holds while A doesn't cannot be maintained. In a classical setting, this amounts to showing that it is not the case that all the sentences in Γ can be assigned value 1 while A is assigned value 0 by the corresponding tableau. However, in CC/TT and DF/TT we have three values, two of which are designated, so this is not enough: we have to exclude that all the sentences in Γ can be assigned a designated value, that is either 1 or 1 /2, while A is assigned value 0. And this requires to consider all the possible combinations of assignments of values 1 and 1 /2 to sentences in Γ (keeping the assignment of value 0 to A fixed). Of course, as soon as Γ contains more than 1 sentence, showing that A follows from Γ requires more than one tableau-more precisely, it requires 2 k tableaux, for k the cardinality of Γ.
In addition, tableau calculi are given for inferences with finite sets of premises. They can be generalized to the case of infinite sets of premises, but this results in an infinitary formalism, namely a formalism in which one either constructs infinitely many tableaux, or infinitary tableaux, that is, well-founded trees of transfinite ordinal lengths. 3 For these reasons, we now present another formalisms to capture CC/TT-and DF/TT-validity: many-sided sequent calculi, in particular three-sided sequent calculi. Three-sided sequent calculi are a generalization of standard sequent calculi: instead of building derivation trees labeled with sequents, the rules of the calculus generate derivation trees labeled with triples of sets of sentences, called three-sided sequents. Unlike tableaux, sequent calculi are not refutation calculi, and therefore any derivation of A from Γ establishes that A is provable from Γ. In addition, sequent calculi handle arbitrary sets of premises, including infinite ones. They can also handle (possibly infinite) sets of conclusions, and therefore generalize CC/TT-and DF/TT-validity to multiple conclusions. All these advantages have little costs for the intuitiveness of the calculus. Even though one cannot represent in a sequent calculus all the possible outcomes of assigning a given value to a set of sentences, the sequent rules that we are going to use are very close to the tableau rules, and mirror closely the evaluations of their target sentences according to the CC and DF truth tables.

Three-sided sequent calculi
In this section, we introduce sound and complete three-sided sequent calculi for CC/TT and DF/TT. Since both CC/TT and DF/TT are super-logics of LP (they extend the latter with a new conditional), we can obtain a sequent calculus by extending an existing calculus for LP, in particular the three-sided sequent axiomatization of LP provided by Ripley 2012. A three-sided sequent, or a sequent for short, is an object of the form Γ | ∆ | Σ where Γ, ∆, and Σ are sets of formulae. As above, we focus on the calculus for CC/TT, and then indicate how to adapt it to the case of DF/TT.

Three-sided sequent calculus for CC/TT
Let CC/TTm be the calculus given by the following principles: Axiom: Rules: A derivation of a sequent Γ | ∆ | Σ in CC/TTm is a tree labeled with sequents, whose leaves are axioms of CC/TTm and whose remaining nodes are obtained from their predecessors by applying the CC/TTm-rules. Let Γ ⊢ CC/TTm ∆ be a shorthand for 'there is a derivation of Γ | ∆ | ∆ in CC/TTm'.

Definition 2.1 (Satisfaction and Validity).
The following lemma, adapted from Ripley 2012, is immediate from the definition of satisfaction and validity.
Lemma 2.2. For every sets of formulae Γ and ∆: Before establishing soundness and completeness for CC/TTm, we provide an example of how one can reason with this calculus. More precisely, we show the equivalence of A → ¬B and ¬(A → B) within it. By the above lemma, this amounts to deriving the sequents ¬ In the following examples, we use the empty set symbol ∅ only in order to make the derivations more readable. The following derivation establishes the first sequent: The following derivation establishes the second sequent: We now proceed to establish soundness and completeness for CC/TTm. Proof. By induction on the length of the derivation of Γ | ∆ | ∆.
To prove completeness, we prove the following more general result.
Proposition 2.4. For every triple of sets of formulae Γ, ∆, and Σ, exactly one of the two following cases is given: Proof. We employ the method of Schütte's search trees, adapted to CC/TTm. 4 For every sequent Γ | ∆ | Σ, such method provides the means to construct a tree labeled with sequents which either constitutes a derivation of Γ | ∆ | Σ in CC/TTm or can be used to extract a countermodel to Γ | ∆ | Σ.
We begin by defining three inductive jumps, that extend a given directed tree labeled with sequents by applying all the rules of CC/TTm. Formally, such a tree is constituted by a pair N, S , where N is the set of nodes and S is the set of edges, together with a labeling function, that is, a function from N to their labels (that is, sequents). To simplify our presentation, we identify nodes with their labels, and pairs of nodes with pairs of labels. For every labeled directed tree N, S , define the following sets by positive elementary induction: The jumps + , † , and ‡ correspond to the operations of extending a given labeled tree to another labeled tree, where the sequents that are added result from applying the rules of CC/TTm 'upside down', that is, going from a sequent to all its possible premises according to the CC/TTm rules. Now we construct a search tree for a every sequent, that is, a labeled tree where the above jumps are systematically applied as many times as possible. For every sequent Γ | ∆ | Σ define (for a limit ordinal δ): Finally, define (where Ord is the class of all ordinals): 5 We now have to show that no formula is in Γ ∞ ∩ ∆ ∞ ∩ Σ ∞ . Suppose that there is a formula A and there are sequents Γ 0 | ∆ 0 | Σ 0 , Γ 1 | ∆ 1 | Σ 1 , and Γ 2 | ∆ 2 | Σ 2 such that A ∈ Γ 0 ∩ ∆ 1 ∩ Σ 2 . We reason by cases: Γ 2 | ∆ 2 | Σ 2 all belong to the same open branch B, then they occur at different heights within B. Suppose without loss of generality that Γ 0 | ∆ 0 | Σ 0 occurs at height n (counting upwards the nodes appearing in B starting from the lowest node, labeled with Γ | ∆ | Σ), that Γ 1 | ∆ 1 | Σ 1 occurs at height n + j, and that Γ 2 | ∆ 2 | Σ 2 occurs at height n + j + k (considering different orders would not make a difference). Since p i ∈ Γ 0 and all the rules of CC/TTm are contextsharing, 6 p i is 'carried upwards' during the construction of successive stages of B. Therefore, at height n + j we have that p i ∈ Γ 1 and p i ∈ ∆ 1 , and at height n + j + k we have that p i ∈ Γ 2 , p i ∈ ∆ 2 , and p i ∈ Σ 2 . But this means that Γ 2 | ∆ 2 | Σ 2 is an axiom of CC/TTm, and that B is closed. Contradiction.
-Suppose A is a complex formula of complexity n + 1, and assume the claim as IH for formulae of complexity up to n. Suppose A is B → C, and that Γ 0 | ∆ 0 | Σ 0 occurs at height n, that Γ 1 | ∆ 1 | Σ 1 occurs at height n + j, and that Γ 2 | ∆ 2 | Σ 2 occurs at height n + j + k. Then:

and one of them is in B.
Therefore one of the following is the case: But all of (i)-(iii) contradict our IH. The cases of the other connectives are similar.
A few observations on the functions w α B are in order. First of all, the definition of w 0 B is in part arbitrary, as other choices of truth value assignments to propositional variables would have been possible. In order to get a countermodel, one just needs a function that (i) assigns to the propositional variables in Γ ∞ , ∆ ∞ , and Σ ∞ a value that is incompatible with the corresponding position of such variables in the union sequent (and clearly there is more than one choice here) and that (ii) is a quasi-C-evaluation.
Notice moreover that the construction of every w α B is by simultaneous induction, but every w α B is inductive in Γ ∞ , ∆ ∞ , and Σ ∞ , since these sets occur also negatively in the definition of w 0 B . 8 This seems unavoidable: there seems to be no definition of 'having value 1 if not in Γ ∞ , 1 /2 if not in ∆ ∞ , and 0 if not in Σ ∞ ' that yields a function and that is positive in Γ ∞ , ∆ ∞ , and Σ ∞ . However, this causes no problem as far as the existence and uniqueness of w B is concerned, since the existence and uniqueness of Γ ∞ , ∆ ∞ , and Σ ∞ is immediate by their definition.
Finally, notice that we gave a simplified inductive construction for w B . More specifically, we define w B directly as a function rather than as a positive elementary set of pairs of sentences and values (then one would have had to show shown that such set is, indeed, a function). Giving a proper positive elementary definition of w B would make it clearer that its construction is by simultaneous induction, but would be significantly less readable.
A completeness theorem for CC/TTm is now immediate from Proposition 2.4.

Three-sided sequent calculus for DF/TT
The three-sided sequent calculus for DF/TT, in symbols DF/TTm, is given by the rules of CC/TTm, with the conditional rules replaced by the following ones: The notions of DF/TTm-derivability, as well as of satisfaction and validity of a threesided sequent are immediate from the corresponding definitions for CC/TTm (Definition 2.1).
Proposition 2.6 (Soundness and completeness). For every set Γ of formulae and every formula A: The proof is entirely similar to the proof of soundness and completeness for CC/TTm.

Algebraic semantics
In this section, we explore the algebraic structures that correspond to DF/TT and CC/TT, and investigate the prospects for an algebraic semantics of these two logics. We begin by recalling some structures, and introducing the algebraic counterparts of DF/TT. We start with DF/TT because, as will be clear in Subsection 3.3, it is algebraically significantly more tractable than CC/TT. We use overlined uppercase Latin letters (A, B, C, . . .) to range over sets (supports of algebraic structures) in order to avoid possible confusions with meta-variables for L → -formulae, and boldface characters to indicate designated elements of the supports of algebraic structures (1, 1, 1 /2, . . .), in order to avoid possible confusions with truth values in truth table semantics.

Definitions
Definition 3.1. A structure A = A, ⊓, ⊔, 0, 1 , where A is a set and 0, 1 ∈ A, is a distributive bounded lattice if for every a, b, c ∈ A: -The lattice conditions are satisfied: -The lattice is bounded: The lattice is distributive: A relatively pseudocompletemented Kleene algebra A = A, ⊓, ⊔, −, , 0, 1 is an Ł3 algebra if for every a ∈ A: There is a distinguished element 1 /2 ∈ A s.t. − 1 /2 = 1 /2, and There is an

Comparisons
Some remarks on De Finetti algebrae are in order. First, we have defined them over Ł3algebrae (also known as Łukasiewicz (or Moisil-Łukasiewicz) trivalent algebrae), but other options are possible, including MV 3 -algebrae. 10 We have adopted Ł3-algebrae both because they are simpler than MV n -algebrae, and in order to better relate our presentation and results to the elegant formalization and the results of Milne 2004. Second, De Finetti algebrae have a paraconsistent flavour, suggested by the behaviour of the element 1 /2. Such flavour is more vividly expressed by noticing that they are both special cases of LP algebrae. In the characterization offered by Pynko (1995), a Kleene algebra A = A, ⊓, ⊔, −, 0, 1 is LP if it has an inconsistent proper filter on its support, that is, if there is an F ⊂ A s.t. for every a, b, ∈ A and for some c ∈ F (i) if a ∈ F and a ⊑ b, then b ∈ F, The definition of Ł3 algebrae follows Milne (2004, 517-518), and so does the characterization of the algebraic counterpart of the De Finetti conditional over them. We note that Milne considers algebrae of conditional events, while we consider arbitrary supports. Nothing crucial hinges on this.
It is easily seen that De Finetti algebrae are LP. Let A be a De Finetti algebra with support A. The set {a ∈ A | 1 /2 ⊑ a} ⊂ A provides the required inconsistent proper filter.
(i) is immediate, because ⊑ is transitive.
As for (iii), notice that both 1 /2 and − 1 /2 are in {a ∈ A | 1 /2 ⊑ a}. A is a D-consequence of Γ, in symbols Γ |= D A, if for every D ∈ D, A is a Dconsequence of Γ.

Algebraic semantics for DF/TT
Notice that, even though De Finetti algebrae include an algebraic counterpart of the Łukasiewicz trivalent conditional, the latter is not used in defining an algebraic evaluation for De Finetti algebrae (and it is not going to be used to construct specific algebraic models of DF/TT either). The reason behind this choice is that we want to isolate the De Finetti conditional, and the respective TT-logic, without including extraneous connectives (such as the Łukasiewicz conditional). However, it would be possible to expand our definition of algebraic evaluations and algebraic consequence to include the Łukasiewicz trivalent conditional, and prove the relative algebraic soundness and completeness theorems by adding suitable multi-sequent rules to DF/TTm.
In order to prove algebraic soundness and completeness for DF/TT, we construct the Lindenbaum-Tarski algebra of a set of formulae, for DF/TTm-deducibility. Therefore, we first isolate the relation of DF/TTm-provable equivalence (where 'equivalence' is formalized via the DF-biconditional).
Definition 3.4. For every Γ ⊆ For, let ∼ df Γ ⊆ For × For be the relation defined as follows: This definition, however, does not partition the set of formulae into equivalence classes, but only into sets that have weaker closure conditions. Lemma 3.5. In general, ∼ df Γ is not an equivalence relation on For × For. Proof. Reflexivity and symmetry hold, since Γ ⊢ DF/TTm A ↔ A, and if Γ ⊢ DF/TTm A ↔ B, then also Γ ⊢ DF/TTm B ↔ A. However, transitivity fails, for otherwise ⊢ DF/TTm would be unsound (consider a DF/TT-evaluation v in which v(A) = 1, v(B) = 1 /2, and v(C) = 0).
As the above proof shows, the failure of transitivity for DF/TTm-provable equivalence is closely connected to the failure of Modus Ponens for DF/TT. However, even though ∼ df Γ is not an equivalence relation on For × For, we will see that it is sufficiently wellbehaved to support an application of the Lindenbaum-Tarski method. Therefore, we proceed with the further steps towards a proof of algebraic completeness.
As shown by Lemma 3.5, ∼ Γ is not an equivalence relation, and the sets [A] Γ are not equivalence classes. Therefore, there is no guarantee that every formula belongs to exactly one of the elements in For/∼ Γ . So, we have to prove that the operations that characterize De Finetti-Lindenbaum-Tarski algebrae, that is ⊓ Γ , ⊔ Γ , − Γ , and ◮ Γ are actually well-defined, and do not depend on the choice of particular formulae: otherwise ⊓ Γ , ⊔ Γ , − Γ , and ◮ Γ might not be operations at all. This is done in the following lemma.
Lemma 3.8 (Independence from representatives). For every set {Γ, A, B, C, D} ⊆ For, the following holds: Proof. We only show the conditional case (the others are similar). Suppose that there is a set {Γ, A, B, C, D} ⊆ For such that A ∼ Γ B and C ∼ Γ D but that it is not the case that . By the completeness of DF/TTm (Proposition 2.6), this means that Γ |= DF/TT A ↔ B and Γ |= DF/TT C ↔ D but Γ |= DF/TT (A → C) ↔ (B → D). Let v be any DF-evaluation that assigns value 1 or 1 /2 to all the sentences in Γ, value 1 or 1 /2 to A ↔ B and to C ↔ D but value 0 to (A → C) ↔ (B → D) (if there are no DF-evaluations that assign values 1 or 1 /2 to all the sentences in Γ, the claim is immediate). A biconditional is assigned value 0 by a DF-evaluation just in case that evaluation assigns value 1 to one side of the biconditional and 0 to the other. Suppose without loss of generality that Proof. It is easy to see that the properties of distributive bounded lattices hold for D(Γ). We do just one case of distributivity in detail.
The line labeled with 'logic' abbreviates the fact that the corresponding identity is proven by the fact that It follows that ⊑ Γ inherits the features of ◮ Γ in D(Γ), and thus the claim is Proof sketch. The left-to-right direction is straightforward. As for the right-to-left direction, suppose that Γ ⊢ DF/TTm A. By Lemma 3.10, this entails that Γ |= D(Γ) A, which in turn entails that Γ |= D A, as desired.
It should be noted that the above proof of algebraic completeness is not, strictly speaking, a genuine algebraic proof: it is parasitic on the Schütte-style completeness proof given in the previous Subsection 2.1. 13 More precisely, the Schütte-style proof is used to construct a countermodel based on the De Finetti algebra with just three elements, 0, 1 /2, and 1, which is then expanded to an evaluation based on D(Γ). A proper algebraic completeness proof would provide a canonical model, namely an algebraic evaluation function that has elements of For/∼ Γ as its values, and that is built from the elements of For/∼ Γ by considering their combinations with the operations of D(Γ) (corresponding to the logical connectives of DF/TT). 14 A canonical model theorem would be more informative, because it would provide a single evaluation that 'encodes' all the inferences having Γ as a premiss that are DF/TT-valid. However, the proof via a canonical model typically requires Modus Ponens, and indeed it breaks down exactly where Modus Ponens is required if attempted for DF/TT. Nevertheless, an algebraic completeness proof is obtained for DF/TT. In the next subsection, we will see that things do not work so well for CC/TT and the TT-logics of Jeffrey conditionals more generally.

An algebraic semantics for CC/TT?
Can we provide a proof of algebraic completeness for CC/TT employing the Lindenbaum-Tarski method, as we did for DF/TT? The CC/TT-conditional appears better-behaved than the DF/TT one-in particular because it obeys Modus Ponensso this would appear prima facie possible.
Let us try to apply the Lindenbaum-Tarski method to CC/TT. First, we need an algebraic counterpart of the Cooper-Cantwell conditional. This is provided by the following definition.
Definition 3.12. An Ł3 algebra A = A, ⊓, ⊔, −, , ⊲, 0, 1, 1 /2 is Cooper-Cantwell if: There is a distinguished element 1 /2 ∈ A s.t. − 1 /2 = 1 /2, and There is an operation ⊲ defined on A × A s.t. a ⊲ b = −w(a) ⊔ (w(a) ⊓ b), where w(a) is a shorthand for −a 1 /2. We then work towards the construction of a Lindenbaum-Tarski algebra for CC/TT. 14 This is essentially the algebraic analogue of a Henkin model in standard completeness proofs for pure (propositional or first-order) logic, and of a canonical model in completeness proofs for normal modal systems.
Since Modus Ponens holds in CC/TT, the relation of CC/TTm-provable equivalence seems better behaved than the one defined for DF/TTm. Lemma 3.14. ∼ c Γ is an equivalence relation on For × For. Proof. Reflexivity holds since Γ ⊢ CC/TTm A ↔ A. Symmetry also holds, since if Γ ⊢ CC/TTm A ↔ B, then also Γ ⊢ CC/TTm B ↔ A. Finally, transitivity holds as well, because if Γ ⊢ CC/TTm A ↔ B, and Γ ⊢ CC/TTm B ↔ C then Γ ⊢ CC/TTm A ↔ C as well.
We now have an equivalence relation, so we can use it to partition the set of formulae into equivalence classes. Since we only work with Cooper-Cantwell algebrae in this subsection, we drop the superscript cc again to improve readability, without risks of confusion. Now, in order to proceed with the proof of algebraic completeness, we would have to define a Cooper-Cantwell version of a Lindenbaum-Tarski algebra. Such a structure would look as follows: However, the construction is blocked, because some of its defining operations turn out to be not well-defined. In particular, the Cooper-Cantwell conditional is not substitutive with respect to negation. This lemma shows that the process of providing an algebraic semantics (via the standard Lindenbaum-Tarski method) for CC/TT stops here: it does not even get off the ground.
In fact, this negative result is more general: it applies to every Jeffrey conditional. Recall that Jeffrey conditionals are required to obey the condition that f → (1, 0) = f → ( 1 /2, 0) = 0. Now, the above proof employs exactly the cases in which a conditional has an antecedent with value 1 and a consequent with value 0, and an antecedent with value 1 and a consequent with value 1 /2. Therefore, no Jeffrey conditional is substitutive with respect to negation-under a TT-notion of validity, and a Strong Kleene interpretation of conjunction and negation. In turn, this means that no 'J-Lindenbaum-Tarski algebra', where 'J' is any Jeffrey conditional, is well-defined, and therefore that no algebraic semantics (via the Lindenbaum-Tarski method) is available for any TT-logic of a Jeffrey conditional.

General Discussion
This two-part paper has reviewed the main motivations for a trivalent semantics for indicative conditionals, interpreting them as conditional assertions, and defining their truth conditions in analogy with the conditions that settle the winner of a conditional bet (i.e., the bet or assertion is declared void when the antecedent is false). Although the idea goes back to de Finetti (1936), and Reichenbach (1935Reichenbach ( , 1944, there have been few explorations of the logics induced by the adoption of that semantic scheme. Beside expounding the historical roots of trivalent semantics for conditionals, our paper has given a systematic survey of the different logics that emerge by (i) choosing a truth table for the conditional operator in agreement with the above rationale, and (ii) determining a specific notion of validity (one vs. two designated truth values, pure vs. mixed consequence relations).
As reviewed in Part I, the trivalent approach yields a fully truth-functional semantics with attractive logical and inferential properties. It also provides the conceptual foundations for a probabilistic theory of assertability and reasoning with conditionals along the lines of Adams (1975). Combining our semantics with defining the assertability of a sentence A as the conditional probability that A is true, given that it has a classical truth value, immediately yields Adams' Thesis that Ast(A → C) = p(C|A). This property highlights the potential of the trivalent approach for guiding an account of the epistemology of conditionals, and explaining how people reason with them (e.g., Baratgin et al. 2013;Baratgin, Politzer, Over, and Takahashi 2018). While the semantics of the trivalent conditional is factual-that is, its truth value is a function of matters in the actual world-no such limits are imposed on the scope of the probability functions in judgments of assertability (e.g., A can be practically unverifiable, but the conditional may still be highly assertable).
With respect to the above challenges (i) and (ii), it quickly transpires that any alternative to a tolerant-to-tolerant (TT-) notion of validity would be either too strong (in the sense of licensing undesirable inferences such as implying the converse condtional) or too weak (in the sense of violating the Identity Law A → A and not having sentential validities). Only the Cooper-Cantwell conditional, where indeterminate antecedents are exactly treated like true ones, satisfies both the full Deduction Theorem and commutation with negation. For conceptual, empirical and logical reasons (the conditional is essentially interpreted as making an assertion upon supposing the antecedent), these are eminently reasonable properties, apparently favoring CC/TT as the best trivalent logic of the indicative conditional.
The results of Part II nuance this judgment. For both DF/TT and CC/TT we can develop sound and complete calculi based on tableaux (Section 1) and three-sided sequents (Section 2). The latter calculi have the advantage of being simpler and more direct: unlike tableau calculi, they do not establish that an inference is valid by showing that it is impossible to assign a designated value to the premises and an undesignated value to the conclusion. Moreover, many-sided sequent calculi make it easier to handle inferences with multiple conclusions, as well as inferences involving infinite sets of sentences.
As soon as we consider the algebraic semantics, however, differences between DF/TT and CC/TT emerge. While provable equivalence fails to be transitive and therefore induces no equivalence relation for DF/TT (Lemma 3.5), we can still use this relation to define a Lindenbaum-Tarski algebra and to show an algebraic soundness and completeness theorem (Proposition 3.11). In other words, A can be derived from Γ using one of the above calculi (e.g., many-sided sequents) if and only if a consequence relation holds between Γ and A in the associated de Finetti algebrae. The failure of Modus Ponens for DF/TT however, blocks the construction of a canonical algebraic model.
Things look bleak, by contrast, for CC/TT and other TT-logics based on a Jeffrey conditional. While provable equivalence induces an equivalence relation for these logics, the construction of a Lindenbaum-Tarski algebra does not get off the ground because provable equivalence fails to be substitutive under negation. More precisely, the Cooper-Cantwell biconditional ↔ falls short of expressing CC/TT-equivalence since A ↔ B |= CC/TT ¬A ↔ ¬B. Which means that there is not, and cannot be, a fruitful algebraic treatment of Jeffrey conditionals. In fact, this is grounded in a defining property of Jeffrey conditionals: to preserve Modus Ponens and to yield a full Deduction Theorem, a trivalent conditional based on the "defective" truth table needs to obey f → (1, 0) = f → ( 1 /2, 0) = 0. It is exactly this property which makes substitution under negation fail (Lemma 3.16), and prevents a proper algebraic semantics for Jeffrey conditionals.
Clearly, the failure of substitution under negation is closely related to the failure of contraposition in Jeffrey conditionals-an inference that does not fail in DF/TT. Indeed, the same evaluation provides the counterexamples employed in proving both Proposition 5.6 (Part I) and Lemma 3.16 (Part II). So it turns out that what has been a strength of Jeffrey tolerant-tolerant logics, and CC/TT in particular, at the level of desirable conditional principles, comes at the price of the algebraic semantics. Importantly, the lack of an algebraic semantics is not a mere technical fact, but it has philosophical consequences as well. In particular, in every Jeffrey tolerant-tolerant logic, even if it is the case that A ↔ B, the same equivalence does not hold in general for logically complex sentences that result by uniform substitutions of A and B (see Lemma 3.8 for a formally precise version of this property). Therefore, Jeffrey conditionals do not provide a workable notion of equivalence.
Of course, the limitations of Jeffrey conditionals just reviewed arise from the combination of the semantics of Jeffrey conditionals, TT-validity, and Strong Kleene conjunction, disjunction, and negation: one might therefore wonder whether they can be improved on by altering some of these parameters. However, as the results of Part I show, adopting an alternative to TT-validity does not seem promising. As for the semantics of the other connectives, it should be noted that Cooper himself adopts alternative truth tables for conjunction and disjunction, while retaining the K3 table for negation (see also Humberstone 2011, §7.19, 1044 and following). Cooper's conjunction and disjunction, however, have some rather perplexing behaviors: for instance, in Cooper's original system, one cannot in general infer A ∨ B from A, for when A has value 1 /2 and B has value 0, A ∨ B has value 0 as well-Cooper's conjunction displays similar oddities. Alternatively, one might inquire into what happens to the interaction of Jeffrey conditionals with a non-K3 negation. To be sure, the K3-negation squares particularly well with the philosophical motivation for de Finettian conditionals: when a conditional assertion A → C is "called off" because A is false, the same should happen for the negation of that assertion (i.e., the sentence A → ¬C, thanks to the commutation scheme). Nonetheless, it might be worth investigating how Jeffrey logics (keeping a tolerant-tolerant notion of validity) fare when coupled with what Chemla and Égré (2018) call a "Gentzen-regular" negation, that is a negation obeying the Gentzen sequent calculus rules. While a Gentzen-regular negation might avoid some of the above problems, it would lose the commutation of conditional and negation, and the attached connexive principles (see Subsection 5.2 of Part I). In conclusion, there seem to be structural limitations, or at least unavoidable tradeoffs, that affect Jeffrey conditionals, when it comes to their interaction with other connectives.
We therefore believe that it is not easy to justify a clear preference between the two logics CC/TT and DF/TTthat we have isolated as most promising amongst trivalent logics of indicative conditionals. Both have attractive properties, both have limitations-but they agree in essential properties such as the valuation of classical sentences, the Import-Export principle, the analysis of paradoxes of material implication, their connexive nature, and the connection to a theory of assertability. To solve the limitations highlighted throughout the paper, one would probably have to give up one or more of these features. So while there is perhaps no perfect trivalent semantics for indicative conditionals, they need to be considered carefully between two-valued logic and modal logics of conditionals. In any event, they give rise to a promising research program, and we shall support this claim by sketching some future projects that build on our work in this paper.
Firstly, we would like to extend the current framework to predicate logic and to investigate how the trivalent conditionals fare in that context, including how they interact with a naïve or a compositional truth predicate. Secondly, we would like to apply trivalent semantics to McGee's famous challenge to Modus Ponens, applying our accounts of logical consequence and probabilistic assertability (McGee 1985; Stern and Hartmann forthcoming). Thirdly, one should review the intuitions and inference schemes which fuel connexive logics (e.g., Aristotle's Thesis, Boethius' Thesis) from a trivalent perspective, and conduct a more detailed comparison. Fourthly, we need to develop more precise criteria as to which inferences should be validated by a trivalent logic of conditionals, based on the concept of supposition, and which inferences can be relegated to a probabilistic theory of assertability grounded in the truth conditions. In other words, we have to formulate a precise account of how the truth conditions of indicative conditionals relate to reasoning with them.
Finally, there is the question of how a trivalent semantics integrate into a general theory of conditionals, including those in the subjunctive mood. Extending a de Finettian treatment of indicative conditionals yields the consequence that all conditionals with false antecedents-in particular, all counterfactuals-have indeterminate truth value. The difference between them is only a difference in assertability (because their conditional probabilities p(C|A) can be different and will typically vary with context). This perspective is close to Jeffrey's view who qualifies counterfactual questions either as "nonsense" or as "colorful ways of asking about conditional probabilities" (Jeffrey 1991, 164). On this picture, the traditional view that indicative conditionals are epistemic and counterfactuals are metaphysical (Lewis 1973b,a;Edgington 1995;Khoo 2015) would be reversed: while indicatives are factual statements (i.e., conditional assertions) with non-trivial truth conditions, counterfactuals come out having trivial truth conditions and differ only in their epistemic import, that is, their assertability conditions. Whether the proponent of a trivalent semantics for indicative conditionals should be committed to such far-reaching philosophical consequences is, of course, a question that we have to postpone to future research.

A Appendix: some questions about algebraizability
We conclude with a discussion of the prospects for a full algebraizability of DF/TT, offering some conjectures and sketching strategies to answer them in future work. Questions about algebraizability appear appropriate in order to explore the viability of DF/TT as a candidate model for the indicative conditional, because they involve the formalization of notions that are relevant for an interesting indicative (bi)conditional, as it will become clear in the following discussion. In this respect, however, the DF/TT conditional reveals non-negligible limitations.
The notion of algebraizability, introduced by Blok and Pigozzi (1989), generalizes the link between a logic and its algebraic semantics, imposing stricter conditions than those that are required for algebraic completeness. 15 Let us first introduce some notational conventions. We can now formulate the notion of algebraizability. (A1) is a generalization of algebraic completeness, where the right-hand side expresses in the object-language, via pairs of sentences, the semantic requirement that A has a designated value whenever all the sentences in Γ do. To see this, consider a logic whose only designated value is 1, and whose algebraic counterparts are structures featuring a top element 1, such that the corresponding algebraic evaluations send ⊤ to 1. Classical logic and Boolean algebrae are a case in point. In order to express via equations the idea that A follows from Γ in classical logic, let the function f be s.t. f (A) = A ≈ ⊤. Letting B be the class of Boolean algebrae, f [Γ] |= B f (A) becomes B ∈ Γ B ≈ ⊤ |= B A ≈ ⊤, which expresses, via equations, the idea that whenever every sentence in Γ has value 1, so does A.
(A2) ensures that the solvability of equations 18 is fully captured by some formula of the object-language. Let us consider the same example as above. In classical logic, two sentences B and C have the same value if and only if the corresponding biconditional B ≡ m C (where ≡ m denotes the material biconditional) has value 1. This fits easily into 16 In other words, D validates the equational inference from E to C ≈ D if every De Finetti evaluation based on D that assigns the same value in D (the support of D) to all the pairs of sentences that constitute the equations in E, also assigns the same value to C and D. More succintely, every De Finetti evaluation based on D that solves all the equations in E, also solves C ≈ D.
17 Let ' f [Γ]' be a shorthand for { f (B) ∈ P (For × For) | B ∈ Γ}. 18 Which is on the left-hand side of B ≈ C |= |= A f (g(B ≈ C)) (A2), letting g be s.t. g(E) = {B ≡ m C ∈ For | B ≈ C ∈ E}-in other words, g is the function that sends B ≈ C to B ≡ m C. In this way, B ≈ C |= |= B f (g(B ≈ C)) becomes B ≈ C |= |= B (B ≡ m C) ≈ ⊤ which clearly express the idea that B and C have the same value whenever the corresponding biconditional holds (i.e. has value 1) in classical logic. (A1) and (A2) generalize algebraic completeness, but they do not sit very well with the conditional of DF/TT. In the framework of DF/TT, the right-hand side of (A1) expresses that whenever all the sentences in Γ have value 1 or 1 /2, so does A. Letting which might be plausibly thought to express, via equations, the idea that whenever every sentence in Γ has value 1 or 1 /2, so does A. Now, suppose that we take again g to be the function that turns every equation B ≈ C into B ↔ C (this time with the De Finetti biconditional), that is, g(E) = {B ↔ C ∈ For | B ≈ C ∈ E}. In this way, (A2) becomes which, however, does not express the idea that B and C have the same value whenever B ↔ C holds in DF/TT. In fact, it is not the case that, in order for B ↔ C to have the same value as ((B ↔ C) ↔ ⊤) ∨ ((B ↔ C), B has to have the same value as C; an evaluation e such that e(B) = 1 and e(C) = 1 /2 provides a counterexample. This translates into the algebraic semantics, considering a De Finetti algebra D and an algebraic evaluation e based on D s.t. e(B) = 1 and e(C) = 1 /2 (for 1, 1 /2 ∈ D). Of course, this observation tells us only that the schema B ↔ C does not express the fact that B and C have the same value-which is not surprising, given the 1-and 1 /2-rows of the truth table of the De Finetti conditional. However, it is the general idea of mapping identity of semantic values to a defining formula that expresses 'having a designated value' that seems at odds with the conditional of DF/TT, because DF/TT does not distinguish between 1 and 1 /2 when it comes to valid inferences (and hence designatedness), nor does its conditional, because the DF/TT biconditional takes a designated value even when its two sides have values that differ by 1 /2. In a toleranttolerant semantics, validity (together with the corresponding conditionals) does not depend on the identity of the semantic values that are preserved from premises to conclusion, but on their similarity: 1 and 1 /2 are not identical, but similar enough for DF/TT not to distinguish them. In conclusion, condition (A2) can only be satisfied via a formula (which should replace B ↔ C in (1)) that captures a notion of validity based on the identity of semantic values, hence not one that is encoded by the tolerant reading of the De Finetti conditional. Clearly, this does not show that DF/TT is not algebraizable: here, we leave the question open. However, the above observations suggest a seemingly promising strategy to prove non-algebraizability: if one can show that a truth-function expressing identity of truth values is not definable in the truth table semantics for DF, this would translate into the algebraic semantics, and establish non-algebraizability. 19 Even if DF/TT actually turns out to be non-algebraizable, the question remains whether it becomes algebraizable over other logics. A natural choice here would be to look at Łukasiewicz trivalent logic (with a TT-notion of validity), because De Finetti algebrae are defined over Ł3 algebrae. We leave open the question whether DF/TT plus the Ł3 conditional is actually algebraizable. However, we notice that there are reasons to expect a positive result here, namely that the Łukasiewicz trivalent conditional can be used to express the identity of semantic values, for example via the schema ¬((A ↔ B) → ¬(A ↔ B)). In this way, (1) would become B ≈ C |= |= D (¬((B ↔ C) → ¬(B ↔ C))) ≈ ((¬((B ↔ C) → ¬(B ↔ C))) ↔ ⊤) ∨ ((¬((B ↔ C) → ¬(B ↔ C))) ↔ (⊥ → ⊤)) 19 We mention this proof strategy because it seems both simpler and more informative than a proof via Isomorphism Theorems, which are the standard results employed to prove non-algebraizability (see Font 2016, Chapter 3.5). A proof via functional incompleteness would be more informative because it would establish that the notion of 'having the same value' is inexpressible in DF/TT, which seems an interesting fact to know about potential candidates for the indicative biconditional.