Bias and Modality in Conditionals: Experimental Evidence and Theoretical Implications

The concept of bias is familiar to linguists primarily from the literature on questions. Following the work of Giannakidou and Mari (Truth and Veridicality in Grammar and Thought: Modality, Mood, and Propositional Attitudes, University of Chicago Press, Chicago, 2021), we assume “nonveridical equilibrium” (implying that p and ¬p as equal possibilities) to be the default for epistemic modals, questions and conditionals. The equilibrium of conditionals, as that of questions, can be manipulated to produce bias (i.e., reduced or higher speaker commitment). In this paper, we focus on three kinds of modal elements in German that create bias in conditionals and questions: the adverb wirklich ‘really’, the modal verb sollte ‘should’, and conditional connectives such as falls ‘if/in case’. We conducted two experiments collecting participants’ inference about speaker commitment in different manipulations, Experiment 1 on sollte/wirklich in ob-questions and wenn-conditionals, and Experiment 2 on sollte/wirklich in wenn/falls/V1-conditionals. Our findings are that both ob-questions and falls-conditionals express reduced speaker commitment about the modified (antecedent) proposition in comparison to wenn-conditionals, which did not differ from V1-conditionals. In addition, sollte/wirklich in the antecedent of conditionals both create negative bias about the antecedent proposition. Our studies are among the first that deal with bias in conditionals (in comparison to questions) and contribute to furthering our understanding of bias.


3
These questions are negatively biased: the speaker disbelieves that the addressee has spoken to Mary or that Mary helped. In both cases, bias arises arguably because the speaker decides to use not a simple unadorned question, but to augment the basic option with the use of particular devices (or even mere intonation such as with rising declarative questions, falling interrogatives, etc.).
Crucially, we talk about speaker bias rather than question bias because bias is rooted in the speaker's choice to go beyond the equilibrium information-seeking mode. 1 Bias reflects the intrusion of the speaker's assumptions and expectations (i.e., for the audience to agree with them), and is, of course, optional: the speaker doesn't have to choose the biased option, see (1) vs. (2), or (4) vs. the sentences without NPIs. They do so because they have reasons to be more committed to the positive proposition (positive bias) or the negative one (negative bias). Bias, then, encodes the speaker's doxastic and epistemic commitments that force them to abandon neutrality (presupposing, of course, that they are sincere). In the case of positive bias, the speaker seems to be more committed to p, and in the negative bias less committed to p. Speaker commitment 2 is a notion we borrow from the modality vocabulary of Mari (2018a, b, 2021a, b)'s veridicality theory that we discuss in Sect. 2, and is a handy way to talk about the speaker's attitude towards the veridicality of a sentence.
The same manipulation of nonveridical equilibrium by speaker commitment is observed with conditionals, and it involves the use of modal devices such as tense and mood choices (see a.o., Iatridou, 1991;Fintel, 1999Fintel, , 2007Fintel, , 2011Arregui, 2005), evaluative adverbs and discourse particles (Grosz, 2012;Liu, 2012) and NPIs (Liu, 2019(Liu, , 2021. A hitherto less studied area concerns the choice of conditional connectives (CCs), see e.g., Ippolito & Su (2014) and Jiang (2019) on the Mandarin counterfactual CC yaobushi 'if-not', Hoeksema (2012) on unless, and Reis & Wöllstein (2010) and Liu (2019Liu ( , 2021 on the German CCs wenn vs. falls. it outside rains stays Susanne at home 'If it is raining outside, Susanne will stay at home.' 1 There is also contextual bias, or bias relating to evidence available in the context ('evidential bias', Büring & Gunlogson, 2000;Romero & Han, 2004;Sudo, 2013;Northrup, 2014;Domaneschi et al., 2017), or 'answer bias' (Krifka, 2015;Malamud & Stephenson, 2015). We assume that such 'biases' are part of the belief basis of the speaker upon asking the question, i.e., the speaker is biased because contextual evidence allows it. We will not worry about how belief is formed (see Giannakidou & Mari, 2021a, 2021b for recent discussion on belief formation). 2 The Giannakidou and Mari notion of commitment we use is subjective, i.e., it represents the set of propositions held by individual linguistic agents (e.g., the speaker, the subject of the sentence in embedding which Giannakidou (1997) and Giannakidou & Mari call individual anchors, recycling a term from Farkas, 1992), it therefore differs from Krifka's use of 'commitment state' which is "modelled as a set of propositions, containing the propositions that are publicly shared by the participants [emphasis ours]" (Krifka, 2015: 328-329). Krifka's commitment corresponds to common ground assumptions, and the goal of speech acts is to "change a commitment state". Speaker commitment in our view can be entirely private, even solipsistic (Giannakidou & Mari, 2016, 2018a, 2018b, 2021a, 2021b, as is the case with verbs of belief (but also imagination, and dreaming).
According to Liu (2019), the at-issue content of (5) is that of the regular conditional in both cases, but falls adds to the sentence a non-at issue content that the speaker takes it to be unlikely or does not take it to be likely that it is raining outside. 3 Liu continues that falls expresses a weaker speaker commitment towards the antecedent than wenn and indicates that the speaker does not take p as likely. Liu (2021) discusses a number of diagnostics to make negative bias clear, of which we will mention one: wenn but not falls can be used in factual (premise) conditionals, as shown in (6) Falls cannot be used if it is known that the antecedent is true or if the speaker intends to accommodate the antecedent proposition; rather, the choice of using falls instead of wenn conveys a negative bias. The falls vs. wenn contrast is observed in other languages too (such as Italian or Chinese, which we elaborate on later), and is correlated with the fact that many CCs have a modal source, e.g., the Modern Greek an/ean 'if' (Chatzopoulou, 2019(Chatzopoulou, , 2021, which starts out as a modal particle in Homeric and Classical Greek.
In conditionals, modal verbs and modal adverbs can also be used. As shown in the naturally-occurring examples across three different languages (English, German and Chinese) in (7), the presence of modal verbs should/sollte/yao and the adverb really/wirklich/zhende seems to convey a lower-than-equilibrium degree of speaker commitment towards the antecedent proposition, in comparison to their unmodalized variants such as If it comes to this, then farewell, humanity! in (7a). In this paper, we will focus on the interpretive effects of modal verbs, the adverb really and CCs in conditionals. For German, Reis & Wöllstein (2010: 137) have also noted the following relation between the modal verb sollte and the CC falls: "But what we can already say now is that sollte does roughly the same thing as falls: falls is also strictly limited to hypothetical conditionals and excludes counterfactual use 8 ; falls-conditionals are often also marked with sollte (Zifonun et al., 1997(Zifonun et al., : 2281. In particular, falls, via its literal meaning 'in the case that …', refers to the possibility of the realization of the antecedent proposition, just as sollte does." 9 The CC then is also understood, as we are suggesting, in modal terms. While the intuition is clear that with modal modifications the equilibrium is manipulated by the speaker to reveal that they consider p less likely (or they are less certain about p, which we will get back to when we discuss the nature of the bias), these facts have hardly been tested experimentally in order to gain clarity about the size of the effects and interactions. We set out to examine the effect of modal elements in creating bias in this paper. Is the effect observable? How does negative bias arise? What is the contribution of the modal verbs and what is the contribution of the adverbs? How do they interact with each other and with conditionals?
The present paper is an attempt to address these questions and to draw some preliminary conclusions that can guide further research. The modality strategies observed in Germanwhich we focus on from now on-make it necessary to assume that there is a modal structure in the conditional, following Giannakidou (2021) (see Kaufmann, 2005 for an earlier analysis to that end), and that this modal structure is responsible for the bias. In Sect. 2, we outline the framework of modality by Giannakidou & Mari (2018a, b), which we use as the basis for our analysis, and define some predictions made by the system. We proceed with our two experiments in Sects. 3 and 4, followed by discussion and conclusion in Sects. 5 and 6.

Formal Aspects and Experimental Hypotheses of Bias and Modality
Modal expressions in natural languages are the common devices to reflect the speaker's epistemic or doxastic stance towards the truth of a proposition. Almost all analyses of modality assume that non-alethic modal expressions are nonveridical (Kratzer, 1977(Kratzer, , 1991Giannakidou, 1997Giannakidou, , 1998Giannakidou, , 1999Giannakidou, , 2013Condoravdi, 2002;Portner, 2009;Beaver & Frazee, 2016, Giannakidou & Mari, 2016, 2018a, b, 2021a, b, pace von Fintel & Gillies, 2010, i.e., they do not entail that the proposition is true. The function of modal expressions is to convey the nonveridical attitude of the speaker: upon hearing or reading a modal sentence, the audience understands that the speaker cannot be fully, i.e., veridically, committed to the truth of a proposition. This is an important assumption for the experiments we will report in the following sections. For example, consider the following declarative sentences. (8a) uses the present tense, and (8b,c) contain the modal verbs (must and may/might): (8) a. It is raining.
b. It must be raining. c. It may/might be raining.
Let us call the tensed unmodalized sentences such as (8a) "bare". In semantics and pragmatics, we assume that in asserting a bare sentence the speaker is saying something that they know or believe to be true-they are, in other words, veridically committed to it. Giannakidou & Mari (2018a, b, 2021a; see also Giannakidou, 1998Giannakidou, , 2013 call this the veridicality principle of assertion (and it follows from abiding by Gricean Quality, which is fundamental to co-operative conversation). Thus, upon hearing an unmodalized sentence the hearer understands that the speaker knows, or has grounds to believe that it is raining or that it rained. On the other hand, when a speaker chooses to use a modal, they take a nonveridical stance, which means that the speaker is uncertain about whether it is raining or not-and this uncertainty is typical also of questions and conditionals. The uncertainty is in nonveridical equilibrium with possibility modals: with may or might as in (8c), raining is considered a mere possibility, and the speaker has no reason to believe It is raining is closer to what is the case than It is not raining. Possibility modals are thus very much like conditional protasis and information-seeking questions in this regard.
When a necessity modal such as must/should as in (8b) is used, the equilibrium is manipulated towards raining being considered more likely by the speaker. Giannakidou and Mari coin the term 'biased modals' to characterize necessity modals: the speaker is biased in favor of the prejacent proposition, though they still are not veridically committed to it. Modal bias reveals an epistemic stance supported by evidence in favor of the proposition, but it does not mean that the speaker knows p to be true. Modals remain indicators that the speaker reasons with uncertainty and that they leave both options, p and ¬p, open. But with biased modals, the speaker appears to be more committed to p than to ¬p, and this commitment is itself gradient (i.e., must expresses more bias toward p than should or ought to, see Portner, 2009 and Sode & Sugawara, 2019 on German sollte).

Formal Aspects
The gradience described above is sometimes captured by positing secondary ordering sources for the modals (Portner, 2009;Portner & Rubinstein, 2016). Giannakidou & Mari (2018a, b, 2021a argue instead that bias is an additional parameter of evaluation for all modals, and it comes in the form of a ranking metaevalution function O. It is not necessary for the purposes of this paper to go through the entire system, but let us offer the necessary details relevant to conditionals, which will motivate our experimental hypotheses in Sect. 2.2. MUST associates with an epistemic modal base M(i), where i is the speaker, which is partitioned by a stereotypical ordering S into Ideal and non-Ideal worlds. Ideal S is a function over M(i)(t u )(w 0 ), and the output Ideal S is a subset of M(i)(t u )(w 0 ): (9) M(i)(t u )(w 0 ) = λw'(w' is compatible with what is known by the speaker i in w 0 at t u ) (10) Ideal S (M(i)(t u )(w 0 )) = {w' ∈ M(i)(t u )(w 0 ): ∀q ∈ S (w' ∈ q)} So defined, Ideal S delivers the worlds in the modal base where all the propositions in S are true. S is a set of propositions that corresponds to common ground norms. The truth condition for MUST says that p is true in the Ideal set of M(i). We give here the truth condition with NONPAST which is the future-shifting tense found in conditionals: (11) Given a set Ideal S and the utterance time t u , [[must/sollen(NONPAST(p))]] M,i,S is defined only if M(i) is nonveridical and is partitioned into Ideal S and ¬Ideal S worlds. If defined, [[must/sollen(NONPAST(p) The MUST prejacent will be true in Ideal worlds at a future time that includes t u (t u , ∞). For more discussion on future shifting and the nonveridicality of MUST, see Giannakidou and Mari for details; for a related analysis of the modalized tense above, see Kaufmann (2005). The key observation here is that only in Ideal worlds is p true.
In addition, Ideal worlds are privileged by the ranking function O which is an evaluative function that ranks the Ideal worlds as better possibilities (in the sense of Portner, 2009 andKratzer, 1986) producing positive bias. According to Giannakidou and Mari, a modal adverb is typically the realization of O: When modal verbs and modal adverbs co-occur, such as in (13), the literature sometimes talks about 'modal concord' (Geurts & Huitink, 2006;Huitink, 2012Huitink, , 2014Grosz, 2010;a contrario Anand & Brasoveanu, 2010;Zeijlstra, 2007;Huitink, 2012;Lyons, 1977;Giannakidou & Mari, 2018b). Yalçın (2007: 994) claims that "iterating epistemic possibility operators add no value in the semantics". In comparison, Giannakidou & Mari (2018b), using the term "modal spread" hold the view that multiple expressions of modality have one semantic role and that the adverb presents the ordering source of the modal. In more detail, the epistemic modal structure involves three ingredients: (1) a nonveridical modal base M(i), (2) a secondary modal base S that partitions M(i) into Ideal S and ¬Ideal S subsets, relying on stereotypical assumptions, and (3) a meta-evaluation O triggered by stereotypicality that ranks the Ideal S worlds as better possibilities than ¬Ideal S worlds in M(i). The preference for higher ranking of Ideal S is lexically specified, and MUST and MIGHT differ in their lexical preferences (both use S, but higher ranking of Ideal S is only a feature of MUST). Giannakidou  REALLY contributes the definedness condition of O for better possibility, and we can think of it as a felicity condition attached to the speaker. Because it is a definedness condition of the speaker it can be objected to by the hearer who might not share the bias, and this captures the intuition that bias can be 'cancelled' or 'ignored', though it is in fact objected to by the hearer's commitments. Now the propositions are reversed: ¬p worlds are better possibilities, resulting in negative bias. We can understand the positivity in the condition (15-ii) for REALLY as a contextual bias, and the negative bias as the speaker bias, see (16). Both take widest scope, as they project out of conditionals, an entailment-cancelling context (Simons et al., 2010). (16) If Anne really becomes a lawyer, she will open a law firm.
a. contextual positive bias: Anne will become a lawyer. b. speaker negative bias: It is not the case that Anne will become a lawyer.
The effect of really is similar to its effect in questions as in (2c), where the speaker presupposes a contextual positive bias and conveys their negative bias. Giannakidou & Mari (2021b) further argue that the use of an epistemic modal verb in a question indicates increased uncertainty because ∩ O ⊃ M(i), i.e., extending the set of possibilities beyond the modal base (i.e., those worlds considered by the speaker), thereby making it harder to think of what would be a 'correct' answer. This accounts for why modal questions seem openended and with increased uncertainty. In conditionals, we have both REALLY and modal verbs creating bias-and there may be a detectable difference between increased uncertainty and negative bias, a point to which we return in the discussion of our experiments.
Regardless of the precise contribution of each, the fact is that O reverses the positivity of REALLY and MUST/sollte.
Does really reverse the positivity also in declarative sentences? Our intuition says that while its degree modifying use can be in the scope of negation, the bias use cannot, see (17). (17) a. It is really (not) raining. / It is *not really raining. (polarity-focusing really) b. It is (not) really raining. / It is really *not raining. (degree-modifying really) We think bias is observed also in declarative sentences. In a sentence such as Agnes really passed the exam, we propose that really (a) requires a negative contextual bias (e.g., the speaker's prior or a contextual salient agent's attitude) for ¬p over p possibilities, and (b) asserts p (creating a speaker positive bias). The requirement (a) is lacking in a sentence without really. The oddity of really following negation is probably due to scope conflicts between really and negation (Liu, 2012). Focusing on nonveridical contexts only, we assume that the negative bias is present not only in questions but also in conditionals, as we illustrated. 11 Now recall that in the conditional domain, as we showed, not only adverbs but also CCs can convey different commitments towards the antecedent proposition, including the speaker's doxastic, deontic or emotional evaluation towards the antecedent or the consequent. For example, Visconti (1996) claims that the Italian CC casomai 'if-ever' (made up of a simple CC caso 'in case, if' and a NPI mai 'ever') differs from nel caso che 'in the case that' in terms of the speaker's attitude towards the antecedent p that is expressed at the level of conventional implicatures: While nel caso che is epistemically neutral, casomai conveys a negative bias, namely, 'improbable(p)'. In more recent literature, Liu (2019Liu ( , 2021 as we mentioned in Sect. 1, have shown that the German CC falls expresses a weaker speaker commitment towards the antecedent proposition than wenn. In one of the reported experiments using the forced lexical choice task, the author found that the participants chose falls significantly more than wenn in the context of negative priors, e.g., when the protagonist does not believe the antecedent proposition, see (18) for an example of the used stimuli; a reverse pattern was found in the context of positive priors (see Liu & Wang, 2021 on Mandarin CCs in this regard). (18)  ______ it tomorrow rains stay I at home ('Kathi has taken tomorrow off. She {believes / doesn't believe} that it will rain tomorrow and is thinking: ______it rains tomorrow, I will stay at home.')

Experimental Hypotheses
In the above, we saw that certain CCs express a weaker speaker commitment than others. Modal verbs such as must/should in general express a positive bias (high degree of speaker commitment) towards the modified proposition p, i.e., a strong speaker commitment about p (but weaker speaker commitment than their unmodalized variant, which is veridical) in declarative sentences (veridicality principle); in nonveridical contexts with nonveridical equilibrium presupposed, we expect that the positive bias towards p weakens in comparison to variants without them. In conditionals, the adverb really expresses a weaker commitment than the variants without them as well.
In addition to modal verbs, adverbs, and CCs, word order variation within conditionals might also have an effect on distribution and semantics, see the discussion of verb-first (V1) conditionals in comparison to wenn/falls-conditionals and questions in German (Reis & Wöllstein, 2010). While we know about the wenn/falls contrast in this aspect based on Liu (2019Liu ( , 2021, it is unclear how they compare with V1-conditionals. In the rest of the paper, we will deal with multiple expressions of modality in (different kinds of) conditionals and questions in German, with a focus on the adverb wirklich 'really' and the modal verb sollte 'should'. With regard to their interpretive effects, the theoretical framework of veridicality and bias we have outlined allows us to formulate the hypotheses in (19): (19) Hypotheses of wirklich and sollte in conditionals and questions a. Hypothesis 1. Different sentence types and subtypes convey different degrees of speaker commitment about the modified (antecedent) proposition. b. Hypothesis 2. The use of the adverb wirklich indicates reduced speaker commitment about the modified (antecedent) proposition. c. Hypothesis 3. The use of the modal verb sollte indicates reduced speaker commitment about the modified (antecedent) proposition.
We implemented two experiments to test these hypotheses. The two rating experiments in German tested the interpretive effects of sentence types (conditionals vs. questions), conditional types (wenn vs. falls vs. V1-conditionals), modal verbs (sollte/würde) and the adverb wirklich 'really', which we report in Sects. 3 and 4. We collected subjects' inference about their belief whether the prejacent p is true in different manipulations, based on the assumption that the comprehender's inference of the speaker's assumptions reveals the meaning of the used modal expressions. We will discuss the results of both experiments and their implications as well as limitations in Sect. 5.

Experiment 1
In this experiment, we tested the speaker's doxastic (i.e., belief-based) commitment in relation to sentence types, the modal verb sollte and the adverb wirklich. We take the comprehender's belief judgment to be indicative of speaker (i.e., individual anchor) commitment, and a weaker commitment relates to negative bias. Differing degrees of speaker commitment do not affect the semantic (at-issue) content of the given question or conditional, since, as we said, bias is a precondition on the question or conditional. For example, the high negation such as in Isn't Agnes a vegetarian? in (2a) does not affect the question meaning of the sentence Is Agnes a vegetarian? but contributes a speaker meaning at a separate (non-at-issue) dimension.

Method
Participants 83 adult German native speakers (33 female, 50 male, mean age = 30.2, SD = 9.7) participated in the study. The experiment was approved by the ethics committee of the German Linguistic Society.

Design and Materials
Experiment 1 was based on a 2 × 2 × 2 factorial design, with three within-participants and within-items factors: SentenceType (wenn-conditional vs. ob-question) × Adverb (with or without wirklich 'really') × ModalVerb (with sollte/würde 'should/would' or without), see an example in (20). We controlled the stimuli in such a way that wirklich is used as a polarity-focusing adverb but not as a degree modifier. The reason why we did not use the same modal verb across all the conditions is that sollte in embedded questions only gets a deontic or bouletic reading, instead of the doxastic reading which we target. In total, we used 40 target items. Each item consisted of a context-setting sentence (S1), a critical sentence (S2), a critical question (S3) about the speaker's belief in the prejacent (i.e., the proposition in the conditional antecedent or the question), and a final comprehension question (S4) for attention check. The complete list of target items is included in Appendix 1. In addition, we used 48 filler items of similar structure. (20) S1: Paula denkt über ihre Zukunft nach. (Paula is thinking about her future.)

S2: a.
Sie denkt: "Wenn ich eine Arbeit finde, kaufe ich mir ein Mac Book." (She thinks to herself: 'If I find a job, I will buy myself a Mac Book'.) b.
Sie denkt: "Wenn ich wirklich eine Arbeit finde, kaufe ich mir ein Mac Book." (She thinks to herself: 'If I really find a job, I will buy myself a Mac Book'.) c.
Sie denkt: "Wenn ich eine Arbeit finden sollte, kaufe ich mir ein Mac Book." (She thinks to herself: 'If I should find a job, I will buy myself a Mac Book'.) d.
Sie denkt: "Wenn ich wirklich eine Arbeit finden sollte, kaufe ich mir ein Mac Book." (She thinks to herself: 'If I should really find a job, I will buy myself a Mac Book'.) e.
Sie fragt sich, ob sie eine Arbeit findet. (She asks herself if she will find a job.) f.
(She asks herself if she will really find a job.) g.
(She asks herself if she would find a job.) h.
(She asks herself if she would really find a job.) S3: Glaubt Paula, dass sie eine Arbeit findet? (Does Paula believe that she will find a job?) S4: Denkt Paula über Ihre Oma nach?
(Is Paula thinking about her grandmother?) While "nonveridical equilibrium" is assumed to be the default for both conditionals and questions in Giannakidou & Mari (2018a, b, 2021a, it is to note that wenn-sentences in German are ambiguous between temporal and conditional interpretations. In relation to this duality, it is feasible to assume that wenn-conditionals may carry a positivity that embedded ob-questions do not. Based on this and the literature we outlined above, we formulated the following specific predictions for Experiment 1. E1.P1 (see 19a): Ob-questions would receive lower ratings of speaker commitment than wenn-conditionals (i.e., concerning the antecedent proposition). We thus expected an effect of SentenceType. E1.P2 (see 19c): The adverb wirklich would lower the ratings of speaker commitment in both questions and conditionals. We thus expect an effect of the adverb in the overall data as well as in the sub-analysis of the wenn-conditional or the ob-question dataset.
E1.P3 (see 19d): The modal verb sollte would lower the ratings of commitment. We thus expected an effect of the modal verb in the sub-analysis of the wenn-conditional conditions. We did not have specific predictions for the verb würde, but as it is strictly speaking not a modal verb, we expected it to be different from sollte.

Procedure
Participants took part in the study at the online crowd-sourcing platform Prolific (https:// www. proli fic. co/) for small payments. They started with four practice trials. For the target items, they read S1 and S2, and then answered the question in S3 on a 7-point Likert scale (with labelled endpoints, i.e., 1 = Stimmt gar nicht 'absolutely no', 7 = Stimmt vollkommen 'absolutely yes'), which we take to reflect the degree of the speaker or the doxastic agent's, e.g., Paula's commitment in (20), to the given proposition in S2. They also answered the polar question in S4 with "yes" or "no". The experiment was programmed and hosted on Ibex Farm (Drummond, 2013). Each participant saw all 40 target items and 48 filler items presented in a pseudorandom order. The total experimental duration was approximately 25 min.

Data Analysis
First, we assessed participants' response accuracy on the comprehension questions. The data of one participant with a response accuracy below 85% was ruled out. In total, the data of 82 participants were analyzed. We analyzed the rating data via cumulative link mixed models for ordinal regression implemented in the ordinal package (Christensen, 2019) in R (R Core Team, 2019). We conducted an analysis of the entire data set. In addition, we also conducted two separate analyses of Adverb and ModalVerb in the wenn-conditional type and the ob-question type. For all models, the factors were manually sum coded with ± 0.5, that is, SentenceType (wenn-conditional 0.5, ob-question − 0.5) × Adverb (without wirklich 'really' 0.5, or with it − 0.5)× ModalVerb (without sollte/würde 'should/would' 0.5, or without it − 0.5). The cumulative link mixed model included the logit-link function. We used the most parsimonious model approach (Bates et al, 2015): The model including random by-subject and by-item intercepts was chosen as it fit the data best. 12 The p-values were obtained through model comparison via likelihood ratio tests between the model without the respective effect against the full model (Christensen, 2019). In the case of significant effects, we provide the p-values rounded to three decimals unless they are smaller than 0.01; in the case of nonsignificant effects, p-values are rounded to two decimals. We report the results below.

Results
The rating responses are visualized in Fig. 1, with the descriptive statistics in Table 1.
The results of the entire data set (see Table 2) are the following: (1) There was a significant effect of SentenceType with lower ratings for the ob-question conditions than for the wenn-conditional conditions ( β =0.82, p < 0.0001), confirming E1.P1. (2) There was a significant effect of Adverb with lower ratings for the conditions with wirklich than without wirklich ( β =0.43, p < 0.0001), confirming E1.P2. (3) There was a significant effect of ModalVerb in general with lower ratings for the conditions with ModalVerb than without it ( β =0.18, p = 0.007).
While we did not have specific predictions about the interaction of the tested effects, the model shows that the three-way interaction between SentenceType × Adverb × Modal-Verb was not significant. The two-way interactions between SentenceType × ModalVerb or Adverb × ModalVerb were not significant either, but the two-way interaction between Sen-tenceType × Adverb was significant. Given the significant SentenceType × Adverb interaction, we conducted two sub-analyses looking into the effect of Adverb in questions and conditionals separately. We included the factor of ModalVerb in the model despite the lack of the SentenceType × ModalVerb interaction to further explore the data. The results show that the effect of Adverb was greater in questions ( β =0.69, p < 0.0001) than in conditionals ( β =0.28, p = 0.003), explaining the interaction effect. The effect of sollte in wenn-conditionals ( β =0.21, p = 0.03) was slightly different from the effect of würde in ob-questions ( β =0.17, p = 0.07), explaining the lack of interaction. In the rest of the paper, we will only deal with sollte in conditionals.

Experiment 2
In Experiment 1, we tested the wenn-conditionals in comparison to ob-questions, with the finding that the questions convey lower speaker commitment to the modified proposition than conditionals. Furthermore, we found a commitment-weakening effect of the adverb wirklich across both sentence types, as well as a similar effect of the modal verb sollte in conditionals. We decided to look closely at bias and modality in conditionals in a follow-up study, in particular, because different kinds of conditionals have been argued to differ in semantics and pragmatics, and thus might convey different degrees of speaker commitment: Among others, Liu (2019Liu ( , 2021, for example, provide distributional and experimental evidence that falls-conditionals convey lower speaker commitment to the antecedent proposition than wenn-conditionals. Reis & Wöllstein (2010), as we introduced earlier in the paper, suggest that V1-conditionals might be semantically related to questions and that they are different from wenn-conditionals. In order to examine whether the effects of wirklich and sollte we found in Experiment 1 obtain across different types of conditionals (wenn/ falls/V1-conditionals) and how they interact with one another, we conducted a second experiment of an explorative nature. It is meant as a first step that might lead to hypotheses for further testing in a more strict and comprehensive way. We considered three types of indicative conditionals in German, one with the CC wenn, the other with the CC falls, which has been argued to indicate a lower degree of commitment (Liu, 2019). The third type are the so-called V1-conditionals (with the finite verb appearing in the initial position of the antecedent clause) which Reis & Wöllstein (2010), among others, discuss. We will not discuss the formal aspects of these expressions in detail but only the following aspects related to the experiment we conducted.
First, it has been noted that in the case of hypothetical use of falls, it often co-occurs with the modal verb sollte, such as in (21). Second, in V1-conditionals in the hypothetical use, sollte is often used as well, such as in (22). This contrasts with the strong necessity verbs muss/müsste which cannot be used here, see Sode & Sugawara (2019) and the corpus findings of Hacquard & Wellwood (2012) for the English strong necessity verb must, which in conditionals almost exclusively has root interpretations instead of epistemic ones. (21) Reis & Wöllstein (2010) argue that V1-conditionals (as well as falls-conditionals) are different from wenn-conditionals, and based on distributional facts, suggest tentatively that V1-conditionals and questions might have a common semantic core. While their focus is on the structural properties of V1-conditionals, in Experiment 2, we tested these three conditional types and their interaction with sollte and wirklich focusing on the interpretive effects with regard to speaker commitment about the antecedent proposition.

Method
Participants 89 adult German native speakers (32 female, 57 male, mean age = 29.4, SD = 8.4) participated in the study. The experiment was approved by the ethics committee of the German Linguistic Society.

Design and Materials
Experiment 2 was based on a 2 × 2 × 2 factorial design, with three within-participants and within-items factors: ConditionalType (falls-conditional vs. V1-conditional), Adverb (with or without wirklich) and ModalVerb (with or without sollte), see an example in (23). In addition, we also used Condition 1 from Experiment 1 with wenn as a control condition. In total, we used 45 target items, with 40 from Experiment 1 and 5 new items in order to create counter-balanced sets; the complete list of target items is included in Appendix 1. As in Experiment 1, each item consisted of a context-setting sentence (S1), a critical sentence (S2) and a critical question (S3) about the speaker's belief in the antecedent proposition in S2 and a final comprehension question (S4) for attention check. In addition, we used the same 48 filler items as in Experiment 1. (23) S1: Paula denkt über ihre Zukunft nach. (Paula is thinking about her future.)

S2: a.
Sie denkt: "Wenn ich eine Arbeit finde, kaufe ich mir ein Mac Book." (She thinks to herself, 'If I find a job, I will buy myself a Mac Book.') b.
Sie denkt: "Falls ich eine Arbeit finde, kaufe ich mir ein Mac Book." (She thinks to herself, 'In case I find a job, I will buy myself a Mac Book.') c.
Sie denkt: "Falls ich wirklich eine Arbeit finde, kaufe ich mir ein Mac Book." (She thinks to herself, 'In case I really find a job, I will buy myself a Mac Book.') d.
Sie denkt: "Falls ich eine Arbeit finden sollte, kaufe ich mir ein Mac Book." (She thinks to herself, 'In case I should find a job, I will buy myself a Mac Book.') e.
Sie denkt: "Falls ich wirklich eine Arbeit finden sollte, kaufe ich mir ein Mac Book." (She thinks to herself, 'In case I should really find a job, I will buy myself a Mac Book.') f.
Sie denkt: "Finde ich eine Arbeit, kaufe ich mir ein Mac Book." (She thinks to herself, 'If I find a job, I will buy myself a Mac Book.') g.
Sie denkt: "Finde ich wirklich eine Arbeit, kaufe ich mir ein Mac Book." (She thinks to herself, 'If I really find a job, I will buy myself a Mac Book.') h.
Sie denkt: "Sollte ich eine Arbeit finden, kaufe ich mir ein Mac Book." (She thinks to herself, 'Should I find a job, I will buy myself a Mac Book.') i.
Sie denkt: "Sollte ich wirklich eine Arbeit finden, kaufe ich mir ein Mac Book." (She thinks to herself, 'Should I really find a job, I will buy myself a Mac Book.') S3: Glaubt Paula, dass sie eine Arbeit findet? (Does Paula believe that she will find a job?) S4: Denkt Paula über Ihre Oma nach?
(Is Paula thinking about her grandmother?) The literature we reviewed above only allows to make predictions about main effects, see (19), thus, we formulated the following specific predictions for Experiment 2.
E2.P1 (see 19b): Different kinds of conditionals would receive different speaker commitment ratings. Thus, we expected a difference between the wenn-vs. falls-conditional or V1-conditional conditions (i.e., Condition 1 vs. 2/6 in Table 3 below) in that fallsconditionals would receive lower speaker commitment ratings than wenn-conditionals, and possibly V1-conditionals would also receive lower speaker commitment ratings than wenn-conditionals. We did not have clear predictions about an effect of ConditionalType (falls-conditionals vs. V1-conditionals, i.e., Condition 2-9 in Table 3 below). E2.P2 (see 19c): The adverb wirklich would lower the ratings of speaker commitment across different kinds of conditionals. We thus expected an effect of the adverb in the overall data as well as in the sub-analysis of the falls-conditional or the V1-conditional dataset.
E2.P3 (see 19d): The modal verb sollte would lower the ratings of speaker commitment across different kinds of conditionals. We thus expected an effect of the modal verb in the overall data as well as in the sub-analysis of the falls-conditional or the V1-conditional dataset.

Procedure
The procedure of Experiment 2 was identical to Experiment 1. Participants took part in the study online at Prolific for small payments. They started with four practice trials. For the target items, they read S1 and S2, and then answered the question in S3 on a 7-point Likert scale (with labelled endpoints, i.e., 1 = Stimmt gar nicht 'absolutely no', 7 = Stimmt vollkommen 'absolutely yes'), which we take to reflect the degree of the doxastic agent's commitment to the given proposition in S2. They also answered the polar question in S4. The experiment was programmed and hosted on Ibex Farm (Drummond, 2013). Each participant saw all 45 target items and 48 filler items presented in a pseudorandom order. The total experimental duration was approximately 30 min.

Data Analysis
First, we assessed participants' response accuracy on the comprehension questions. Four participants had a response accuracy < 85%, and their data were not included in the analysis.
We analyzed the rating data via cumulative link mixed models for ordinal regression implemented in the ordinal package (Christensen, 2019) in R (Core Team, 2019). First, we conducted a one-factorial analysis of Conditions 1, 2 and 6 (see Table 3 and Analysis 2a below), which correspond to the three conditional types without wirklich or sollte. For these, we used treatment coding with wenn as reference level plus a slope for falls and another slope for V1. Second, we conducted an analysis of Conditions 2-9 (Analysis 2b) based on a 2×2×2 factorial (i.e., ConditionalType×Adverb×ModalVerb) within-participants and within-items design. In the model, the factors were manually sum coded with ± 0.5, i.e., ConditionalType (V1-conditional 0.5 vs. falls-conditional -0.5), Adverb (without wirklich 'really' 0.5 or with it − 0.5) and ModalVerb (without sollte 'should' 0.5 or with it -0.5). The cumulative link mixed model included the logit-link function. We used the most parsimonious model approach (Bates et al., 2015) and added random by-subject or by-item intercepts for the effects, or both if possible (i.e., when the better fitting model   converged). 15 The p-values were obtained through model comparison via likelihood ratio tests between the model without the respective effect against the full model (Christensen, 2019). They are reported as in Experiment 1. We report the results below.

Results
The rating responses are visualized in Fig. 2, with the descriptive statistics in Table 3.

Analysis 2a
The results of the one-factorial analysis of Condition 1, 2 and 6 (see Table 4) are the following.
(2) There was a significant effect of Adverb with lower ratings for conditions with wirklich than without it ( β =0.25, p < 0.0001). 4) There was a significant effect of ModalVerb with lower ratings for conditions with sollte than without it ( β =0.16, p = 0.02).
(3) There was no three-way interaction or any two-way interactions.

Discussion
In the current study, we tested degrees of doxastic commitment about a given proposition as indication of speaker bias. The observed effects are a result of diverse manipulating sources having to do with modality: sentence types (wenn-conditionals vs. ob-questions) and subtypes (wenn/falls/V1-conditionals), the modal verbs (sollte) and the adverb (wirklich). The results are summarized in Table 6 and the list of (24).  (19): a. Hypothesis 1 was confirmed: Different sentence types and subtypes convey different degrees of speaker commitment about the modified (antecedent) proposition.
b. Hypothesis 2 was confirmed. The use of the adverb wirklich indicates reduced speaker commitment about the modified (antecedent) proposition. c. Hypothesis 3 was confirmed. The use of the modal verb sollte indicates reduced speaker commitment about the modified (antecedent) proposition.
In the following, we will discuss the results concerning the factors one by one.

Sentence Types and Subtypes
In Experiment 1, we found a significant effect of SentenceType, with higher ratings for the wenn-conditional conditions than for the ob-question conditions. Our results show a higher rating in wenn-conditionals with mean ratings of 4.44 on a 7-point scale, and 4.57 for Condition 1 without sollte or wirklich in the antecedent; ob-questions have mean ratings of 4.00 on a 7-point scale, and 4.22 for Condition 5 without sollte or wirklich, so equal to or slightly above the median 4. If we relate "nonveridical equilibrium" (Giannakidou & Mari, 2018a, b, 2021a to the median, the reason for the higher rating with wenn-conditionals might lie in the fact that asking a question places the speaker in a de facto position of informational neutrality while in the conditional protasis speaker and comprehender treat p as a condition for the antecedent to be true, hence as a 'given' or even a cause for the consequent to happen. Thus, while both questions and conditional protases are logically in a nonveridical equilibrium, the equilibrium is masked by the contentful relation between antecedent and consequent, while questions are monoclasual and the issue does not arise. Additionally and more importantly, as we mentioned in Sect. 2, we must note that the CC wenn in German is also used as a temporal connective, such as in the ambiguous example in (25). In relation to this duality, it is feasible to assume that wenn-conditionals may carry a positivity that embedded questions do not, in line with the experimental results. (25) Wenn wir das Spiel gewinnen, verraten sie uns ihr nächstes Ziel. 16 if/when we the game win tell they us their next goal 'If/When we win the game, they'll give us the next goal.' For this reason, we took a closer look at the stimuli, which reveals that some of the wennsentences are ambiguous between a temporal and a conditional reading. 17 For example, the test item (1), stated in (26a), is ambiguous between a temporal and a conditional reading, but the item (3), stated in (26b), can only have a conditional reading, see Appendix 1. (26)  In this regard, world knowledge plays a role too: most careers will probably see at least one promotion, so the temporal reading of the item (7) in the stimuli, stated in (27a), is quite easily accommodated. In comparison, it is certainly possible for a football team to never win a championship, so a temporal reading for the item (9), stated in (27b), requires enriching the context with the assumption that the protagnist's team is good. (27)  Thus, it is worthwhile to systematically check for the availability of temporal readings in the materials in follow-up studies. However, we need to be cautious with claiming that the ambiguity is the (sole) reason for the difference between ob-questions vs. wenn-conditionals, as in Experiment 2 we did not find a difference between wenn-conditionals and V1-conditionals even though the latter do not have a temporal reading.
Our experimental results also raise the question whether wenn is the default CC in German. While wenn is the most frequent CC in German, researchers do not have a consensus regarding the question whether wenn or falls is the prototypical CC (Breindl et al., 2014). It is further worth noting that in our stimuli we used embedded ob-questions, which might differ from default unembedded questions such as Sie denkt: "Finde ich eine Arbeit?" or Sie denkt: "Ob ich eine Arbeit finde?" (She thinks to herself: Will I find a job?). We will leave the last point for future investigation as well.
In Experiment 2, we found a significant difference for the wenn-conditionals vs. the falls-conditionals in that the latter received lower ratings in line with our predictions based on Liu (2021). In addition, the falls-conditionals received lower ratings than the V1-conditionals which did not differ from wenn-conditionals. While nonveridical equilibrium is assumed to be the default for conditionals, our results also show variation between different conditional types. The interpretive effect of V1-conditionals in comparison to the others (i.e., similar to wenn, but different from falls) has not been previously elicited experimentally, to our knowledge, and this is relevant for our understanding of V1-conditionals in general. For example, Reis & Wöllstein (2010), focusing on morphosyntactic questions of conditionals, argue that V1-conditionals (as well as falls-conditionals) are different from wenn-conditionals, but our data provides a different picture from a pragmatic perspective. While the wenn vs. falls contrast is oftentimes attributed to the temporal interpretation of wenn which falls does not have, the lack of a difference between wenn vs. V1-conditionals with no temporal reading casts doubt on the idea that temporality is the sole explanation for the wenn vs. falls contrast.

The Adverb wirklich
In both experiments, we found a significant effect of wirklich in that conditions with it received lower ratings than without it, showing a weakening effect of doxastic commitment by the speaker, as understood by the comprehender. This holds across different sentence or conditional types, which was consistent with our predictions. There are several aspects in the results that we will discuss here briefly.
First, we did not find any interaction effects in both experiments, except for a significant two-way interaction between SentenceType x Adverb in Experiment 1 in that the effect of Adverb was greater in questions than in conditionals, that is, wirklich lowered the ratings to a greater extent in questions than in conditionals. We do not have a definite explanation for this, but note that questions (P?) and conditionals (If P, Q) address different questions under discussion (QUDs)-the former about P, and the latter relating to two different kinds of QUDs, namely "Under which conditions Q?" and "What follows from P?", see Fintel (2001) and Arregui & Biezma (2016). The question "Wirklich P?" address the same QUD as one without wirklich. On the other hand, conditionals with wirklich seem to relate to only the QUD "What follows from P?", that is, with Q being the more salient proposition at-issue than P, unlike in questions. This might explain the SentenceType x Adverb interaction in Experiment 1 and the absence of an interaction effect in Experiment 2 as the latter only tested conditionals.
A reviewer suggested to us that wirklich is anaphoric-an observation consistent with our analysis of REALLY mentioned earlier. Recall from earlier discussion that for REALLY we proposed a contextual and a speaker bias, repeated in (28). (28) If Anne really becomes a lawyer, she will open a law firm.
a. contextual positive bias: Anne will become a lawyer. b. speaker negative bias: It is not the case that Anne will become a lawyer.
The contextual bias of REALLY, by the speaker and hearer via their previous mental state in the context, renders it anaphoric, and the speaker bias produces the negativity. REALLY is thus different form a purely modal element such as sollte, which lacks the contextual bias. In this regard, REALLY is also distinct from falls, which is not anaphoric either. This difference needs to be considered in future studies targeting the interaction between these expressions (e.g., in relation to modal spread).
A reviewer also pointed out to us that the presence of wirklich in wenn-conditionals makes a temporal reading impossible. We agree with this intuition; however, we do not think this is the reason for the effect of wirklich, as its effect in questions in Experiment 1 was greater, and also because there was an effect of wirklich in the falls-and V1-conditionals in Experiment 2, even though neither type of conditionals has a temporal interpretation.
On an additional note, the future semantic investigation of wirklich (really/truly) might benefit from including their close relatives such as tatsächlich (actually/indeed). While they look similar at first sight, there are distributional differences that hint at potential differences: for example, all the native speakers we consulted share the intuition that the effect of tatsächlich 'in fact, actually' in conditionals is comparable to that of wirklich, but it has been pointed out to us (Manfred Krifka, p.c.) that while tatsächlich can occur in the prefield in German, wirklich cannot.

The Modal Verb sollte
In both experiments, we found a significant effect of sollte in that conditions without it received higher ratings than with it, showing that it has a weakening effect in terms of speaker commitment.
As an alternative to the "weakened commitment" analysis, Sode & Sugawara (2019), argue that both sollte, in their term "on its deliberative use" in contrast to weak necessity modals, and falls introduce a use condition that takes the truth of the antecendent proposition p as "a truly open possibility against a given conversational background" (see Reis & Wöllstein, 2010 for a similar point, as we presented in Sect. 1). By this, they stand in contrast to V1-/wenn-conditionals, which we can then take to be positively biased. Either perspectives are in line with the lower ratings with sollte in our experimental data.
A related aspect we plan to investigate in future studies concerns the effect of sollte (and possibly also wirklich) in falls-conditionals. Since both falls and sollte express negative bias towards the antecedent proposition as our studies show, and they often co-occur (Reis & Wöllstein, 2010), the question is whether they work as one modality structure, as expected by the Giannakidou and Mari framework of modal spread we adopted.

The Nature of Bias
We provided experimental evidence for bias triggered by modal devices, and in closing, we want to offer some more comments on the nature of bias as it emerges from our findings and hypotheses. Bias, as we said, is individually anchored, i.e., to the speaker or comprehender who makes assumptions about the speaker's belief state. As we said at the beginning, bias goes in different directions (either positive or negative) and is itself a doxastic state, i.e., the belief or credence of the individual anchor, prior to asserting the conditional or asking a question, that the proposition will play out in the positive or negative direction. This credence is the result of all the factors that form belief, i.e., the anchor's knowledge, beliefs, expectations plus contextual biases that are given by the context. So, what is the semantic or pragmatic status of bias? Is it a presupposition, an implicature, or a felicity condition? One must admit that the bias belongs to the category of non-at-issue content, and is speaker-oriented (Liu, 2012;Potts, 2005), except for the contextual bias in the case of wirklich, which is not always anchored to the speaker. We have suggested, following Giannakidou (2021), that speaker bias is best understood as a felicity condition attached to the speaker, e.g., like specificity conditions (Ionin, 2006, Giannakidou & Quer, 2013. Felicity conditions are definedeness conditions that are not motivated in the common ground. We might think of them as being similar to conventional implicatures in the sense of Potts (2005) or even weaker, as lexically triggered conversational implicatures.
In the course of the discussion, we have been characterising bias both as "weakened commitment" and "increased uncertainty" of the anchor. While we used explicitly the anchor's belief (or credence) as the measurement in both experiments, it would be interesting to see whether testing on uncertainty will yield similar or different results-by, for example, asking subjects how certain Paula is that she will find a job. However, the certainty measure can probably work well for conditionals but it might not work for questions, or at least might result in very low ratings for them. This is a methodological question worth further investigation.
An open question concerns the strength of the bias. There are likely nontrivial differences (effects of gradience) between the bias-creating devices we mentioned in the paper: high negation, negative tag, NPI, modal verbs, adverbs, conditional connectives. We are, unfortunately, not able to address the differences in the present study, but we hope to have offered a solid rationale for how to address bias experimentally that can be used to design similar experiments with other devices. It is to be expected, we think, that, just like with modals, the properties of the different classes of expressions might affect the manifestation or degree of bias. For instance, the bias of minimizer NPIs (such as Does he give a damn about what I say?) is distinctively stronger than that of a simple NPI Have you ever been to London? In addition, the bias in the latter might be optional (that is, absent in some contexts, e.g., in the sentence Have you ever been to London, by any chance?). More empirical study is needed to establish such patterns in conditionals.
Finally, another open question concerns the interaction between the broad discourse context and the local sentence context where bias-triggering expressions occur. For example, a reviewer pointed out to us that S1 in the experiments is differently biased with respect to the target conditional between items: doing a marathon training (item (2), see Appendix) usually culminates in running a marathon, i.e., being able to clear the distance, so this item seems positively biased. Conversely, someone who often has to work late (item (6), see Appendix) will probably not have Friday evening off without any further evidence, so this item seems negatively biased. We checked the ratings on these items: Item (2) received the overall rating of 4.41 and item (6) 4.09, which is in line with the reviewer's comment. On the other hand, a closer look at e.g., the respective ratings in Condition 1 (the conditional condition without wirklich or sollte) show an opposite pattern with 4.78 for item (2) and 5.10 for item (6). Thus, we will leave a systematic check for S1 bias and subsequent item analysis for future work.

Conclusion
In this paper, we investigated bias in conditionals and questions with a focus on the adverb wirklich, the modal verb sollte, and different sentence types (ob vs. wenn) and conditional subtypes (falls vs. wenn/V1) in German. We hypothesized that sollte/wirklich/falls all create negative bias about the antecedent proposition. The two experiments we conducted show effects in the predicted directions. One of the clearest bias generating effects we observed was the effect of the metaevaluation function O (Giannakidou & Mari, 2018a, b, 2021a exhibited by wirklich. In addition, we also found differences between the wenn-conditionals and the ob-questions in Experiment 1 and between the wenn-, falls-and V1-conditionals in Experiment 2, with wenn-and V1-conditionals being similar in this aspect. Our studies are among the first that address bias in conditionals (in comparison to questions), and we expect our findings to deepen the understanding of what bias is, and how it is produced both generally and specifically in conditionals.
Funding Open Access funding enabled and organized by Projekt DEAL. M.L.'s work on the project was supported by a DFG (German Research Foundation) grant for the project "The Semantics and Pragmatics of Conditional Connectives: Cross-linguistic and experimental Perspectives" (project number: 367088975). S.R.'s work was supported by a DFG grant for the project "Pragmatic Functions and Effects of Register Variation and Switch: a Register approach to negation and polarity" (project number: 416591334). A.G.'s work was supported by a research fellowship from the Franke Institute for Humanities at the University of Chicago which she gratefully acknowledges.
Data Availability All data and code associated with the experiment reported in this paper are available at the following data repository: https:// osf. io/ qkvtf/.

Conflict of interest
The authors have no conflicts of interest to declare.
Ethical Approval All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. Informed consent was obtained from all individual participants included in the study. The study was approved by the ethics committee of the German Linguistics Society.
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