Alternatives and attention in language and reasoning: A reply to Mascarenhas & Picat 2019

In this paper, we employ an experimental paradigm using insights from the psychology of reasoning to investigate the question whether certain modals generate and draw attention to alternatives. The article extends and builds on the methodology and ﬁndings of Mascarenhas & Picat (2019). Based on experimental results, they argue that the English epistemic modal might raises alternatives. We apply the same methodology to the English modal allowed to to test different hypotheses regarding the involvement of alternatives in deontic modality. We ﬁnd commonalities and differences between the two modals we tested. We discuss theoretical consequences for existing semantic analyses of these modals, and argue that reasoning tasks can serve as a diagnostic tool to discover which natural language expressions involve alternatives.


Introduction
Recent efforts to seek convergence between natural language semantics and the psychology of reasoning have led to articulated theories of interpretive processes and general-purpose reasoning.In particular, the erotetic theory of reasoning of Koralus & Mascarenhas (2013, 2018) incorporates linguistic insights into a variant of the mental models theory of reasoning (Johnson-Laird 1983) to account for a wide range of failures of deductive reasoning.At the core of both the mental models and erotetic theories of reasoning is the idea that attentional mechanisms structure our mental representations of states of affairs, and that a semantics of alternatives is required to model the effects of attention in reasoning.
Disjunction and indefinites are generators of alternatives par excellence.They induce a particular kind of illusory inference whose extant accounts all agree must be due to the presence of alternatives, driving attention in ways that render tractable the space of possibilities to be considered, but that introduce opportunities for fallacious reasoning under certain well-understood conditions.
In this paper, we advance our understanding of the connection between the semantics of alternatives in language, attention, and failures of reasoning by looking at reasoning with two different modal constructions in English.Mascarenhas & Picat (2019) have shown experimentally that epistemic might gives rise to illusory inferences much like those associated with indefinites.The explanation they discuss has two key components.First, following Ciardelli, Groenendijk & Roelofsen (2009), might introduces alternatives as in inquisitive semantics or Hamblin semantics.Second, following the erotetic theory, reasoning with these alternatives is based on question-answer dynamics.Together, these two components predict the inferences they observe with might.They conjecture that all and only linguistic elements that generate alternatives as in inquisitive or Hamblin semantics will give rise to the particular class of fallacies that have been observed so far with disjunctions, indefinites, and might.The present paper examines this conjecture by extending their experimental paradigm to the English deontic possibility modal allowed to, which no extant theory argues is a generator of alternatives of the relevant kind.Our experiment includes both might and allowed to, providing a direct comparison with previous results and with each other.
We largely replicate the findings of Mascarenhas & Picat (2019) for might, but we observe a more complex picture for allowed to.On the one hand, it 2:2 Alternatives and attention in language and reasoning shows fallacious conclusions structurally parallel to those found with might.On the other hand, allowed to fails to pattern with might, disjunctions, and indefinites in other well-established experimental results on the fallacious structures of central interest.
We explore the theoretical implications of these findings.Given the stability of the effect for might, we argue against the idea that the new data present evidence against the erotetic view.Rather, we outline how different extant theories of modality can account for the contrast we observe.Specifically, we discuss how certain possibility modals are conventionally associated with the generation of alternatives and that theories which incorporate this property in their analysis fare better when accounting for our data.As a result, we provide arguments from the psychology of reasoning that help winnow the conceptual space of theories of possibility modals in novel ways.1

Illusory inferences from disjunction
The erotetic theory of reasoning of Koralus & Mascarenhas (2013) incorporates linguistic insights into a variant of the mental-models theory of reasoning (Johnson-Laird 1983) to account for a class of attractive fallacies known as illusory inferences from disjunction.Consider the example in (1), greatly simplified from the original examples discovered by Walsh & Johnson-Laird (2004). (1) Premise 1: John speaks English and Mary speaks French, or else Bill speaks German.
The reasoning problem in (1), along with a number of structurally identical problems, have acceptance rates between 80% and 85% (Walsh & Johnson-Laird 2004, Mascarenhas & Koralus 2017, Koralus & Mascarenhas 2018).Yet, the deductive inference in (1) is a fallacy.Suppose Bill speaks German (modeling premise 1) and John speaks English (premise 2), but Mary does not speak French.This is a model of the premises but not the conclusion, and the inference in (1) is thus invalid.

Alternatives, mental models, and illusory inferences
Building on the mental-models explanation of these fallacies by Walsh & Johnson-Laird (2004), the erotetic theory of reasoning offers an account of this and related illusory inferences proposing that a question-answer dynamic is at the core of these fallacies.For ease of exposition, let us consider the logical structure behind the example in (1).
(2) Premise 1: ( ∧ ) ∨  Premise 2:  Fallacious conclusion:  Following Hamblin semantics (Kratzer & Shimoyama 2002, Alonso-Ovalle 2006) and inquisitive semantics (Groenendijk 2008, Mascarenhas 2009), the erotetic theory of reasoning takes the disjunction in the first premise of (2) to raise an issue, putting forth two alternatives: ∧ and .The reasoner is now effectively entertaining a question, and will seek to find the most expedient way of answering or dispelling it.As it turns out, the second premise offers hints at an answer: the second premise  is related to the first alternative  ∧  rather than the second , and the reasoner rushes to pick an answer: The right alternative from the first premise is  ∧ , whence  follows by a mental-models analog of conjunction elimination.2

Alternatives in language beyond disjunction
If the attractiveness of fallacious schemata as in (2) is to be explained in terms of the presence of alternatives in the interpretation of the first (disjunctive) premise, then we expect other linguistic items that have been argued to raise alternatives of a similar kind to produce similar illusory inferences.
In the traditions of both Hamblin semantics (Kratzer & Shimoyama 2002) and inquisitive semantics (Ciardelli 2009), indefinites are akin to wh-questions. 3An expression such as some pilot is assigned roughly the same mean-ing as which pilot.Accordingly, a sentence like Some pilot writes poems is analyzed as the set of all propositions of the form  writes poems, for  a pilot.In other words, the meaning of the question Which pilot writes poems?In this analysis, the erotetic theory predicts that the example below should give rise to an illusory inference. (3) Premise 1: Some pilot writes poems.
Premise 2: John is a pilot.Fallacious conclusion: John writes poems.
Mascarenhas & Koralus (2017) found that inferences like (3) were in fact attractive, with acceptance rates around 35%, significantly above the baseline for mistakes established by invalid controls without alternative-generating elements.This demonstrates the existence of the predicted illusory inferences with alternative-generating linguistic operators besides disjunction.

Reasoning vs. interpretation
Illusory inferences from disjunction as in (1) have acceptance rates around 85%, while illusory inferences with indefinites as in (3) are accepted around 35% of the time.Why the different acceptance rates, if both disjunctions and indefinites produce alternatives according to our best semantic theories?Mascarenhas (2014) showed that the disjunction inferences are amenable to a pragmatic account.As an alternative to mental-models accounts we can derive the observed conclusion assuming an entirely classical reasoning module, acting on pragmatically strengthened meanings of the premises.Under most modern theories of scalar implicature (e.g.Sauerland 2004, Spector 2007) the first premise of the illusory inference from disjunction in (4a) is predicted to be interpreted as in (4b), a fact Spector (2007) already observed outside the context of reasoning problems.
Assuming (4a) is interpreted as in (4b), and incorporating the second premise , the conclusion  is no fallacy at all.It follows classically from the conjunction of the two premises.Crucially, no absolving implicature is predicted tive semantics and Hamblin semantics.For example, we make no predictions for inference patterns with premises involving focus alternatives (Rooth 1996).
by any existing theory of scalar implicature for illusory inferences with indefinites.Recruiting experimental paradigms from psychosemantics into the domain of reasoning problems, Picat (2019) argues that, rather than being competing accounts, the erotetic theory and the scalar-implicature account are two possible routes leading to the same inference-making behavior in the case of the illusory inference from disjunction.For the indefinites case, only the erotetic theory predicts a fallacy, the scalar-implicature route is blocked, explaining its lower endorsement rate.

Deduction, norms of rationality, and probabilistic reasoning
The inference in (1) has the form of a deductive problem, and the experimental methodologies used to investigate it and related data points ostensibly rely on deduction and logical validity.Yet, a whole group of approaches to human reasoning exists that rejects the idea that logical validity plays any role in and of itself in human reasoning.Work in the "New Paradigm" tradition (see in particular Oaksford & Chater 2007) holds that the functional aim of the human reasoning faculty is not to track logical validity, but instead to deliver rational answers to decision problems solved under uncertainty.In this view, human reasoning relies on probabilistic inference, and the rational norm is the classical probabilistic calculus, instead of classical validity.
Deductive validity emerges as a special case of the probabilistic calculus, but it is by no means the goal of the system.One particularly fruitful model, for example, proposes that naive reasoners consider the probability of the putative conclusion conditional on the conjunction of the premises, and check whether that posterior probability is higher than a contextually defined standard.If it is, they respond that the conclusion is indeed valid.Another view uses the notion of -validity, where a conclusion will be deemed valid if its probability is greater than that of its premises, across probability distributions.
Interestingly, accounts in this vein of inferences as in (1) above are impossible, unless they make the same non-classical commitments as the erotetic theory regarding the interpretation of disjunctions and the dynamics of questions and answers.Consider the logical form of the inference as schematized in (2).The conjunction of the premises is as in (5a) below, which is equivalent in classical logic to (5b).Since the classical probabilistic calculus is built atop classical logic, the equivalence above holds in the relevant probabilistic approaches.Consequently, we will get the following equality for the joint probability of the premises, where the comma ',' represents premise conjunction.
(( ∧ ) ∨ , ) = (( ∧ ) ∨ ( ∧ )) However, this means that this approach cannot distinguish between the observed conclusion  and an unobserved and altogether unintuitive conclusion of , unless their prior probabilities were significantly different, which while possible is entirely unwarranted, certainly for the myriad examples of these inferences available in the literature.Consequently, a view of human reasoning that aims to interpret apparent mistakes as rational probabilistic inferences purely in terms of prior or posterior probabilities takes no account of these illusory inferences from disjunction. 4 A final word on terminology is in order.As we have explained, the idea that there is such a thing as properly deductive reasoning among naive humans is a matter of debate.Our goal here is to discuss the crucial importance of alternatives and the semantics that generate them in reasoning and in language, and accordingly we are altogether agnostic as to whether the human faculty for reasoning targets probabilistic reasoning, deduction, or both.We will continue to use terms like "deduction" and "validity" in a descriptive sense, however, referring to the at least superficial nature of the tasks at hand, which are ostensibly about deduction and validity, and not about probabilistic inference.Additionally, we continue to refer to deviations from the classical norm of deduction as "fallacies," a term we also mean purely descriptively.
4 There are, however, good reasons to think that a probabilistic component is part of the phenomenon.Sablé-Meyer & Mascarenhas (2021) have shown that illusory inferences are possible with second premises that do not entail an element of the first premise, but are instead merely probabilistically connected to it.They show that the complete story involves combining the erotetic theory with a probabilistic strategy for selecting an alternative from the first premise based on the information in the second premise.Crucially, the role of alternatives as studied in linguistic semantics is no less central in their theory, and the probabilistic component is one of Bayesian confirmation theory, rather than the standard of rationality in "New Paradigm" probabilistic approaches.This more complex version of the erotetic theory is not needed for the cases of interest in this article, so we refrain from presenting it.(6) John might be in London.
Ciardelli, Groenendijk & Roelofsen (2009) argue for a version of inquisitive semantics where a sentence may give rise to alternatives displaying proper inclusion between them.They propose that the meaning of ( 6) should be seen as identical to (7a) in their version of inquisitive semantics, where ⊤ is the tautology.Accordingly, they argue that (6) gives rise to the alternative set in (7b).
(7) a. John is in London ∨ ⊤ b. {John is in London, ⊤} Crucially, the sentence as a whole is not a tautology in inquisitive semantics.It is informationally idle in that an update with this sentence will not exclude possibilities from any common ground.Nevertheless, it contains two distinct alternatives, it is not identical to the interpretation of the tautology, and is therefore not equivalent to it in inquisitive terms.Ciardelli, Groenendijk & Roelofsen (2009) argue that the semantic contribution of a might sentence is not to provide information or raise a bona fide question, but simply to draw attention to a single possibility.Since one of the alternatives is the trivial alternative, to which it makes no sense to draw attention, a might-sentence in this semantics offers one alternative for consideration, namely: in the case of ( 6), the proposition that John is in London. 55 Ciardelli (2009) and Ciardelli, Groenendijk & Roelofsen (2009) stress that their analysis is not meant to capture epistemic uses of might.They argue that this may be an advantage of their analysis, as might has been argued to differ from other epistemic operators.However, we see no principled reason not to assume that different epistemic operators differ in their sensitivity to alternatives.

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Alternatives and attention in language and reasoning Consider now a conjunction embedded under might as in (8a), which in the view just presented is to be analyzed as (8b).If the fallacious conclusion in ( 10) is an illusory inference resulting from erotetic reasoning, a certain pattern is predicted, also based on what previous research on indefinites and disjunction has revealed.First, if the fallacy is indeed attractive, the rate of "yes"-responses to the conclusion of critical targets containing modals should be higher than the rate of "yes"-responses to invalid control problems not containing might.
The well-established cases of the fallacy of interest display order effects where the attractiveness of the fallacy is mitigated when the order of the premises is reversed, as would be expected if the fallacy relies on a questionanswer dynamic (Mascarenhas & Koralus 2017, Koralus & Mascarenhas 2018).Thus, an effect of order is predicted for the cases at hand should erotetic reasoning be involved.Additionally, fallacious conclusions should disappear altogether if the two premises are combined into one sentence, "flattening" the question-answer structure.
To test these predictions, Mascarenhas & Picat (2019) included reasoning problems with the structure in (10) in their study, as well as items with the order of the two premises reversed, and a "flat" structure version that 2:9 combines the truth-conditional information in the two premises into a single sentence.
Additionally, they considered the possibility that the first premise alone might suffice to prompt the conclusion, a plausible hypothesis since the first premise certainly raises the probability of the embedded conjunction.If fallacious conclusions in the canonical case in (10) were entirely attributable to the first premise alone, then these fallacious conclusions would not constitute a case of the illusory inferences of interest.
Summing up, they tested the four conditions in (11) in a between-subjects design, due to the high degree of similarity between the stimuli in the conditions.
(11) a. Canonical They found that canonical and reversed targets showed significantly more "yes"-responses than "no"-controls, displaying the fallacious conclusion they predicted.Canonical and reversed targets were also more accepted than flat targets, which in turn were no different than invalid controls, showing as expected that the mistake disappears if the question-answer dynamic is destroyed.They also found that canonical targets were more accepted than P1 targets, proving that the fallacy cannot simply be explained in that the first premise raises the probability of the embedded conjunction.They failed to find an effect of the order of premises 1 and 2 being reversed, however; canonical and reversed targets were not significantly different.Mascarenhas & Picat (2019) tentatively attribute this null result to their experiment's being underpowered to detect an effect of the size witnessed in earlier studies for canonical vs. reversed illusory inferences, a drop in endorsement of the fallacy in the order of ten percentage points (Mascarenhas & Koralus 2017).Lastly, they found that the rate of fallacies with might was comparable to that with indefinites, which suggests, they argue, that the fallacy is based on erotetic reasoning alone and not on scalar implicatures.Mascarenhas & Picat (2019) conclude that the findings are in line with Ciardelli's (2009) proposal that might introduces inquisitive/Hamblin alternatives but could also be made sense of from of a Kratzerian view on possibility modals together with the assumption that existential quantifiers raise alternatives.

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Alternatives and attention in language and reasoning

Illusory inferences with deontics
Mascarenhas & Picat (2019) argue convincingly that might produces fallacious inferences.With the exception of the null result for order effects between premises, these fallacious inferences have properties particular to illusory inferences with disjunction and indefinites of interest, lending weight to the idea that the same mechanism is at play.However, their results are merely compatible with the strong hypothesis that all and only inquisitive-semantics/Hamblin-style alternatives give rise to illusory inferences of the relevant kind in reasoning.To test this more ambitious hypothesis, we need to extend the methodology of connecting alternative generation with inference-making behavior to more cases.In particular, it is important to find a case that is close enough to the existing cases to justify a suspicion that illusory inferences might be found, while being distinct enough analytically to support substantive theoretical considerations.We propose adding deontic modals to the mix.
Extending Mascarenhas & Picat's (2019) paradigm to deontic modality addresses several questions left open by their study, and can potentially inform our theories of modality and reasoning.To our knowledge, no theory of deontic possibility modals exists that argues that they are alternative generators in the sense introduced above.Yet, they are very close in form and meaning to epistemic modals (Kratzer 1977(Kratzer , 1991(Kratzer , 2012)), for which such a theory might in principle be formulated, as outlined by Mascarenhas & Picat (2019).We return to these theoretical considerations at the end of this article.Like Mascarenhas & Picat (2019), we used an inference-making task to test a series of predictions related to the alternativegenerating power of epistemic and deontic possibility modals.

Design and Materials
The target reasoning problems involved one of two modals, epistemic might or deontic allowed to.They instantiated one of the three structures in (12).We treated structure as a between-subjects factor due to the sentences in the three conditions being very similar.We fully crossed this factor with the factor modality, which we treated as a within-subjects factor.As 2:11 Nadine Bade, Léo Picat, WooJin Chung, Salvador Mascarenhas a result, a given participant saw only one of the structure conditions in ( 12), but saw both modals, in different items. 6  (12) a. Canonical might have/was allowed to ( ∧ ),  ⊢  b.Flat  ∧ might have/was allowed to ( ∧ ) ⊢  c.Reversed , might have/was allowed to ( ∧ ) ⊢ We give examples of each structure with epistemic and deontic modality in (13).
(13) a. John {might have stolen / was allowed to steal} from the rich and give(n) to the poor; John stole from the rich ⊢ John gave to the poor.Canonical b.John stole from the rich, and he {might have stolen / was allowed to steal} from the rich and give(n) to the poor ⊢ John gave to the poor.Flat c. John stole from the rich; John {might have stolen / was allowed to steal} from the rich and give(n) to the poor ⊢ John gave to the poor.Reversed We also varied which of  vs.  from the first premise appeared in the second premise, controlling for any potential order effects from the embedded conjunction.This last factor order was tested within subjects.
In sum, we manipulated 3 factors: modal with two levels (deontic or epistemic), structure of the reasoning problem with three levels (canonical, flat, reversed) and in which order the two conjuncts in premise 1 appeared in premise 2/conclusion (order "left": left conjunct of premise 1 appeared in premise 2, right conjunction in conclusion; order "right": right conjunct of premise 1 appeared in premise 2, left conjunct appeared in conclusion).Thus, we had 2 × 2 × 3 = 12 conditions.We created 6 items per withinsubjects condition (4 in total, deontic-left, deontic-right, epistemic-6 There were two minimal changes in the material compared to that tested by Mascarenhas & Picat (2019).First, we minimally changed the flat structure condition.This was because the more extreme version of the flat condition a and modal(b) used previously prompts additional inferences in the deontic case that are absent from the epistemic case.Specifically, a and allowed to b may give rise to the implicature not allowed to a given that a is a salient excludable alternative.This reading is not available for might, as not might a contradicts a, and therefore cannot be excluded innocently (Fox 2007).We also changed the tense to the past form.We did this to avoid the ambiguity of the present tense in English, and a possible interference with type of modality.We did not expect this change to affect epistemic might.

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left, epistemic-right).These 24 critical items were distributed across 4 experimental lists for each group factor (structure) separately, resulting in 12 sub-experiments that participants were randomly assigned to.
Besides target items, we used 12 control items and 12 baseline items.Control items served the purpose of eliminating unfocused participants.They used valid and invalid conclusions based on modus ponens and disjunctive syllogism.Baseline items used conjunction elimination and were used to establish a baseline for error rates.

Participants and Procedure
We recruited 183 participants via Prolific; 56% were female and their mean age was 33 (ranging from 20 to 60, SD = 13.2).Participants were assigned to one of 12 experimental lists and each solved 48 reasoning problems.The procedure was the same as used in Mascarenhas & Picat 2019.Participants were presented with premises and a proposed conclusion.They had to evaluate whether the conclusion followed from the premises.Before the core of the experiment began, we explained and exemplified the concepts of valid and invalid conclusion.
Our exclusion criteria were the same as in Mascarenhas & Picat 2019.We excluded people who reported having often taken notes, having taken at least one graduate-level course in semantics/pragmatics, or who answered fewer than 50% of control questions correctly.With these exclusion criteria we removed 26 participants (14.2%) and analyzed data from 157 participants.

Results
We analyzed the results with generalized linear mixed-effects models, using the lme4 package and glmer function in R. We report the most complex converging model.Fixed effect factors were modal with levels deontic/epistemic (reference level: deontic), order of conjuncts with levels left/ right (reference level:left), structure with levels canonical, flat, reversed (reference level: canonical).All fixed effects were dummy-coded.We calculated contrasts with pairwise comparisons using the emmeans package in R and used Holm correction for multiple comparisons.We report both corrected and uncorrected -values.
First, we looked at whether deontic modals give rise to fallacious inferences of the form modal(a and b); a ⊢ b.We compared the rate of falla-

Table 1
Summary of output from the model testing the influence of inference type on responses.
cies with canonical structures to "no"-baselines for both modals separately. 7 Our analysis shows that the rate of "yes"-responses differs significantly from "no"-baselines for both modals; see Figure 1 and Table 1.
To test whether the two modals are affected differently by structure, we looked at the interaction between structure and modal in our analysis.We compared the maximally converging model with the interaction term (RESP ∼ MODAL*STRUCTURE + (1 SUBJECT) + (1 ID)) to a model without the interaction term (RESP ∼ MODAL+STRUCTURE + (1 SUBJECT) + (1 ID)).The nested model comparison via log likelihood ratio tests revealed that the interaction term is justified ( 2 (2)=33.887).The interaction shows that structure affects epistemic and deontic modals differently.Zooming in on contrasts, we observe a significant difference between canonical versus flat structures for both modals, with the effect being more pronounced for epistemics.Canonical and reversed structures, however, do not differ significantly from each other for deontics.The contrast is only marginally significant for epistemics (Table 2).This result for epistemics is consistent with previous experimental results showing that the effect of order of premises is very subtle, and only visible with a very high number of participants (Mascarenhas & Koralus 2017).The presence of a significant interaction together with the difference in estimates, however, is suggestive of epistemics being affected by reverse versus canonical order of premises, whereas deontics are not.Zooming in further, we looked at the contrast between "no"-baseline items and flat structure items, and "no"-baseline items versus reversed target items, for both modals separately. 8We see that, for epistemics, flat structures do not differ significantly from "no"-baselines (Table 3).However, for deontics, they do (Table 4).Furthermore, there is a significant difference between reversed targets and "no"-baselines for both modals.The contrasts we find for each modal are displayed and highlighted in Figures 2 and 3.

Table 3
Output of the model looking for the effect of structure (flat and reversed) on responses for epistemics ("no"-baselines as reference level)

Table 4
Output of the model looking for the effect of structure (flat and reversed) on responses for epistemics ("no"-baselines as reference level) We also investigated the contrast between modals for each structure.We summarize this contrast in Table 5.We observe that there are significantly more "yes"-responses for deontics than for epistemics for each structure. 9

Table 5
Contrasts between epistemic (reference level) and deontic modality for each structure.

Summary of experimental results
We replicated the findings for epistemics from Mascarenhas & Picat 2019.There are fewer fallacies for reversed and flat targets than there are for canonical targets.Furthermore, reversed and canonical targets differ from "no"-baselines, whereas flat targets do not.We observed a significant rate of fallacies with deontics as well, as evidenced by a significantly higher rate of "yes"-responses to canonical targets than "no"-baselines (main effect).However, the rate is significantly higher 9 We also checked for a possible effect of order of conjuncts (Was the left or right conjunct of premise 1 mentioned in premise 2/conclusion?) by comparing a model RESP ∼ MODAL*ORDER*STRUCTURE + (1+MODAL*ORDER subjectId)+ (1+ORDER id) to MODAL+ORDER+STRUCTURE + (1+MODAL*ORDER subjectId)+ (1+ORDER id).We found the interaction term to be justified ( 2 (6)=13.683).Since we had no specific predictions regarding the effect of order of conjuncts, we refrain from offering post-hoc speculative explanations.The presence of an interaction, however, suggests that following the linear order of presentation of the conjunctions in premise/conclusion is not (one of) the main sources of the overall effect.

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Nadine Bade, Léo Picat, WooJin Chung, Salvador Mascarenhas than for epistemics and they are much less sensitive to structure (interaction).Specifically, we found no difference between reversed and canonical targets with deontics, and we found that deontic flat structures produce a fallacy.

General Discussion
At first glance, our results appear to challenge the conjecture that the fallacies of interest are intrinsically connected with inquisitive/Hamblin alternatives: we found what look like the epistemic illusory inferences in deontics.This is surprising because, unlike in the case of epistemics, no extant theories of deontics propose that they generate the relevant kinds of alternatives.However, standard possible-worlds accounts see epistemic might and deontic allowed to on par with one another and the common core semantics traditionally proposed for both cases, and indeed for possibility modals in general, is existential in nature: There is some (properly restricted) accessible world where the prejacent proposition is true.If this existential quantifier in the truth conditions for possibility modals were interpreted within inquisitive/Hamblin semantics, it would give rise to alternatives, roughly corresponding to the question "Which accessible possible world is such that the prejacent is true in it?"This idea can be developed into an account of illusory inferences with epistemics quite naturally.
Assuming the relevant ordering between possible worlds is well founded (the limit assumption), Kratzer's (1991) semantics for modality predicts that a sentence might() will be true just in case there is a -world among the best-ranked worlds.Along with a few assumptions, there is a strategy for accounting for epistemics in a relational semantics such as Kratzer's. (14) a. Assertion as truth in the actual world When asserting a proposition , a speaker communicates (their belief) that  holds in the actual world.b.

Inquisitiveness of existentials and erotetics
The existential quantifier that occurs in the truth conditions of a might-sentence is inquisitive.Human reasoning is erotetic, and in particular inquisitive existential quantifiers raise questions that are resolved along the lines outlined in Section 1.3 for illusory inferences with indefinites.

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Under these assumptions, the first premise of epistemic fallacies might( ∧ ) provides some information about the existence of a possible world, and draws attention to the question "Which is this best-ranked  ∧ -world we're discussing?" Asserting the second premise then states that the actual world is an -world.Now erotetic mechanisms kick in: The actual world is an world, and this points in the direction of answering the question at hand with "The actual world is the best-ranked  ∧ -world in question."From here,  follows immediately.
Extending this account to deontics in order to assimilate them to epistemics is in principle possible and constitutes an important theoretical avenue to explore.The main trouble is that the assumption of reflexivity in (14b), entirely standard for epistemic modality, is generally unwarranted in deontic modality.Indeed, reflexivity in a deontic frame guarantees that everything that is in fact the case is ipso facto permitted.This is highly implausible as an assumption that experimental participants should make when reasoning about deontic problems, but it remains to be seen whether there are other ways to leverage the shared existential semantics of epistemics and deontics and extend the relational account of epistemics we just sketched to a general theory that predicts illusory inferences for all possibility modals.In the remainder of this discussion section, however, we pursue a very different approach.
Further experimental and theoretical research is needed that would lie outside the scope of this reply, but we submit that there are two orders of reasons to suspect that the fallacies we found with deontics are in fact of a different kind than the ones of interest, found with disjunctions, indefinites, and epistemics.In other words, the deontic fallacies are generated by a different mechanism (expanded on below) than the ones with epistemics, and specifically deontic fallacies do not originate from the same alternativegenerating and alternative-handling mechanisms as epistemic fallacies.Below, we furthermore propose a novel operationalization of the inferencemaking phenomenon of interest that defines a more complex signature for these fallacies, going beyond simply looking at the rate of acceptance of canonical cases.Our proposed definition of "illusory inferences with alternatives" puts together previously observed properties of the clear and experimentally robust cases with disjunctions and indefinites.We submit that the picture that emerges allows us to distinguish epistemics from deontics, thereby rehabilitating Mascarenhas & Picat's (2019) hypothesis that illusory inferences of this kind are inextricably connected to inquisitive/Hamblin al-

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Nadine Bade, Léo Picat, WooJin Chung, Salvador Mascarenhas ternatives as studied by semanticists, and without pursuing the radical and unlikely hypothesis that all possibility modals should give rise to illusory inferences.

Another theory of deontic "fallacies"
We found that about 50% of the time our participants drew the fallacious conclusion in (15).( 15) Premise 1: allowed to ( ∧ ) Premise 2:  Fallacious conclusion:  There is an independently observed property of deontics that would produce behavior as in (15) for deontics, without making such predictions for epistemics.Witness first that the inference in (16a) does not necessarily go through for the sentence in ( 16).This effect has been dubbed a package deal by the literature on deontics (Merin 1992, van Rooy 2000).( 16) John is allowed to steal from the rich and give to the poor.
a. ↛ John is allowed to steal from the rich.
The effect can be described as a sentence 'allowed to  and ' only giving permission to do both or neither  or , which van Rooy (2000) calls biconditional permission. 10For reasons of space we cannot go into the details of the suggested analyses.Still, we would like to propose that the package deal reading could play a role in the rate of fallacies we observe, despite the fact that it does not make the inference that John stole from the rich a valid one: Given that there is no reflexivity with deontic modals, it is not granted that John only did what he was allowed to do.However, the first premise suggests that he partly did what he was allowed to do, making it more plausible that he is following the rules.This, in turn, means that he likely respects the package deal, and wouldn't perform one action without also performing the other, making the observed conclusion a very attractive one.Crucially, this attractiveness stems from a mechanism entirely independent from alternative generation or question-answer dynamics.This line of explanation clearly requires more investigation as the reasons for drawing the fallacious conclusion could be multi-factorial, with package deal effects only being one contributing factor.

Anatomy of illusory inferences from alternatives
The most conspicuous property of the illusions of interest is of course drawing the target fallacious conclusion.Yet, the well-studied data pertaining to disjunctions and indefinites strongly suggest that these inferences have a more complex signature.We propose that the conjunction of four points is required to diagnose the illusory inferences of interest.
A "Fallacy" More fallacious conclusions than for invalid controls.The hallmark of illusory inferences from alternatives is a fallacious conclusion that picks one of the available possibilities (alternatives) raised by the first premise on the basis of merely partial overlap with the information in the second premise.
B "Order of premises" Fewer fallacious conclusions with reverse order of premises.The erotetic theory of reasoning explains illusory inferences from alternatives in terms of a question-answer dynamic.Reasoners consider the disjunctive or indefinite first premise as raising a question, to which the second premise provides a hint at an answer.It should be harder for participants to engage in question-based reasoning if the order of the two premises is reversed.Indeed, illusory inferences with disjunction and with indefinites both show a drop in acceptance of about 10 percentage points when the order of the premises is reversed (Mascarenhas & Koralus 2017, Koralus & Mascarenhas 2018).Crucially, the same experimental studies did not find a drop in acceptance in control inferences with premises of comparable syntactic complexity.
C "Dynamics" No fallacious conclusions without dynamics.A much more radical mitigation of the phenomenon occurs if the dynamic structure of the stimulus is altogether destroyed (as in the flat structures in this article and its 2:21 Nadine Bade, Léo Picat, WooJin Chung, Salvador Mascarenhas precursor).This is a novel criterion we propose due to the findings presented in this paper and Mascarenhas & Picat 2019.When the order of the premises is reversed and combined into a single premise with a conjunction, we see no illusory inferences from alternatives.

D "Rate" Rate of fallacious conclusions depends on available mechanisms.
As explained in the introduction, we expect there to be a lower rate of fallacies if the mechanism behind arriving at the conclusion is solely based on erotetic reasoning.Illusory inferences with disjunctions have alternative accounts in terms of scalar implicature, while illusory inferences with indefinites lack such alternative accounts, explaining their diverging acceptance rates.
In view of these criteria, the fallacies with might constitute an instance of the illusory inferences of interest, for they satisfy all points A-D, as do the paradigmatic cases of disjunctions and indefinites.Yet, the fallacies we found with deontics do not satisfy all of these points.Specifically, they fail B and C, and have a doubtful status for D. Accordingly, we cautiously propose that the fallacies with deontics are not an argument in favor of their inquisitive/Hamblin-alternative potential, but the product of altogether different mechanism(s).

Conclusions
We extended experimental work on reasoning with alternatives from epistemic modals to deontic modals.Our goal was to investigate the hypothesis that all and only linguistic elements that generate inquisitive/Hamblin alternatives produce the illusory inferences the literature has found with disjunctions and indefinites.We found fallacies with deontics reminiscent of the ones with epistemics, but we failed to find other characteristic properties of the illusory inferences with disjunctions and indefinites.Our results are compatible with two main interpretations.The first is that deontics produce alternatives just like epistemics.From a theoretical standpoint, it would be most natural to leverage the shared existential semantics of all possibility modals in relational semantics and give a unified account of the alternativegenerating potential of epistemics and deontics.The central challenge in this connection will be to discover plausible assumptions about the accessibility relation for deontics that would derive this result.From an experimental per-2:22 spective, the challenge will be to explain the sharp differences we found between epistemics and deontics once one zooms out from the main fallacious structure of interest to consider well-established properties of these fallacies with disjunctions and indefinites, the paradigmatic cases of alternative generation and illusory inferences.
The second possibility is that only epistemics generate alternatives and participate in the illusory inferences of interest.The fallacies we observed with deontics would be generated by an entirely unrelated mechanism.We proposed that the notion of package deals, independently proposed in the literature on deontics, would predict the inference-making behavior we observed without postulating alternatives.Additionally, this analysis would explain the fact that fallacies with deontics have a different broader signature than fallacies with epistemics, since they would be generated by entirely independent mechanisms.

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(5) a. (( ∧ ) ∨ ) ∧  b. ( ∧ ) ∨ ( ∧ ) 2:6 (2019)  pursued the hypothesis that there is a strong connection between inquisitive-semantics/Hamblin-style alternatives and illusory inferences by testing the English epistemic possibility modal might.They based their conjecture on an analysis of might byCiardelli, Groenendijk & Roelofsen (2009) in the framework of inquisitive semantics, involving attentive content, which would draw the listener's attention to a single proposition.Consider the sentence in (6).
(8) a. might( ∧ ) b. ( ∧ ) ∨ ⊤ The logical structure in (8b) is a special case of the first premise of standard illusory inferences from disjunction reviewed above: ( ∧ ) ∨ .If this analysis is on the right track, then we expect the schema in (9) to give rise to illusory inferences.(9) Premise 1: might( ∧ ) Premise 2:  Fallacious conclusion:  Mascarenhas & Picat (2019) used reasoning tasks to test the hypothesis that might introduces alternatives and thus invites illusory inferences.The target reasoning problems had the structure in (9), instantiated in (10).
(10)   Miranda might play the piano and be afraid of spiders.Miranda plays the piano.Fallacious conclusion: Miranda is afraid of spiders.

Figure 1 Figure 2
Figure 1Proportion of "yes"-responses by inference type; error bars indicate the standard error of the mean.Black lines and stars indicate significant contrasts.

Figure 3
Figure 3 Proportion of "yes"-responses by structure versus "no"baselines for deontics; error bars indicate the standard error of the mean.Black lines and stars indicate significant contrasts.