Jean Baratgin
email: jean.baratgin@univ-paris8.fr

David E Over

Guy Politzer

Uncertainty and the de Finetti tables 1

Keywords: New paradigm psychology of reasoning, Indicative conditionals, Bets on Conditionals, Uncertainty

The new paradigm in the psychology of reasoning adopts a Bayesian, or probabilistic, model for studying human reasoning. Contrary to the traditional binary approach based on truth functional logic, with its binary values of truth and falsity, a third value that represents uncertainty can be introduced in the new paradigm. A variety of three-valued truth table systems are available in the formal literature, including one proposed by de Finetti. We examine the descriptive adequacy of these systems for natural language indicative conditionals and bets on conditionals. Within our framework the so-called "defective" truth table, in which participants choose a third value when the antecedent of the indicative conditional is false, becomes a coherent response. We show that only de Finetti's system has a good descriptive fit when uncertainty is the third value.

The new Bayesian, or probabilistic, paradigm in the psychology of reasoning [START_REF] Oaksford | Bayesian rationality: The probabilistic approach to human reasoning[END_REF][START_REF] Over | New paradigm psychology of reasoning[END_REF][START_REF] Over | New paradigm psychology of reasoning[END_REF][START_REF] Pfeifer | The conditional in mental probability logic[END_REF] implies a close parallel relationship between assertions of the indicative conditional of natural language, if A then C, and other uses of conditionals, particularly bets on conditionals, I bet that if A then B. Politzer, Over, and [START_REF] Politzer | Betting on conditionals[END_REF] explain how this predicted relation goes back to [START_REF] Ramsey | Truth and probability[END_REF]Ramsey ( /1990[START_REF] Ramsey | General propositions and causality[END_REF]Ramsey ( /1990) and de Finetti (1936de Finetti ( /1995de Finetti ( , 1937de Finetti ( /1964) and provide experimental evidence that this parallel relation exists (see also Baratgin, Over, & Politzer, in press).

There are two aspects to the parallel relation. First, the subjective probability of the conditional assertion is the subjective conditional probability of C given A, P(if A then C) = P(C|A), and the subjective probability that the conditional bet will be won is also P(C|A).

Second, the conditional assertion will be true, and the conditional bet will be won, when A and C are both true, and the conditional assertion will be false, and the conditional bet lost, when A is true and C is false. When A turns out to be false, no indicative assertion or bet is made. The assertion and bet are void: the indicative conditional if A then C is neither true nor false, and the bet on it is neither won nor lost. The finding that the indicative conditional has such a three-valued table goes back to [START_REF] Wason | Reasoning[END_REF], and psychologists long called it the defective truth table, as if there were something wrong with the table or the participants for producing it (see [START_REF] Evans | [END_REF], for a review). Psychologists who used "defective" in a negative sense were apparently unaware that this table is found in the formal systems of a number of philosophers, logicians, linguists, and mathematicians, including de Finetti (1936de Finetti ( /1995)), [START_REF] Cooper | The propositional logic of ordinary discourse[END_REF], [START_REF] Jeffrey | On indeterminate conditionals[END_REF], [START_REF] Farrell | Implication and presupposition[END_REF], [START_REF] Calabrese | An algebraic synthesis of the foundations of logic and probability[END_REF], and [START_REF] Goodman | Conditional inference and logic for intelligent systems a theory of measure-free conditioning[END_REF] (see Baratgin, Politzer, & Over, under review, for a survey of these formal conditional tables). A conditional if A then C for which P(if A then C) = P (C|A) has been called a probability conditional [START_REF] Adams | A primer of probability logic[END_REF][START_REF] Oaksford | Bayesian rationality: The probabilistic approach to human reasoning[END_REF][START_REF] Over | New paradigm psychology of reasoning[END_REF]) and a conditional event (de Finetti, 1936(de Finetti, /1995;;[START_REF] Pfeifer | The conditional in mental probability logic[END_REF]. We will prefer the term "conditional event" when referring to the three-valued "defective" table, which should anyway be called the 2x2 de Finetti table for this conditional as he was the first to construct it in a major theory of the conditional. See Table 1.

- --------------------------------------Insert Table 1 about here.

- ---------------------------------------The identity, P(if A then C) = P(C|A), has such deep consequences for the study of conditional reasoning that it has been called the Equation by both philosophers [START_REF] Edgington | On conditionals[END_REF] and psychologists [START_REF] Oaksford | Bayesian rationality: The probabilistic approach to human reasoning[END_REF][START_REF] Over | New paradigm psychology of reasoning[END_REF]. Psychologists have provided strong support for the Equation as a description of people's actual probability judgments about conditionals [START_REF] Douven | The Adams family[END_REF]Evans, Handley, Neilens, & Over, 2007;[START_REF] Evans | Conditional and conditional probability[END_REF][START_REF] Fugard | How people interpret conditionals: Shifts toward conditional event[END_REF][START_REF] Oberauer | The meaning(s) of conditionals: Conditional probabilities, mental models and personal utilities[END_REF][START_REF] Over | The probability of causal conditionals[END_REF][START_REF] Politzer | Betting on conditionals[END_REF]. The most common account of how people make the judgment, P(if A then C) = P(C|A), is that they use the Ramsey test [START_REF] Edgington | On conditionals[END_REF][START_REF] Ramsey | General propositions and causality[END_REF]Ramsey, /1990)). They hypothetically suppose A, making whatever mental changes are necessary to preserve consistency, and judge to what extent C follows. Some more specific proposals have been made about how people implement the Ramsey test, as in the suppositional account of [START_REF] Evans | [END_REF], but we do not need to take a stand on these hypotheses in this paper.

Our aim here is to extend our study of if A then C to a context in which its components, A and C, are uncertain [START_REF] Politzer | Betting on conditionals[END_REF]Baratgin et al., in press). Previous studies of the three-valued, "defective" table, going back to [START_REF] Wason | Reasoning[END_REF], have had the limitation that they were conducted under conditions of certainty. Consider the conditional studied in [START_REF] Politzer | Betting on conditionals[END_REF]:

(1) If the chip is square (S) then it is black (B).

A distribution of chips was displayed for the participants in the study. There were two black round chips, one white round chip, three black square chips, and one white square chip. The participants were told that "the chip" referred to the chip that was going to be randomly selected from this distribution, and they were also asked to consider the following bet on the conditional:

(2) I bet that if the chip is square (S) then it is black (B). Politzer et al. followed the standard, and indeed universal, practice of presenting what were in effect the four cells of 2x2 truth table -S true and B true, S true and B false, S false and B true, and S false and B false -in conditions of certainty. The participants were to say whether, in each of these cells, the conditional (1) was true, false, or neither true nor false, and whether the conditional bet (2) was won, lost, or neither won nor lost. In similar truth table tasks, cards have been used with letters and numbers, or colours and shapes, and a conditional was given about these items [START_REF] Evans | [END_REF]. In all these tasks, the participants were asked to suppose that the four cells successively held, or were successively shown items exemplifying these cells, and asked for their judgments about the conditional. Under these conditions of certainty, the responses were generally consistent with the three-valued, "defective" table, adding to the evidence of a close relation between indicative conditionals and bets on conditional bets (see also [START_REF] Oberauer | The meaning(s) of conditionals: Conditional probabilities, mental models and personal utilities[END_REF].

The new paradigm, however, stresses that most judgment and reasoning in everyday life and in science takes place in a context of at least some uncertainty [START_REF] Oaksford | Bayesian rationality: The probabilistic approach to human reasoning[END_REF][START_REF] Over | New paradigm psychology of reasoning[END_REF]. For example, the antecedent and consequent of the following indicative conditional are uncertain for most people:

(3) If global warming continues (W) then London will be flooded (L).

We could, following the traditional paradigm of studying truth table tasks under certainty, ask people for their judgments about (3) after asking them simply to assume that both W and L are true, that W is true and L is false, and so on. But what are their judgments given that both W and L are to a degree uncertain? Such questions about uncertainty have not yet been addressed in psychological research. The only previous study of extended truth tables in the psychological literature is [START_REF] Elqayam | The collapse illusion effect: A semantic-pragmatic illusion of truth and paradox[END_REF], which is concerned with the important but distinct topic of the semantic paradoxes.

The question of participants' judgments in such a 3x3 task was even not raised in the traditional paradigm, and apparently no predictions follow for it in this paradigm (Evans & Over, 2004, report no work at all on 3x3 truth table tasks in their review). However, the new paradigm, with its emphasis on uncertainty, does imply what the responses should be in the 3x3 task, in an extended table worked out by de Finetti himself. Thus we can distinguish between the 2x2 de Finetti table, see Table 1 again, and the 3x3 de Finetti table, see Table 2, for the conditional event. In fact, de Finetti (1936/1995) also proposed three-valued tables for negation, not-A, conjunction, A & C, disjunction, A or C, and the material conditional, equivalent to not-A or C, when A and C can be true, false, or uncertain (Baratgin et al., under review). These tables can be called de Finetti tables for the connectives, but the study we report here focuses on whether participants' judgments about the natural language indicative conditional and the bet on the conditional conform to the 3x3 de Finetti table for what he called the conditional event or to some other formal table found in the philosophical, logical, linguistic, and mathematical literatures.

- -------------------------------Insert Table 2 about here - --------------------------------Before going on to our experimental work, let us clarify the notion of uncertainty that we are focusing on and introduce some terminology. To make the example more concrete, let us suppose that the assertion of (3) is understood in context to be about the next ten years.

Suppose it turns out to be definitely false that global warming has continued, and so we judge that P(W) = 0. Then the indicative conditional (3) is void, because there is no indicative fact to settle the matter, and we would switch to a counterfactual conditional, If global warming had continued, then London would have been flooded. Even though P(W) = 0, we can still say that the probability of this counterfactual is the conditional probability, P(L|W), with this value being high or low depending on a Ramsey test for the counterfactual [START_REF] Gilio | The psychology of inferring conditionals from disjunctions: A probabilistic study[END_REF][START_REF] Over | The probability of causal conditionals[END_REF]. In effect, the uncertainty of the indicative conditional is transferred to the counterfactual when P(W) = 0. (See Pfeifer, this issue, and Pfeifer & Kleiter, 2010, for more on conditionals with antecedents having 0 probability.) 3 for all these cells.

In a 2x2 table generally for

- ----------------------------------------Insert Table 3 about here. - ----------------------------------------There has been no psychological research on Table 3 when the third value is uncertainty as we are using the term (but see again [START_REF] Elqayam | The collapse illusion effect: A semantic-pragmatic illusion of truth and paradox[END_REF], and more specifically on the five important uncertain states of mind corresponding to the a/b, c/d, a/c, b/d and a/b/c/d cells. In the psychological literature, the role of uncertainty in premises of deductive arguments has been studied (see for a review [START_REF] Politzer | Reasoning with conditionals[END_REF]. Some studies have outlined how far participants endorse a conclusion derived from uncertain premises for Modus Ponens and Modus Tollens [START_REF] George | The endorsement of the premises: Assumption-based or belief-based reasoning[END_REF][START_REF] George | Reasoning from uncertain premises[END_REF][START_REF] George | Evaluation of the plausibility of a conclusion derivable from several arguments with uncertain premises[END_REF][START_REF] Politzer | Deductive reasoning from uncertain conditionals[END_REF][START_REF] Stevenson | Deduction from uncertain premises[END_REF]. The main result is that the rate of endorsement is linked to the degree of uncertainty related to the premises. Participants attach a degree of uncertainty to the conclusion when at least one premise has a degree of uncertainty, and also that there is a relationship between the degree of uncertainty in the premises and the degree of uncertainty for the conclusion of both Modus Ponens, and Modus Tollens. However, no experiment conducted directly on Table 3 has investigated how the evaluation of a conditional as a whole -as true, false, or uncertaindepends on the uncertainty of its antecedent and consequent. For a full account of people's understanding of conditionals and uncertainty, it is necessary to discover how they respond in the five new cells. This is the main ambition of our study. Following de Finetti (1936/1995), we can also think of the de Finetti conditional event table as a description of the outcomes of a bet on a conditional. Let us say that we are arguing with other people about climate change, asserting (3) and then offering them the bet in (4):

(4) We bet that, if global warming continues (W), then London will be flooded (L).

Keeping the context about the next ten years, we win the argument to support our assertion of (3), and the bet on the conditional (4) we make as part of it, when the time passes and we observe that global warming continued and London has flooded, the a cell; and we lose both when we observe that global warming has continued but London has not flooded, the b cell.

During much of the ten year period, we may well be uncertain whether global warming will continue and London will flood, and then we cannot yet settle the argument or the bet. But in the c, c/d and d cells, when global warming stops, we do not win or lose the argument or the bet about the indicative conditional, for there is then no indicative fact that makes (3) true or false and that settles the argument and the bet. Although the indicative (3) and the bet on it (4) are now void, there is still uncertainty about what would have happened had global warming continued, about the counterfactual in effect, and the c, c/d and d cells can be thought of as having uncertainty in this sense. In this paper, we will look for participants' evaluations in the five newly introduced cells, and examine whether the close parallel relation between assertions of conditionals and bets on conditionals extends to these cells. We will use novel experimental materials and generalise the results of [START_REF] Politzer | Betting on conditionals[END_REF] to all nine cells of the 3x3 table.

METHOD Materials

The stimuli were individually presented on a computer screen. Participants saw a computer representation of a "game" composed of two opaque boxes, one on top of the other, with some open and clear space between them. The top box had square or round chips in it that were black or white. A random chip would sometimes fall from the top box to the lower box, and then a photo would be taken of this chip while it passed through the space between the boxes. The photos were sometimes taken through filters that had the effect of concealing the shape or the colour of the chips, or both the shape and the colour (see Figure 1).

- ----------------------------------------Insert Figure 1 about here -----------------------------------------Consequently the shape of the chips the participants could see had three possible values: the chips could appear as square, round, or blurred. The colour of the chips the participants could see could also take three values: black, white, or grey. There were overall nine possibilities (3x3 combinations) corresponding to the nine cells of Table 3. See Figure 2.

- ----------------------------------------Insert Figure 2 about here -----------------------------------------An elementary scenario described two children playing a game with this apparatus. A child, Marie, selected a chip at random and let it fall from the top, and another child, Pierre, had to guess which kind of chip (the shape and colour) this was. He could say, for instance, If the chip is square, then it is black, or I bet that if the chip is square, then it is black, so reproducing the two conditions -indicative conditional (IC) and bet on conditional (BC)studied by [START_REF] Politzer | Betting on conditionals[END_REF]. The task for the participants was to decide, for each of the nine "photographs", whether what Pierre said was true, false, or neither (for IC), or whether the bet was won, lost, or neither (for BC). The option "neither" here covers the senses of uncertainty that we discussed above.

- ----------------------------------------Insert Figure 3 about here -----------------------------------------Other connectives were also studied (e. g., conjunction, The chip is square and black), and the simple affirmation, The chip is black. The latter was always presented first and was used to familiarise participants with the task.

Participants

A total of 192 French native speakers, students (remote teaching) at the University Paris 8 volunteered for the experiment. Their mean age was 37. They already held a degree and resumed their studies (social sciences). They had no specific background in logic or probability theory.

Design and procedure

The participants were randomly allocated to the two conditions: the Indicative Conditional (IC) or the Bet on conditional (BC), using the same indicative conditional, if the chip is square, then it is black. They were required to answer nine randomly ordered questions (see Figure 2), each question corresponding to a cell of the table. The question was presented in a 3-option multiple choice format, to which they answered by selecting one option (Figure 4).

- ----------------------------------------Insert Figure 4 about here -----------------------------------------

RESULTS

Data analysis

For the simple affirmation, all participants correctly answered. This is evidence that they recognized and understood the iconic representation of the three possibilities for the antecedent, that is, the square, "blurred", and round shapes were visually well distinguished, with the "blurred" icon representing uncertainty about the shape; and similarly for the consequent: the black, grey, and white colours were visually well distinguished, with the grey shade representing uncertainty about the colour.

For the conditional questions, the tables produced by participants will be analysed in two stages.

The first stage casts the results in terms of the traditional classification. We categorize the answers in the four "old" cells of the traditional table that correspond to the four cases where the antecedent and consequent are either true or false (a, b, c, d cells of Table 3). The results are presented in Table 4.

- ----------------------------------------Insert Table 4 about here -----------------------------------------The patterns of response for conditions IC and BC were nearly identical (Χ 2 = 3.92, df = 4, p = .41) and reproduce traditional observations (at least qualitatively) in the literature (for a review see [START_REF] Evans | [END_REF]). The table traditionally called "defective", Table 1, was participants' modal interpretation in both conditions (41.7%). The second most frequent response was the conjunction table (34.9%). A much smaller minority (13%) of the participants displayed the material conditional table, and a still smaller minority the material bi-conditional table and the bi-conditional table, which represents the bi-conditional event in de Finetti's terms.

In the second stage of the analysis, we further characterize the tables by considering all nine cells. The observed three-valued tables are compared with the relevant three-valued formal tables in the logical, linguistic, philosophical, and mathematical literatures. This analysis now focuses on the five "new" cells of the table (the c/d, a/c, b/d, a/c, and a/b/c/d cells of Table 3). Each participant's observed table is classified by considering the formal table to which it is the closest. Our criterion of "closeness" or "distance" is as follows. A participant's table is taken to be a perfect instance of a formal table N when it is identical to N. A participant's table is a close instance of N when it differs from N just by one cell (and from any other formal table by more than one cell). If a participant's table differs equally (by one cell) from two (or more) formal tables, it is close but classified as ambiguous between these tables (these are equally likely). Finally, if a participant's table differs by two or more cells from all formal tables, then it is classified as indeterminate as it differs too much to make a reliable identification. The results are presented in Table 5.

- ----------------------------------------Insert Table 5 about here -----------------------------------------Almost all of the participants' tables coincide with a system of three-valued tables proposed in the formal literature.

The patterns in the two conditions, IC and BC, are parallel and almost identical for the three main types of interpretation of the conditional: conditional event, conjunction, and material conditional (all chi-square values yield p >.36, df=4).

For each of the three major interpretations observed -conditional event, conjunction, and material conditional -the majority of the observations coincides with a table coherent with de Finetti's three-valued logical system (de Finetti, 1936(de Finetti, /1995)). In other words, more than half of the conditional event interpretations coincide with de Finetti's specific conditional event (while the remaining conditional event interpretations are distributed over other conditional event tables in the formal literature, such as Farrell's, and Cooper's) and the same obtains for the conjunctive and material conditional interpretations of the conditional). 1 For both conditions IC and BC pooled together, 57.7% of the observations identified as conditional event tables coincided with de Finetti's specific conditional event table and approximately 20% were a Farrell conditional table [START_REF] Farrell | Implication and presupposition[END_REF][START_REF] Goodman | Conditional inference and logic for intelligent systems a theory of measure-free conditioning[END_REF], which differs from de Finetti's table only for the b/d cell (see Table 5). The two other types of table with a frequency exceeding 10% coincide with the Cooper conditional event table (Cooper, 1968, Calabrese;[START_REF] Calabrese | An algebraic synthesis of the foundations of logic and probability[END_REF], in which the a/c and b/d cells differ from de Finetti's table, and an ambiguous table which differs only by one cell from de Finetti's table (a/c cell) and Cooper's table (b/d cell). De Finetti's conjunction table was chosen by 52.2% of the participants who had a conjunctive interpretation (averaged across both conditions). Porte's conjunction table [START_REF] Porte | Schémas pour le calcul des propositions fondé sur la conjonction et la negation [Schemas for a conjunctive-negative calculus of propositions[END_REF] was the second most frequently observed table (28.4% of the participants). This table differs from de Finetti's conjunction table in the b/d cell. In addition, there were many de Finetti conjunction tables in the category "ambiguous conjunction table" (that is, one out of the two candidate tables was a de Finetti table).

In the IC condition, 43% of the participants who had a material conditional response produced de Finetti's material conditional table. The McCathy material conditional table [START_REF] Mccarthy | A basis for a mathematical theory of computation[END_REF] represents 35% of participants' answers out of those who gave a material conditional response. McCathy's table only differs from de Finetti's in the a/c cell (see Table 5). In the BC condition, de Finetti's material conditional table predominates (72.7%) out of those giving a material conditional interpretation.

Among the very few participants who had a material bi-conditional interpretation and a bi-conditional interpretation, de Finetti's material bi-conditional table and de Finetti's biconditional (event) table were the majority answer, respectively.

DISCUSSION

In this experiment, we have categorized participants' response tables by analysing how close they are to relevant tables in the formal literature (see Baratgin et al., under review, for more on this). We wish to single out three results here as worthy of special note.

The first result is that there are a limited number of tables that actually account for people's responses. The 2x2 "defective" truth table can be theoretically extended with a third value to a 3x3 table in 243 (3 5 ) ways. However, the responses of our participants are by no means scattered among all these possibilities. Our evidence is that people respond to uncertainty in a limited number of ways, and these are found in the formal literature (Baratgin et al., under review).

The second main result is that a majority of participants conform to the 3x3 de Finetti conditional event table in both the indicative conditional , IC, and the bet on conditional, BC, conditions, and in the same proportion. This result reinforces the new paradigm argument that participants are not "defective" but coherent. The finding that there is not a difference between the two conditions generalizes [START_REF] Politzer | Betting on conditionals[END_REF]. People treat questions about the truth or the falsity of the assertion of indicative conditionals as very similar to questions about winning or losing bets on conditionals.

The third main result is that there is a minority response conforming to the conjunctive table in both the IC and BC conditions. A minority conjunctive response has been found in experiments on participants' judgments about the probability of conditionals like (1), but not (or not so much) like (3) that are more realistic or familiar (compare Evans et al., 2003, with Over et al., 2007). There is evidence that this response is due to a processing limitation [START_REF] Politzer | Betting on conditionals[END_REF]Evans et al., 2007). But another possible explanation for what we have found here is the "matching bias" of [START_REF] Evans | Interpretation and matching bias in a reasoning task[END_REF]. People tend to use "true" for the cases that match items in the conditional statement and "false" for those that do not. Consider (1) again, If the chip is square, then it is black. There is a match with the photo of a square and black chip in the a cell, which makes the conjunction The chip is square and black true. The b, b/d, c, c/d, and d cells are mismatching instances and make the conjunction false. It is visually uncertain whether there is a full match in the remaining cells, the a/b, a/c, and a/b/b/d cells, and these are uncertain cases in de Finetti's table for conjunction.

We can make further points about our experiment and how to develop it by recalling the traditional task in a context of uncertainty, the Wason selection task [START_REF] Evans | [END_REF][START_REF] Wason | Reasoning[END_REF]. Imagine a Wason selection task based on (1). There are four cards describing a chip. One side of a card says whether the chip is square or round and the other side whether it is black or white. The cards are lying on a table with only one side revealed. The S card states that a square chip has been selected, with the hidden side indicating whether it is black or white. The not-S card states that a round chip has been selected, with the hidden side indicating whether it is black or white. The B card states that the chip is black, but it is hidden whether it is square or round. And the not-B card that the chip is white, with it being hidden whether it is square or round. The question that the participants are asked is which cards should be turned over to decide whether or not (1) holds.

There is a clear parallel relationship between such a selection task and our experiment.

The not-S card corresponds to the c/d cell of Table 3. It makes the indicative (1) void by the de Finetti table, and there is no need to turn it over. The related counterfactual, If the chip had been square then it would have been black, is completely uncertain, unless further information is given on the square chips, such as that all, none, or some proportion of square chips are black (see Evans & Over, 2004, on counterfactuals). The other cards keep (1) uncertain as a result of a lack of visual information, and there are clearly good reasons to turn these over given the de Finetti table. The S card corresponds to the a/b cell, the B card to the a/c cell, and the not-B card to the b/d cell. Finally, the a/b/c/d cell in the middle of Table 3, where both S and B are uncertain, would be like a card that is not seen at all and is not used in a traditional selection task.

The traditional, bivalent claim about the selection task was that only the S and not-B cards should be turned over to test whether (1) is true or false [START_REF] Wason | Reasoning[END_REF]. This claim, however, came under serious criticism in some of the first work on the new paradigm [START_REF] Evans | Rationality and Reasoning[END_REF][START_REF] Evans | [END_REF][START_REF] Oaksford | A rational analysis of the selection task as optimal data selection[END_REF]. The traditional presupposition is that conditionals like (1) are material conditionals, logically equivalent to not-S or B, making (1) true when the not-S card, the c/d cell, is shown or the B card, the a/c cell, is shown. In contrast, de Finetti's conditional event -the probability or suppositional conditional -of the new paradigm is uncertain for these cards and cells. We have confirmed the existence of this uncertainty in our study here of 3x3 tables. From a new paradigm point of view, the S, B, and not-B cards should all be turned over, though the S card is the most decisive since the B and not-B cards could result in a void outcome. The most common choice in a selection task is the S card followed by the B and not-B cards [START_REF] Evans | [END_REF]. One way to develop this research would be to construct a selection task for a bet like (2), using the same cards but with the question becoming which cards should be turned over to determine whether the bet is won or lost. The new paradigm again implies that the S, B, and not-B cards should all be turned over. But an additional aspect of a betting-based selection task would be the expected utility of turning over a given card, depending on how much money was bet, and this aspect could also be described in a new paradigm, Bayesian approach (see Evans &[START_REF] Evans | [END_REF]Oaksford &[START_REF] Oaksford | A rational analysis of the selection task as optimal data selection[END_REF], on deontic selection tasks).

Our materials are close to the way in which de Finetti (1936Finetti ( /1995Finetti ( ,1937Finetti ( /1964) described his table. Some of what de Finetti writes on his tables is open to more than one interpretation, but he is clear that the uncertainty value is "superimposed" on the underlying classification of A and B as either true or false (de Finetti, 1936(de Finetti, /1995)). As part of his account of subjective probability, the entries in his tables are epistemic states. People can definitely know, or be certain, that A or that B are true or false, or be uncertain to some extent of A or of B. Our participants were in these epistemic states. They could see clearly the shape or colour of a chip, or they could be uncertain of the shape or of the colour. The third value is not, for de Finetti, on the same "level" as truth and falsity and standing between the two -he compared the uncertainty value in his tables to an entry of "sex unknown" in a statistical survey (de Finetti, 1936(de Finetti, /1995)). In this respect, he is radically different from some other logicians, who have viewed the third value as non-epistemic and objective in some sense. Łukasiewicz (1970), for example, appears to have thought of the future as undetermined in a non-epistemic sense. In his view, it is apparently not simply an epistemic limitation that prevents us from knowing whether it is true or false that London will flood in the next ten years (see [START_REF] Haack | Deviant Logic[END_REF], on his three-valued system, and compare her on Kleene, 1952, who seems to have had a view closer to de Finetti's). We did not, of course, investigate evaluations of future possibilities. In our materials, a chip has a definite shape and colour whether or not these properties are obscured by a filter. The filter introduces epistemic uncertainty into our experiment, and that makes the table of our participants' responses of the same epistemic type as that proposed by de Finetti.

Note also that the subjective value of uncertainty might be progressively divided into finer and finer epistemic distinctions, indicating more or less uncertainty, with the threevalued table becoming four-valued and so on (and interval-valued probability judgments would naturally go with these uncertainty judgments; see [START_REF] Pfeifer | The conditional in mental probability logic[END_REF]. Our study could be developed by making the random chip more or less likely to be square or round and black or white, by having it selected from a frequency distribution. Eventually, the "uncertain" value for if S then B could be replaced by the conditional probability P(B|S) itself [START_REF] Edgington | On conditionals[END_REF][START_REF] Jeffrey | Matter of fact conditionals[END_REF]. The resulting table would go with de Finetti's full manyvalued logic of probability (Finetti (1936(Finetti ( /1995)), and some anomalies of the restricted threevalue system could be avoided. For instance, in the three-valued system, a logically true conditional like if S & B then B will be uncertain by the 3x3 de Finetti Distinguishing different degrees of uncertainty and probability in tables should lead to an investigation of neglected topics in the psychology of reasoning. For example, people will sometimes make abductive inferences (Peirce, 1931(Peirce, -1958) ) to a degree of confidence in A as an explanation of C when they have some degrees of confidence in C and if A then C.

Observing that London is flooded, people might increase their confidence that global warming has continued, because they have high confidence that this explains the flood. The old paradigm generally dismissed this form of reasoning as the fallacy of affirming the consequent, but it can sometimes be a justified inference according to the new paradigm, especially when it is part of dynamic Bayesian belief revision (Oaksford & Chater, 2007, this issue). Abduction can be studied in experiments in which a marginal degree of confidence in A in a table is inferred from confidence in if A then C and C.

Research will also have to be done on the psychological representation and processing of uncertainty. There are some new paradigm hypotheses that radically extend old paradigm mental model and mental logic theories to epistemic or probabilistic accounts (Evans, 2007;[START_REF] Evans | [END_REF][START_REF] Oaksford | Conditional inference and constraint satisfaction: Reconciling mental model theory and the probabilistic approach[END_REF][START_REF] Pfeifer | The conditional in mental probability logic[END_REF]. However, these hypotheses will have to be tested by going far beyond the limitations of traditional studies of the "defective" truth table in the old paradigm to experiments on people's judgments under uncertainty. Two children, Pierre and Marie are playing to make chips fall.

Marie chooses chips at random and drops them one by one, out of Pierre's sight.

Marie chooses one chip at random and drops it. Pierre who did seen the chip being drop says:

Pierre has to guess the shape and colour of every chip.

Indicative conditional condition (IC)

"If the chip is square, then it is black"

Pierre's statement may be true or false. For you to decide, you will have at your disposal photographs taken as just explained.

For each of the following photographs, you will indicate whether Pierre's statement is true or false by selecting the option that coincides with your response.

Conditional Bet condition (BC)

"I bet that if the chip is square, then it is black"

Pierre's bet may be won or lost. For you to decide, you will have at your disposal photographs taken as just explained.

For each of the following photographs, you will indicate whether Pierre's bet is won or lost by selecting the option that coincides with your response.

  if A then C, where A and C are only classified as true or false, the four cells are often labelled with letters. The a cell is where A and C are true, the b cell where A is true and C is false, the c cell where A is false and C is true, and the d cell where A is false and C is false. In the 3x3 table, there are five extra cells. Consider the extra cell between the a and b cells, in which A is true but C is uncertain. We will call this the a/b cell, because the a cell or alternatively the b cell will result once the uncertainty about C is resolved, i.e. C is found to be certainly true or certainly false. There is also clearly the c/d cell, the a/c cell, and the b/d cell. At the centre of the table, there is the final extra cell, the a/b/c/d cell, when both A and C are uncertain. See Table

Figure 1 .Figure 3 .

 13 Figure 1. The game presentation, with the different possible photographs

Figure 4 .

 4 Figure 4. Example for the situation in which the antecedent is uncertain and the consequent

  table supposing S & B is uncertain, but of course P(B|S & B) = 1, and if S & B then B is never intuitively uncertain. Such oddities do not arise supposing the value of if S & B then B is P(B|S & B) when S & B is uncertain.

Table 1 .

 1 The 2x2 de Finetti, "defective" table for if A then C: T = true, F = false, V = void.

			C
			T	F
		T	T	F
	A	F	V	V

Table 2 .

 2 The 3x3 de Finetti table for if A then C: T = true, F = false, U = uncertain.

				C
			T	U	F
		T	T	U	F
	A	U	U	U	U
		F	U	U	U

Table 3 .

 3 The cells of the 3x3 table.

				C
			T	U	F
		T	a	a/b	b
	A	U	a/c	a/b/c/d	b/d
		F	c	c/d	d

Table 4 .

 4 The tables produced considering only two values for A and C -the first stage of our analysis -and the percentage of their occurrence (N = 192).

					IC	BC	All
	Conditional event	38.6	45.1	41.7
	C|A T	U	F
	T	T		F
	U
	F	U		U
	Conjunction	35.6	34.1	34.9
	A∧C	T	U	F
	T	T		F
	U
	F	F		F
	Material conditional	13.9	12.1	13
	A⊃C	T	U	F
	T	T		F
	U
	F	T		T
	Material bi-	5.9	1.1	3.6
	conditional
	A⇔C T	U	F
	T	T		F
	U
	F	F		T
	Bi-conditional	1	4.4	2.6
	C||A T	U	F
	T	T		F
	U
	F	F		U
		Others		5	3.3	4.2

Table 5 .

 5 Table answers (in %) in a three-valued framework (second stage analysis, N = 184*).

		Conditions		IC	BC	All
	Number of differences	0	1	Total	0	1 Total
						Conditional event	N = 80
		de Finetti	33.3 23.1 56.4 51.2 7.3 58.5	57.5
	C| F A T	U	F
	T	T	U	F
	U	U	U	U
	F	U U	U
			Farrell	17.9 2.6 20.5 12.2 4.9 17.1	18.8
	C| Fa A T	U	F
	T	T	U	F
	U	U	U	F
	F	U U	U
			Cooper	5.1 7.7 12.8 12.2 2.4 14.6	13.8
	C| C A	T	U	F
	T	T U	F
	U	T	U	F
	F	U U	U
	Ambiguous (Finetti-Cooper)	10.3		10.3 9.8	9.8	10
	C| a A T U F
	T	T U	F
	U	T U U
	F	U U	U
							Conjunction	N = 67
		de Finetti	47.2 5.6 52.8 41.9 9.7 51.6	52.2
	A∧ F C T	U	F
	T	T	U	F
	U	U	U	F
	F	F	F	F
			Porte	19.4 8.3 27.8 25.8 3.2 29	28.4
	A∧ P C T	U	F
	T	T	U	F
	U	T	U	F
	F	F	F	F
			Others			11.1	16.1	13.4
	Ambiguous + indeterminate			8.3	3.2	6
						Material conditional	N = 25
		de Finetti	42.9		42.9 72.7	72.7	56
	A⊃ F C T	U	F
	T	T	U	F
	U	T	U	U
	F	T	T	T
		McCarthy	35.7		35.7 9.1	9.1	24
	A⊃ Mc C T U	F
	T	T U	F
	U	U U	U
	F	T T	T
			Others			14.3	18.2	16
	Ambiguous + indeterminate			7.1	4

Financial support for this work was provided by the French ANR agency under grant ANR Chorus 2011 (project BTAFDOC). We should like to thank Paul Egré, Jean-Louis Stilgenbauer and Tatsuji Takahashi for much discussion and other help in our research.

Note

1 For each of the three interpretations, the table ranked second (Farrell's for the conditional event, Porte's for the conjunction, and McCarty's for the material conditional) differs from de Finetti's tables by just one cell. If these responses are regarded as de Finetti's tables with one error, de Finetti's tables exceed 75% of the responses.


Over, D Jean & Baratgin
email: jean.baratgin@univ-paris8.fr

G Politzer

New psychological paradigm for conditionals and general de Finetti tables

The new Bayesian paradigm in the psychology of reasoning aims to integrate the study of human reasoning, decision making, and rationality. It is supported by two findings. One, most people judge the probability of the indicative conditional, P(if A then B), to be the conditional probability, P(B|A), as implied by the Ramsey test. Two, they judge if A then B to be void when A is false. Their three-valued response table used to be called 'defective', but should be termed the de Finetti table. We show how to study general de Finetti truth tables for negations, conjunctions, disjunctions, and conditionals.

Introduction

Bayesian models of cognition have become increasingly prominent in the psychology of reasoning in recent years, and this trend has resulted in the development of a new Bayesian paradigm [START_REF] Manktelow | Paradigms shift: Jonathan Evans and the science of reason[END_REF][START_REF] Oaksford | Bayesian Rationality: The Probabilistic Approach to Human Reasoning[END_REF], 2009;[START_REF] Over | New paradigm psychology of reasoning[END_REF][START_REF] Pfeifer | The conditional in mental probability logic[END_REF]. The older binary paradigm in the psychology of reasoning focused on deduction from assumptions. Propositions were classified as either true or false, or consistent or inconsistent with each other. Conclusions either followed necessarily or not at all. If the participants in a reasoning experiment did not assume the premises, but relied on their subjectively relevant beliefs, or they inferred conclusions that did not necessarily follow, they were deemed to be biased and to commit fallacies. This older paradigm inspired many theoretical models (mental rules, mental models, rules or schemas, and others) but limited the field of research to the study of the human deduction ability [START_REF] Evans | Questions and challenges for the new psychology of reasoning[END_REF]. The new paradigm recognizes that most inferences in everyday affairs and science, whether logically valid or not, are from uncertain beliefs or hypotheses and not assumptions, and that reasoning must take account of this uncertainty if it is to be useful for acquiring rational beliefs and making rational decisions. The basic goal of the new paradigm is to integrate the psychology of reasoning and the psychology of judgment and decision making and provide a full account of human reasoning, decision making, and rationality.

Consider the valid inference of Modus Ponens (MP), inferring B from if A then B and

A. Let if A then B here be:

(1)

If Italy leaves the Euro this year (I) then the Euro will decline in value (D).

We cannot make a rational decision about what to do with our Euros by simply assuming that if I then D and that I, in order to infer D by MP, that the Euro will decline in value. We must at least assess our degree of confidence in (1), P(if I then D), and the prediction that I, P(I).

We can then try to have a rational degree of belief in the conclusion of MP, P(D), for our decision making. There was a significant step towards the new paradigm when experiments showed that people's confidence in the premises of conditional inferences does affect their confidence in the conclusions in systematic ways [START_REF] George | The endorsement of the premises: assumption based or belief-based reasoning[END_REF][START_REF] George | Reasoning from uncertain premises[END_REF][START_REF] Politzer | Uncertainty and the suppression of inferences[END_REF][START_REF] Politzer | Deductive reasoning from uncertain conditionals[END_REF][START_REF] Stevenson | Deduction from uncertain premises[END_REF], 2001).

The Equation And Restricted De Finetti Table

The new paradigm was given an even greater boost by the finding in experiments that people judge the subjective probability of the natural language indicative conditional, P(if A then B), to be the conditional probability of B given A, P(B|A). Philosophers have long argued for this relation between the conditional and conditional probability, P(if A then B) = P(B|A). It has been seen as so important normatively that it has simply been called the Equation [START_REF] Douven | The probabilities of conditionals revisited[END_REF]. Logical and philosophical support for it can be traced back to [START_REF] Ramsey | Truth and probability[END_REF]/1990[START_REF] De Finetti | The logic of probability. Translated in R. B. Angell, The logic of probability[END_REF]/1995[START_REF] De Finetti | Foresight: Its logical laws, its subjective sources[END_REF][START_REF] De Finetti | Foresight: Its logical laws, its subjective sources[END_REF], the founders of contemporary subjective probability theory [START_REF] Politzer | Betting on conditionals[END_REF]. The Equation can be studied in probability tasks as the conditional probability hypothesis and has been confirmed for a very wide range of conditionals [START_REF] Douven | The Adams family[END_REF][START_REF] Evans | Thinking about conditionals: A study of individual differences[END_REF][START_REF] Evans | Conditionals and conditional probability[END_REF][START_REF] Fugard | How people interpret conditionals: Shifts towards the conditional event[END_REF][START_REF] Oberauer | The meaning(s) of conditionals: Conditional probabilities, mental models and personal utilities[END_REF][START_REF] Politzer | Betting on conditionals[END_REF], including examples like (1) that could be called 'causal' conditionals [START_REF] Over | The probability of causal conditionals[END_REF].

Judging that P(if A then B) = P(B|A) is the majority response in these experiments. This strong confirmation of the conditional probability hypothesis provides substantial empirical support for a Bayesian account of conditional reasoning. The indicative conditional can be represented by the conditional probability and rational degrees of belief in conclusions of conditional inferences can be calculated from subjective probabilities for the premises [START_REF] Adams | A Primer of Probability Logic[END_REF][START_REF] Gilio | The psychology of inferring conditionals from disjunctions and its probabilistic analysis[END_REF][START_REF] Hahn | The rationality of informal argumentation: A Bayesian approach to reasoning fallacies[END_REF][START_REF] Oaksford | Bayesian Rationality: The Probabilistic Approach to Human Reasoning[END_REF], 2009;[START_REF] Over | Conditionals and non-constructive reasoning[END_REF][START_REF] Pfeifer | The conditional in mental probability logic[END_REF]. A Bayesian account can also be applied to what have been called utility conditionals [START_REF] Bonnefon | A theory of utility conditionals: Paralogical reasoning from decisiontheoretic leakage[END_REF].

An important point to note is that a significant number of participants on some probability tasks give the conjunctive response, P(if A then B) = P(A & B), and not the majority conditional probability response, P(if A then B) = P(B|A). It has been found that participants who give the conjunctive response are of lower cognitive ability than the participants who respond with the conditional probability [START_REF] Evans | Thinking about conditionals: A study of individual differences[END_REF], 2010;[START_REF] Politzer | Betting on conditionals[END_REF]. There is also a developmental trend: young children respond with the conjunctive probability, but this response tends to decline with age, and the conditional probability response increases until it becomes the majority response in undergraduate students [START_REF] Gauffroy | Heuristic and analytic processes in mental models for conditionals: An integrative developmental theory[END_REF]. The most recent finding, superseding earlier work, is that the conjunctive response tends to decline and be replaced by the conditional probability response as more probability tasks are done [START_REF] Fugard | How people interpret conditionals: Shifts towards the conditional event[END_REF]. It is hard to see how people can make rational decisions if they think that P(if A then B) = P(A & B), but yet another point about this response is that it does not occur significantly for the most realistic conditionals like (1) and, for an example actually studied, 'If global warming continues then London will be flooded' [START_REF] Douven | The Adams family[END_REF][START_REF] Over | The probability of causal conditionals[END_REF].

More research is needed to understand fully, not only on the conjunctive response for conditionals, but also on how uncertainty affects judgments about conjunctions, disjunctions, negations, and conditionals more generally.

Another strong body of support for the new paradigm comes from experiments on truth table tasks and the discovery of what came to be called the defective truth table in psychology (see [START_REF] Evans | If[END_REF], for review up to that date). If (1) were a material conditional (as in much of the older paradigm), (1) would be equivalent to not-I or D and would be true when I & D is true, false when I & not-D is true, and true for the two false antecedent, not-I cases, when not-I & D is true and when not-I & not-D is true. However, participants in truth table task experiments diverged from this table in the false antecedent cases. They would say that these cases were 'irrelevant' for determining the truth or falsity of (1). More recently, participants have responded that the conditional is neither true nor false in these cases [START_REF] Politzer | Betting on conditionals[END_REF]. From a new paradigm point of view, these participants are not producing a 'defective' truth table in any negative sense. Psychologists in the old paradigm were apparently unaware that de [START_REF] De Finetti | The logic of probability. Translated in R. B. Angell, The logic of probability[END_REF][START_REF] De Finetti | The logic of probability. Translated in R. B. Angell, The logic of probability[END_REF][START_REF] De Finetti | Foresight: Its logical laws, its subjective sources[END_REF][START_REF] De Finetti | Foresight: Its logical laws, its subjective sources[END_REF] [START_REF] Over | The probability of causal conditionals[END_REF] the relation between counterfactuals and the conditional probability).

The finding that people judge that P(if A then B) = P(B|A) raises the question of how they process conditionals to produce this result. Most psychologists in the new paradigm again follow philosophers in holding that people use the Ramsey test to evaluate conditionals [START_REF] Ramsey | Truth and probability[END_REF](Ramsey, /1990). As modified by later philosophers, the Ramsey test is a process in which people assess P(if A then B) by hypothetically supposing A, making whatever changes are necessary to preserve consistency, and then judging the probability of B given this supposition [START_REF] Douven | The probabilities of conditionals revisited[END_REF][START_REF] Evans | If[END_REF][START_REF] Stalnaker | A theory of Conditional[END_REF]. The result of this process is that P(if A then B) is the probability of B given A, P(B|A). A conditional if A then B with the property that P(if A then B) = P(B|A) has been called a probability conditional [START_REF] Adams | A Primer of Probability Logic[END_REF][START_REF] Oaksford | Bayesian Rationality: The Probabilistic Approach to Human Reasoning[END_REF] and the conditional event [START_REF] De Finetti | The logic of probability. Translated in R. B. Angell, The logic of probability[END_REF][START_REF] De Finetti | The logic of probability. Translated in R. B. Angell, The logic of probability[END_REF][START_REF] De Finetti | Foresight: Its logical laws, its subjective sources[END_REF][START_REF] De Finetti | Foresight: Its logical laws, its subjective sources[END_REF][START_REF] Pfeifer | The conditional in mental probability logic[END_REF]. We will use both of these expressions in this paper for this conditional, but will favour 'conditional event', symbolized by de Finetti as B|A, later in the paper when we are focusing primarily on de Finetti's proposals.

For an account of the natural language indicative conditional, the probability conditional has many advantages over the material, truth functional conditional, logically and psychologically. In particular, the probability conditional does not suffer from the 'paradoxes' of the truth functional analysis of the natural language conditional [START_REF] Douven | The probabilities of conditionals revisited[END_REF][START_REF] Evans | If[END_REF][START_REF] Oaksford | Bayesian Rationality: The Probabilistic Approach to Human Reasoning[END_REF][START_REF] Pfeifer | The conditional in mental probability logic[END_REF] The probability conditional, conditional event also has an advantage over other nontruth functional theories of the conditional, such as that of [START_REF] Stalnaker | A theory of Conditional[END_REF]. [START_REF] Lewis | Probabilities of conditionals and conditional probabilities[END_REF] proved that 'triviality results' follow from assuming that P(if A then B) = P(B|A) for conditionals like Stalnaker's, which are true or false at every possibility. His proof does not apply to the conditional event, which is void and has a truth value gap when its antecedent is false (a fact that Lewis was well aware of; see [START_REF] Mura | Probability and the logic of de Finetti's trievents[END_REF][START_REF] Paneni | Multi-valued conditional events avoid Lewis' triviality result[END_REF][START_REF] Paneni | Multi-valued conditional events avoid Lewis' triviality result[END_REF] as well as [START_REF] Lewis | Probabilities of conditionals and conditional probabilities[END_REF]. And as we have reported above, the evidence is strong, from psychological results on the 'defective', de Finetti table, that people consider indicative conditionals neither true nor false when their antecedents are false. Nevertheless, it is also theoretically possible to avoid the triviality results by holding that people's interpretations of conditionals depend on their belief states (van Fraassen, 1976). This theoretical position does not imply that an underlying binary view has to be given up and has other possible advantages [START_REF] Douven | A puzzle about Stalnaker's hypothesis[END_REF] (1929/1990) and de [START_REF] De Finetti | Foresight: Its logical laws, its subjective sources[END_REF][START_REF] De Finetti | Foresight: Its logical laws, its subjective sources[END_REF][START_REF] De Finetti | Theory of Probability[END_REF] justified their normative theories of subjective probability by referring to bets that were to represent degrees of beliefs. Conditional bets, representing conditional degrees of belief, were central to their theories. The fundamental result is that people who violate normative probability theory, and so are incoherent, are irrational because they are vulnerable to a Dutch book: a series of bets that can only be lost [START_REF] De Finetti | Theory of Probability[END_REF][START_REF] Gilio | The psychology of inferring conditionals from disjunctions and its probabilistic analysis[END_REF]. A conditional bet can be seen as a bet on the conditional event, and the new paradigm implies that there should be a close relation between a degree of belief in an indicative like (1) and a willing to express a conditional bet like:

(2) I bet that, if Italy leaves the Euro this year (I), then the Euro will decline in value (D).

There is clearly a parallel relation between the three-valued de Finetti truth table for (1) and the de Finetti winning table for (2). By these tables, (1) is true and the bet ( 2 = P(D|I), since there is no assertion to be true and no bet to be won in the not-I cases (for a theoretical analysis of logico-algebraic structure see [START_REF] Milne | Bruno de Finetti and the logic of conditional events[END_REF]. [START_REF] Politzer | Betting on conditionals[END_REF] have found evidence that people conform to this parallel relationship for a simple indicative conditional and a conditional bet (see also [START_REF] Oberauer | The meaning(s) of conditionals: Conditional probabilities, mental models and personal utilities[END_REF], for the only other psychological paper on bets about conditionals).

General De Finetti Tables

There is, however, a serious limitation in all the experiments that have been run so far on the 'defective' truth table. The participants have always been given a conditional, if A then B, and then asked about its evaluation for one of the four possible cases,

A & B, A & not-B, not-A & B,
and not-A & not-B, which has always been described or displayed clearly and without any uncertainty. For example, if such an experiment were actually run about ( 1) and ( 2), the participants might be told, say, that Italy was not going to leave the Euro by the end of the year and the Euro was not going to decline in value, a not-I & not-D case, and then asked how ( 1) and ( 2) were to be evaluated. The answer would of course be (on past evidence) that ( 1) is neither true nor false and (2) neither won nor lost. But the point is that it is unrealistic, and in considerable tension with the whole new paradigm approach, to suppose that we are often absolutely certain of which possible case holds for the evaluation of a conditional. We are much more often in a state of uncertainty about such matters as whether or not Italy will leave the Euro by the end of the year I and whether or not the Euro will decline in value D.

We are consequently uncertain about (1) and have a degree of belief P(D|I) about it that is less than 1. The new paradigm is all about uncertainty in our reasoning and how this affects rational inference and decision making. General truth table tasks should therefore be developed that reflect all this uncertainty.

There is fortunately help in the normative literature for a psychological project of developing more realistic truth table tasks. As we have already noted, de [START_REF] De Finetti | The logic of probability. Translated in R. B. Angell, The logic of probability[END_REF][START_REF] De Finetti | The logic of probability. Translated in R. B. Angell, The logic of probability[END_REF] constructed general tables for negation, conjunction, and disjunction, and the conditional event, and in these tables, the two basic propositions, A and B, can be true, false, or have a third value void. For these general tables, the third value can be interpreted as uncertainty in a wide sense and not just as a 'void' sense of neither true nor false. In this interpretation, the general tables for negation, conjunction, disjunction and material conditional correspond to strong Kleene connectives [START_REF] Haack | Deviant logic[END_REF]. The de Finetti general tables were actually proposed by [START_REF] Blamey | Partial logic[END_REF] in his semantic analysis of partial logic (see for a discussion [START_REF] Mura | Probability and the logic of de Finetti's trievents[END_REF], although Blamey did not refer to de Finetti. The de Finetti conditional table also corresponds to the table for the presupposition operator proposed by [START_REF] Beaver | Presupposition[END_REF]. If 'uncertain' means 'unknown', then it is possible that binary values underlie it (and so the approach of Douven &[START_REF] Douven | A puzzle about Stalnaker's hypothesis[END_REF]Verbrugge, under review, andvan Fraassen, 1976, could be taken).

We will now outline one way to develop research on these general de Finetti tables.

Consider an extension of the Politzer et al. ( 2010) study. They gave participants an indicative conditional and a conditional bet about a chip selected at random from an array of square and round chips that were black or white:

(3) If the chip is square (S) then it is black (B).

(4) I bet you 1 Euro if the chip is square (S) then it is black (B).

The participants could then be asked whether the selection of, say, a round and black chip, not-S & B, made (3) true, false, or neither true nor false, and meant that (4) was won, lost, or neither won nor lost. There was no doubt or uncertainty in this experiment about whether the randomly selected chip was square or black. But consider how some uncertainty could be introduced. As well as letting the participants look carefully at the randomly selected chip in good light, there could also be the possibility of letting the participants only get a glimpse of the chip or seeing it in bad light. Now the two basic propositions, here S and B, could be true, false, or have a third value of uncertainty. Suppose the participants were able to see the chip for long enough to be sure that it was square, but the light was so bad that they were uncertain whether it was black or white. With S true and B uncertain, how would the participants evaluate (3), as true, false, or uncertain? If they conformed to the general de Finetti fall from the top box to the bottom box, and then a photo would be taken of these chips in the space between the boxes. The photos were sometimes taken through filters, which could have the effect of concealing the shape or colour of the chips, or both the shape and the colour. The participants were told that children were playing a game about the chips that fell from the top box to the bottom box. One of the children made an assertion about the random chips. We will report here data on the cases in which the child's assertion was an indicative conditional,

(3), a conditional bet, (4), or a negation, not-S, conjunction, S & B, or disjunction, S or B. For example, if the child asserted (3) or (4), and the filter allowed the photo to show a square chip without revealing its colour, then we would have a case like the one that we have described above, when a square chip could be seen but bad light concealed the colour.

Thus in our general truth table task, participants were presented with nine possible photos -S and B true, false, or uncertain -and were asked to say in each case whether an assertion (or conditional bet) by the child was true (won), false (lost), or neither true (won)

nor false (lost). The participants' responses allowed us to construct tables for them for the five different assertion forms that we are considering here -negation, conjunction, disjunction, the indicative conditional, and the conditional bet. The main finding was that a large majority of participants complied with the general de Finetti tables. Most participants responded with the conditional event table for the child's assertion of the indicative conditional and also of the conditional bet (with T interpreted as winning, F as losing, and U as neither winning nor losing). For the child's assertion of a negation, not-S, almost all participants produced the de Finetti table for negation. Similarly, for child's assertion of a conjunction, S & B, slightly more than three quarters of the participants responded with de Finetti's table. This proportion increased to near totality when the answers that differed from de Finetti's conjunction table by a single cell were included (that is, when a single error out of nine answers is allowed). Finally, for the child's assertion of a disjunction, S or B, a large majority of participants conformed to de Finetti's table.

Conclusion

Research on de Finetti's general three-valued tables, which allow for uncertainty in the components of conditionals, negations, conjunctions, and disjunctions, has been totally neglected in the psychology of reasoning, but we hope to have indicated here how to overcome this limitation. Of course, we are well aware that many three-value tables have been proposed by other logicians, philosophers, and AI researchers (see the Appendix of [START_REF] Haack | Deviant logic[END_REF], for a good display of a wide range of tables in the literature up to that date and [START_REF] Goodman | Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning[END_REF], for more recent tables). Milne (in press) also points out that logical validity can be defined in terms of probability, as suggested by the Equation [START_REF] Douven | The probabilities of conditionals revisited[END_REF][START_REF] Adams | A Primer of Probability Logic[END_REF], or directly in terms of preserving values in de Finetti or other tables, and that these two types of definition validate different patterns of conditional inference. More research will be needed to determine the extent to which people conform to any general tables, and to discover which conditional inferences they do endorse. Our object here has been to argue that this research should be done and to explain one way to do it.

The experimental pillars supporting the new paradigm in the psychology of reasoning are two findings for the indicative conditional, if A then B, of natural language. First, that people comply with the Equation, by judging P(if A then B) = P(B|A), which is implied by the Ramsey test. Second, that people conform to the restricted de Finetti table for if A then B.

However, the new paradigm is in the early stages of its development, and there is much to do to investigate and extend these findings. In this paper, we have shown how to extend research to general de Finetti tables and their interpretation, and have argued for the importance of taking this step.

  ) is won when I & D is true, (1) is false and (2) is lost when I & not-D is true, and (1) is void and neither true nor false, and (2) is void and neither won nor lost in the not-I cases. Moreover, the probability that assertion (1) is true and the probability of winning the bet (2) should both be P(I & D|I)

  ; Douven & Verbrugge, under review). It could be argued, however, that the de Finetti table is a better explanation of 'defective' responses in truth table tasks. but for negation, conjunction, and disjunction. See Appendix 1. In these tables, A and B can receive a third value. We will explain later in this paper how to give people general truth table tasks, which have a third value for A and B, and to study for the first time whether these match de Finetti's more general tables.Another much neglected topic in the new paradigm, which is related to both the Equation and the de Finetti table for the conditional, is the conditional bet. Both Ramsey

There is an important point of agreement among Adams, de Finetti,

Lewis, Stalnaker, and van

Fraassen. They all hold that if A then B is true when A & B is true. No epistemic or other connection is required between A and B as a matter of semantics, although such a connection could be a reasonable pragmatic expectation. Some psychological experiments are relevant to this position, but it is still unclear whether people consider if A then B to be false, or only pragmatically infelicitous, when there is no connection between A and B and so A does not raise the probability of B

[START_REF] Over | The probability of causal conditionals[END_REF] Douven & Verbrugge, in press

).

The 'defective', de Finetti table has received far less attention than the Equation in the new paradigm (but see

[START_REF] Evans | Thinking about conditionals: A study of individual differences[END_REF]

the close relation between conditional probability judgments for the conditional, 'defective' responses on truth table tasks, and relatively high cognitive ability, and also

[START_REF] Sevenants | Truth table tasks: Irrelevance and cognitive ability[END_REF]

, on truth table tasks and cognitive ability). Given the negative connotations of the term 'defective', it is better to call this three-valued table simply the de Finetti table

[START_REF] Politzer | Betting on conditionals[END_REF]

. In fact, we should use restricted de Finetti table for the 'defective' truth table, since de Finetti

(1936/1995)

actually proposed what we will call general de Finetti tables, not only for the conditional event,

  table for the conditional event, they would evaluate (3) as uncertain in this case. If they also conformed, in this case, to the general de Finetti tables for negation, conjunction, and disjunction, they would evaluate not-B as uncertain, S & B as uncertain, but S or B as true.Seeing chips more or less quickly in good or bad light might be a reasonable model of some natural circumstances. It would, however, be too hard to construct and control an experiment of exactly this type. What we did was to create a computer representation of two opaque boxes, one on top of the other, with some open and clear space between them. The top box had square or round chips in it that were black or white. Random chips could at times

	And they would have parallel evaluations in terms of winning, losing, and uncertainty about
	(4) and about bets on not-S, S & B, and S or B.

Stevenson, R. and D. Over 2001: Reasoning from Uncertain Premises: The Effects of Expertise and Conversational Context. Thinking and Reasoning, 7, 367-390.van Fraassen, B.C. 1976: Probabilities of conditionals. In W.L. Harper and C.A. Hooker (eds.), Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science (vol.1, pp. 261-301), Dordrecht: Reidel.

Financial support for this work was provided by the French ANR agency under grant ANR Chorus 2011 (project BTAFDOC). We should also like to thank Thomas Charreau and Jean-Louis Stilgenbauer for much discussion and other help in our research,

Appendix 1

The three-valued tables of [START_REF] De Finetti | The logic of probability. Translated in R. B. Angell, The logic of probability[END_REF][START_REF] De Finetti | The logic of probability. Translated in R. B. Angell, The logic of probability[END_REF]), but we have used U here for 'uncertain'.