Online comprehension of conditionals in context: A self-paced reading study on wenn (‘if’) versus nur wenn (‘only if’) in German

1 Introduction

Conditional statements are a widely used linguistic tool that is essential for communicating about possible situations and their probabilities. In a simple conditional of the form “If p, q” (e.g., If it rains, the streets get wet), logically represented by the material implication ‘p → q’, the consequent q (the streets get wet) is stated to be true in the case that the antecedent p (it rains) is true. However, the adequacy of the material implication analysis for simple conditionals has been questioned in a number of ways. In psychology, the mental model theory (Johnson-Laird and Byrne 2002) and the suppositional theory (Evans and Over 2004) both assume that in the case of simple conditionals, only a subset of the four conditional possibilities (p and q; p and ¬q; ¬p and q; and ¬p and ¬q) are activated in the comprehender’s mind. In linguistics, a crucial argument against the material implication analysis is that it yields inadequate interpretations for natural language conditionals (von Fintel 2011). Among others, Kratzer (1986) proposes a semantic analysis called the restrictor analysis in which the conditional connective if in simple conditionals introduces no conditional meaning in itself and if-clauses are used to restrict modal or generic frequency operators. Hence, the sentence If it rains, the streets get wet would contain a covert modal operator MUST and means that all the rain-worlds are worlds where the streets get wet. In abstract terms, a simple conditional of “If p, q” involves universal quantification over worlds or cases and it has the semantics that all p-cases must be q-cases. These different proposals share the assumption that in the presence of a simple conditional “If p, q”, a comprehender will expect the consequent to be true when the antecedent is true. When the antecedent is false, however, matters are more complicated. One factor that contributes to this is the phenomenon called conditional perfection, which we will return to after discussing the meaning of the more complex conditional only if.

A compositional semantic derivation of the meaning of only if requires a combination of the meanings of if, as discussed above, and only. Following Horn (2002), a sentence under the focus particle only – that is, “Only p” – expresses two entailments, the positive entailment p and a negative entailment that can be glossed as “Nothing but p”. The meaning of “Only if p, q” would then contain the entailments “all p-cases are q-cases” and “all ¬p-cases are ¬q-cases”, seemingly making only if a biconditional connective. However, Herburger (2015, 2019 proposes an alternative analysis of only if, in which the foregrounded meaning of “Only if p, q” is negative and involves universal quantification (i.e., “all ¬p-cases are ¬q-cases”), while the backgrounded meaning is positive and has existential force (“some p-cases are q-cases”). Hence, sentences like Only if you work hard do you succeed encode the meaning that “lack of hard work results in lack of success” and, crucially, “some hard work results in success” rather than “all hard work results in success”.

Although it is clear that if and only if have different semantics, the pragmatic phenomenon of conditional perfection needs to be taken into consideration to gain the full picture of their lexical meanings in a situational context. For instance, Geis and Zwicky (1971) claim that by uttering the simple conditional sentence If you mow the lawn, I will give you 5 dollars, the speaker invites the hearer to additionally draw the inference “If you don’t mow the lawn, I won’t give you 5 dollars”. The semantically weaker conditional sentence is then “perfected” to a semantically stronger biconditional one: “If and only if you mow the lawn will I give you 5 dollars”. While there are different accounts for this phenomenon (van der Auwera 1997a, 1997b; Horn 2000; see Noveck 2018: Ch. 4 for a review), all of them treat conditional perfection as a pragmatic inference of a conversational nature, that is, optional and cancelable (see Grice 1989). This means that if-sentences encode the meaning “all p-cases are q-cases” semantically, and “all ¬p-cases are ¬q-cases” pragmatically. Van Canegem-Ardijns and Van Belle (2008), for instance, provide an extensive discussion of the contextual variability of the invited inference, which involves the strengthening of if to only if (see also van der Auwera 1997b). However, we have seen above that only if differs from if in the positive meaning component, and thus is not a biconditional connective but rather a mirror image of if. The question remains whether the existential meaning component in the semantics of only if can be strengthened as well, so that it receives a biconditional meaning. It seems to us that this is also context-dependent, that is, it is more possible in some cases (e.g., Only if it rains do the streets get wet) than in others (e.g., Only if you work hard do you succeed).

We take the analysis by Herburger (2015, 2019 as our working hypothesis, as recent evidence indicates that neither if nor only if are semantically biconditional. In a series of rating studies investigating the semantics of different conditional connectives in German, Liu and Barthel (2021) find that ratings for modus ponens (“If p, q. p. Therefore q.”), which is a valid rule of inference for simple conditionals and can be validated for biconditionals in first-order logic (if we treat iff as a conjunction of ifs), were higher for wenn (‘if’) than for nur wenn (‘only if’), indicating that the material implication p → q is not part of the lexical semantics of nur wenn. Additionally, ratings for affirmation of the consequent (“If p, q. q. Therefore p.”), which is invalid for simple conditionals but can be validated for biconditionals, were higher for nur wenn than for wenn, indicating that the mirrored implication q → p is not part of the lexical semantics of wenn. This means that neither of the two conditional connectives is semantically biconditional, supporting Herburger’s analysis.

From a processing perspective, given these differing lexical meanings of wenn (‘if’) and nur wenn (‘only if’), comprehenders can be assumed to generate different expectations about the probability of the consequent being true when they know about the truth or falsity of the antecedent. Expectation generation, or prediction, is an integral part of language comprehension at sentence and discourse levels (Drenhaus et al. 2014; Grisoni et al. 2020; Köhne and Demberg 2013; Pickering and Garrod 2004, 2013, 2021; Rohde and Horton 2014; Rohde et al. 2011; van Bergen and Bosker 2018; van Berkum et al. 2005; Schwab and Liu 2020; Schwab et al. 2022; Xiang and Kuperberg 2015). Upon encountering a hypothetical conditional statement, the truth probabilities of both antecedent and consequent are unknown. Conditionals are therefore non-veridical contexts, and we assume non-veridical equilibrium to be the default for conditionals (implying that p and ¬p are equal possibilities; see Giannakidou and Mari [2021]). Thus, without contextual manipulations that would trigger any bias (Liu 2019; Liu et al. 2021), comprehenders would not be expected to commit to a prediction of their truth or falsity. If, in addition to the conditional, the comprehender knows about the antecedent being true, they can expect the truth of the consequent in the case of wenn (‘if’) but to a lesser degree in the case of nur wenn (‘only if’), since only in wenn would the truth of the consequent semantically follow from the truth of the antecedent (p → q). In cases where the antecedent is known to be false, on the other hand, the truth of the consequent is uncertain under wenn, making the scenario less predictable. Under nur wenn, however, the consequent can be expected to be false (¬p → ¬q), making the scenario more predictable. We test these predictions and the underlying lexical meanings of the conditional connectives, measuring reading times as an indicator of prediction in conditional scenarios.

The present study consists of two self-paced reading experiments presenting natural language scenarios of conditional situations. In Experiment 1, participants read conditionals with wenn (‘if’) or nur wenn (‘only if’) in identical discourse contexts (“Wenn/Nur wenn p, q”), the confirmed antecedent (p), and then word by word the confirmed or negated consequent (q or not-q) that contained a positive or negative quantifier (ein/kein ‘one/no’). In Experiment 2, participants read “Wenn/Nur wenn p, q. not-p. q/not-q.” with the negated antecedent. In both experiments, the critical dependent measure is the reading time of the quantifier in the final sentence that reveals the consequent to be either true or false. In Experiment 1, we hypothesize that comprehenders generate stronger expectations about the truth of the consequent in wenn than in nur wenn, as the foregrounded meaning of wenn is “all p-cases are q-cases”, while in nur wenn only “some p-cases are q-cases”. We therefore expect shorter reading times for positive quantifiers (ein) in wenn than in nur wenn. In Experiment 2, on the other hand, we expect shorter reading times for negative quantifiers (kein) in nur wenn than in wenn, since the foregrounded meaning of nur wenn is “all ¬p-cases are ¬q-cases”, while in wenn this meaning can only be inferred via conditional perfection.

2 Experiment 1

2.2 Results

All subjects had a correct response rate of greater than 80% in the comprehension questions (mean = 90%, SD = 6%). After a visual inspection of the density distribution of trial lengths, 159 critical trials (6.1%) were discarded because they had a total duration of less than 3 s or more than 15 s.

The following analyses will focus on the reading times of the critical quantifier ein/kein (‘one/no’) that encodes whether the consequent q is true or false in the presented scenarios; that is, we focus on the fifth word in S4. Sixty-six trials (2.1%) were discarded because the reading times of the critical word deviated more than 2.5 SD from the subject mean reading times for the critical word. The critical word in the remaining 2,367 critical trials had a mean reading time of 360 ms (SD = 139 ms).

A Bayesian mixed-effects model on the reading times of the critical word was run in R (R Core Team 2021) using the package brms (Bürkner 2017), with CC and Consequent plus their interaction as fixed effects and as random effects by subject and by item plus the centered length of the critical word and the centered trial number as control variables. CC and Consequent were dummy coded, with “wenn” and “negative Consequent” as reference levels. Table 1 shows the model output. Figure 1 shows the modeled reading times per condition. Bayes factors for hypothesis-specific tests were computed with the hypothesis function of brms. For a guideline to the interpretation of Bayes factors and the labeling categorization of effect strengths, see Andraszewicz et al. (2015).

Table 1:

Bayesian mixed-effects model output on reading times of the critical word in Experiment 1. CrI = credible interval; CC_nonCond = non-conditional control condition.

	Estimate	Lower 95% CrI	Upper 95% CrI
Intercept	378.12	330.33	425.62
length_c	0.93	−7.48	9.52
trialNumber_c	−0.46	−0.54	−0.37
CC_nurWenn	6.53	−5.02	17.99
CC_nonCond	−2.15	−13.44	9.73
Consequent_positive	−14.47	−32.78	3.84
CC_nurWenn:Consequent_positive	−9.96	−25.57	5.69
CC_nonCond:Consequent_positive	0.13	−15.15	15.84

Figure 1:

Modeled reading times for the critical word in Experiment 1 (confirmed antecedent), as extracted from the mixed-effects model presented in Table 1. CrI = credible interval.

Very strong evidence was found for a main effect of Consequent, with negative quantifiers being read more slowly than positive quantifiers (β = −19.45, CrI = [−33.9, −4.85], BF₁₀ = 69.59). No evidence was found for a main effect of CC (β = 3.1, CrI = [−11.78, 17.87], BF₁₀ = 1.79), meaning that reading times for the critical word did not differ between wenn and nur wenn. Moderate evidence was found for an interaction effect of CC and Consequent (β = 9.96, CrI = [−2.95, 23.25], BF₁₀ = 8.58), with the effect of CC being slightly smaller in positive q than in negative q. Breaking this interaction down into simple effects, it was found that while there is moderate evidence for an effect of CC in negative q (β = 6.53, CrI = [−3.13, 16.22], BF₁₀ = 6.57), there is no evidence for such an effect in positive q (β = −3.43, CrI = [−13.49, 6.63], BF₁₀ = 0.4). In these small, moderate effects, Bayes factors are below 10, making the effects unreliable. Therefore, we will not base our interpretations on these effects.

Reading times for the critical word in the critical conditions were overall comparable to reading times of the critical word in the non-conditional control condition (β = −5.73, CrI = [−19.82, 8.53], BF₁₀ = 2.88). In the control condition, a strong effect of Consequent was attested as well (β = −14.34, CrI = [−30.65, 1.99], BF₁₀ = 12.79). This effect was found to be comparable to the effect of Consequent attested for the conditional scenarios (β = −5.11, CrI = [−17.04, 6.34], BF₁₀ = 3.27).

A parallel analysis on the reading times of the post-critical word was conducted to check for potential spill-over effects and yielded the same pattern of results.

3 Experiment 2

3.2 Results

Analyses of results were parallel to Experiment 1. Five subjects were discarded because they had a correct response rate of less than 80% in the comprehension questions, leaving the data of the 24 remaining participants for analysis, who had a mean correctness rate of 93% (SD = 4%). After visual inspection of the density distribution of trial lengths, 153 critical trials (5.9%) were discarded because they had a total duration of less than 3 s or more than 15 s. A further 61 trials (2.3%) were discarded because the reading times of the critical word deviated more than 2.5 SD from the subject mean reading times for the critical word.

The critical quantifier in the remaining 2,367 critical trials had a mean reading time of 327 ms (SD = 112 ms). A Bayesian mixed-effects model on the reading times of the critical word was run using the R package brms, with CC and Consequent plus their interaction as fixed effects and as random effects by subject and by item plus the centered length of the critical word and the centered trial number as control variables. CC and Consequent were dummy coded, with “wenn” and “negative Consequent” as reference levels. Table 2 shows the model output. Figure 2 shows the modeled reading times per condition.

Table 2:

Bayesian mixed-effects model output on reading times of the critical word in Experiment 2. CrI = credible interval; CC_nonCond = non-conditional control condition.

	Estimate	Lower 95% CrI	Upper 95% CrI
Intercept	340.64	300.43	381.01
length_c	7.21	0.52	14.11
trialNumber_c	−0.74	−0.81	−0.67
CC_nurWenn	−12.06	−22.18	−2.09
CC_nonCond	−6.20	−15.89	3.35
Consequent_positive	−4.13	−15.55	7.35
CC_nurWenn:Consequent_positive	12.18	−1.21	25.59
CC_nonCond:Consequent_positive	2.44	−10.83	15.78

Figure 2:

Modeled reading times for the critical word in Experiment 2 (negated antecedent) as extracted from the mixed-effects model presented in Table 2. CrI = credible interval.

Hypothesis-specific tests were conducted using the hypothesis function implemented in brms. No evidence was found for a main effect of Consequent (β = 1.96, CrI = [−5.87, 9.93], BF₁₀ = 0.53). Strong evidence was found for a main effect of CC (β = −11.93, CrI = [−24.6, 0.84], BF₁₀ = 15.3), meaning that reading times for the critical word were shorter in nur wenn than in wenn. Strong evidence was found for an interaction effect of CC and Consequent (β = −12.18, CrI = [−23.57, −1.07], BF₁₀ = 26.59), with the effect of CC being smaller in positive Consequent (ein) than in negative Consequent (kein). Breaking this interaction down into simple effects, it was found that while there is very strong evidence for an effect of CC in negative Consequent (β = −12.06, CrI = [−20.41, −3.81], BF₁₀ = 121.45), there is strong evidence for the absence of a CC effect in positive Consequent (β = 0.13, CrI = [−8.6, 8.86], BF₁₀ = 0.99). Reading times for the critical word in the critical conditions were overall comparable to reading times of the critical word in the non-conditional control condition (β = −3.99, CrI = [−15.38, 7.46], BF₁₀ = 2.54). In the control condition, no effect of Consequent was attested (β = −1.68, CrI = [−11.47, 8.17], BF₁₀ = 1.57), and no interaction of control/critical with Consequent was found (β = 3.65, CrI = [−6.35, 13.56], BF₁₀ = 2.7).

A parallel analysis on the reading times of the post-critical word was conducted to check for potential spill-over effects, yielding no significant effects of interest.

To compare the reading times of Experiments 1 and 2, both data sets were combined and tested for differences in grand mean reading times of the critical quantifier as well as of the pre-critical region preceding the critical quantifier (containing four words). Two Bayesian mixed-effects models were run, with Experiment as a fixed effect and as a random effect by item and a random intercept by subject plus the centered trial number as a control variable. No reliable evidence was found for a main effect of Experiment in the pre-critical region (β = −84.84, CrI = [−242.27, 75.83], BF₁₀ = 4.24), nor in the critical quantifier (β = −34.03, CrI = [−85.35, 17.98], BF₁₀ = 6.03), indicating that any global differences in reading times between experiments are mainly caused by inter-individual variation between the two groups of subjects.

4 General discussion and conclusion

In everyday communication, conditional connectives (CCs) are critical cues in predicting possible developments of situations. Presenting short conditional scenarios, this study investigated the meaning contribution of two different CCs, wenn (‘if’) and nur wenn (‘only if’), and the predictions they trigger in two self-paced reading experiments. In Experiment 1, the conditional was followed by a confirmed antecedent, while in Experiment 2 it was followed by a negated antecedent. The critical word in the ensuing consequent, the quantifier ein (‘one’) or kein (‘no’), revealed differences in reading times when preceded by a confirmed or negated antecedent under the CCs wenn or nur wenn. Negative quantifiers were generally read more slowly than positive quantifiers after confirmed antecedents in Experiment 1, with no differential effect of CC. Interestingly, after negated antecedents in Experiment 2, reading times for confirmed consequents were identical for both CCs, while reading times for negated antecedents were significantly faster after nur wenn than after wenn.

Following a common analysis of the meaning of wenn, the central component of its semantics is that in all cases where the antecedent is true, the consequent must also be true (“all p-cases are q-cases”; Kratzer 1986), with the invited inference that “all ¬p-cases are ¬q-cases” (Geis and Zwicky 1971; Horn 2000; Van Canegem-Ardijns and Van Belle 2008; van der Auwera 1997b). In contrast, according to Herburger (2015, 2019, the semantics of nur wenn is that all cases where the antecedent is false are cases where the consequent is also false (“all ¬p-cases are ¬q-cases”), and that some p-cases are q-cases. Taking these analyses as working hypotheses, comprehenders of a conditional scenario of the form “Wenn p, q. p.”, as presented in Experiment 1, were hypothesized to predict the truth of the consequent q. Comprehenders of a conditional scenario of the form “Nur wenn p, q. p.”, on the other hand, were hypothesized to generate this prediction to a lesser degree. Thus, we expected reading times of the critical quantifier ein (indicating q) after p to be longer for nur wenn than for wenn. For the negative quantifier kein (indicating ¬q), the opposite pattern of reading times was predicted, as “some p-cases are ¬q-cases” is in line with the meaning of nur wenn but not with that of wenn. We thus expected reading times on the negative quantifier kein (indicating ¬q) after p to be shorter for nur wenn than for wenn. In conditional scenarios with negated antecedents (“Wenn p, q. ¬p.”), as presented in Experiment 2, nur wenn would allow comprehenders to generate the expectation of a negated consequent, while wenn should trigger such a prediction not semantically but pragmatically. Therefore, reading times for the negative quantifier kein were predicted to be shorter after nur wenn than after wenn. Reading times for the positive quantifier ein, on the other hand, were predicted to be longer after nur wenn, because of the violation of comprehenders’ expectations.

This predicted pattern of reading times was observed in Experiment 2, with a significant interaction of CC and Consequent. In particular, the results for the negative quantifier kein showed a clear difference between connectives, with faster reading times after nur wenn than after wenn. This shows that the related inference “If ¬p, ¬q” is weaker or more context-dependent for conditionals under wenn, in line with the pragmatic accounts of conditional perfection. Results for the positive quantifier showed no difference in reading times between connectives. While the lack of a difference needs further investigation, it is possible that it results from conditional perfection of wenn so that the affirmation of q given ¬p is equally surprising as in the case of nur wenn.

Contrary to these hypotheses, however, no interaction of CC and Consequent was found on the reading times for the critical quantifier in Experiment 1. While the absence of the expected interaction effect needs to be interpreted with care, two potential explanations come to mind. The first would be that nur wenn is semantically biconditional. In the light of recent analyses and findings of the meaning and interpretation of nur wenn, this possibility seems unlikely. The second, alternative explanation is that nur wenn was pragmatically enriched to a biconditional reading via a generalized conversational implicature, which cannot be easily canceled. This explanation appears reasonable for the scenarios used as stimuli in the present study. In (1), for example, if the bouquets are pretty, Leon is highly likely to buy one, especially as he can control the action himself. This contrasts with cases involving less controllable events; for example, success in the case of hard work, as analyzed by Herburger (2019).

Taken together, comprehenders were shown to use the different meaning nuances of distinct conditional connectives to predict the probabilities of conditional situations and the linguistic expressions that encode them. The different semantics of conditional connectives have been shown to be taken into account by comprehenders during the online processing of conditional scenarios, as the differences in processing costs surface already at the earliest point in a given sentence that encodes the falsity of the consequent. This study corroborates accumulating evidence that comprehenders immediately use available linguistic material to make predictions about the probabilities of upcoming material (Xiang and Kuperberg 2015; van Berkum et al. 2005; Schwab and Liu 2020) – in this case, the conditional background of a given situation – to predict the probability of the consequent.