A probabilistic semantics for modal QAD in standard Arabic

Abstract Gradable Epistemic Modals (GEMs) such as certain or likely have been analyzed to have the semantics of gradable adjectives, which constrain their epistemic modality. This squib analyzes the modal use of the sentential particle qad in Standard Arabic which has the following puzzling behavior: particle qad gives rise to the certainty/likelihood modality meaning only when it is composed with the tense expression of its propositional complement. Modal qad then manifests itself as an extreme, non-gradable modal with no lexically encoded but grammatically derived scale meaning. A probabilistic semantics is proposed to derive the truth conditions of the epistemic modal meaning of qad based on a monotonic, probabilistic time-world branching model. The analysis assigns maximum degree of likelihood to qad in the past tense which triggers certainty. It also assigns a minimum degree of likelihoods to qad in the present tense which triggers possibility.

In almost all languages, modality is an expression of necessities and possibilities that characterizes some state of affairs which is displaced from the actual world such as knowledge, obligation, or ability (Hockett, 1960).In modal logic, a necessity or possibility modal is used to talk about possible worlds which hold true of the modal's propositional complement with the two being analyzed as a universal and an existential quantifier over possible world, respectively (Barcan, 1946(Barcan, , 1947;;Carnap, 1946;Kripke, 1963). 1   Natural language modality comes into at least two types of expression: extreme vs. intermediate/comparative modals.Extreme modals contain only weak and strong auxiliary modals.Their quantificational strength is fixed once and for all to the two extreme forces of quantification so they cannot be modified by degree operators as exemplified in (1).
(1) a. John may be/must be the murderer.
EPISTEMIC POSSIBILITY/NECESSITY b.John may/must go to bed at 8 pm.DEONTIC POSSIBILITY/NECESSITY ABOUT THE AUTHOR Abdel-Rahman Abu Helal is an assistant professor of linguistics working for the Department of English Language anand Literature at Al-Zaytoonah University of Jordan.
Intermediate/comparative modals are readily gradable (e.g., likely and certain).They are associated with a scale 2 and a threshold which can be manipulated by comparative and degree operators such as the examples in (2).
(2) a.It is very likely that Jorge will win the race.
b.It is more likely that Jorge will win the race than it is that Sue will win.
c. How likely is it that Jorge will win the race?d.It is 95% certain that Jorge will win the race.
Two lines of analysis have been put forth to explain the two kinds of modality.On a standard quantificational approach (Kratzer, 1977(Kratzer, , 1981(Kratzer, , 1991)), the two strengths of extreme modals correspond to a duality between an existential and universal quantification over structured sets of possible worlds which are determined by a contextual parameter called conversational background.Conversational backgrounds restrict the worlds under quantification through the interaction of two functions from worlds to the propositions in such a way that the flavor and force of modality being semantically valued.Under this approach, intermediate and comparative modals can be given truth conditions via the so-called comparative possibility which lifts relations on worlds to relations on sets of worlds (i.e., propositions) (Kratzer, 2012).
The other approach is based on a theory that is built around the notion of probability (Lassiter, 2010(Lassiter, , , 2017;;Yalcin, 2007Yalcin, , 2010)).This theory emerges as a consequence of some empirical and theoretical problems facing the Kratzerian semantics of intermediate/comparative modals.To mention but one problem, Yalcin (2010), and independently Lassiter (2010), observed that Kratzer's comparative probability can be used to license invalid inferences with respect to comparative epistemics like the disjunctive inference in (3) which predicts wrong consequences such as (4).
(3) a. α is at least as likely as β b. α is at least as likely as γ c. ∴ α is at least as likely as β ∨ γ (Lassiter 2014: p 5) (4) Context: a fair lottery with 1 million tickets in which every participant including Sam buys only 2 tickets.a. Sam is as likely to win the lottery as anyone else.
b. ∴ Sam is as likely to win the lottery as he is not to win it.
(Lassiter 2014: p 2) Lassiter (2014) showed that a probabilistic semantics to comparative/intermediate modals never encounters this problem; it doesn't license invalid inferences such as (3).It also provides a compositional analysis for intermediate/comparative modals with denotations compatible with the notion of gradability that characterizes the semantics of gradable adjectives (e.g., Kennedy, 2007;Kennedy & McNally, 2005). 3On this view, if we think of the graded epistemic modal (GEM) as a degree measure function which is a mapping from propositions to real probability value between [0,1] and which is additive, 4 we can have a straightforward compositional analysis that derives the correct truth conditions of GEMs in its intermediate and comparative forms and at the same time it doesn't endorse invalid inferences such as the disjunctive inference in (3).Not all certainty/likelihood epistemics exhibit this behavior.Standard Arabic has the sentential particle qad which represents a special case of certainty/likelihood-denoting modality that is neither gradable nor lexically scalar. 5Modal qad only operates on tensed propositions and its certainty/likelihood meanings are grammatically determined by the tense specification of its propositional argument: while the certainty-denoting qad is mainly associated with past tense propositional complement, the likelihood-denoting qad only arises with the present tense complement as exemplified in (5) (Abu Helal, 2023;Bahloul, 2008;Fassi, 2012). 6  (5) a. QAD was ʕ al-a ʔamsi CERTAINTY PRT MODAL arrived-PAST-3SG.M yesterday "He did arrive yesterday." b. QAD y-as ʕ il-u l-yaam-a LIKELIHOOD PRT MODAL arrive-PRESENT-3SG.M the-today "He may arrive now."(Fassi Fehri, 2012: p. 41) In formal semantic terms, the puzzle of modal qad can be rephrased as follows.On the assumption that the meanings of certainty and likelihood are characterized by the maximum and minimum positive interpretations relative to a common scale of probability (Lassiter, 2010 and subsequent work), the two positive interpretations of qad follows from a scale structure that is not inherently encoded in the lexical semantics of qad, but it is grammatically determined by the tense structure of its propositional complement.Therefore, the certainty/likelihood denoting qad is neither gradable nor lexically scalar.To the best of our knowledge, no formal semantic account has been offered to explain this puzzling behavior of modal qad. 7 The raison d'être of this squib is to offer a straightforward solution to the modal qad question by proposing a version of probabilistic semantics based on a particular world-time branching model and to discuss the implications it has for theory of graded modality and measurement (mainly Lassiter 2014).The squib is structured as follows.Section one describes the pattern of epistemic modal qad.Section two outlines two representative approaches to modality represented by Kratzer's comparative possibility and Lassiter's probability.Section three offers a unified analysis to modal qad based on a variant of probabilistic semantics and world-time branching framework.Section four discusses some far-reaching theoretical implications for the theory of graded modality and measurement.The last section concludes the paper. 8

The pattern of Modal QAD
As outlined in the introduction, modal qad exhibits the following two puzzling behaviors: first, qad's modal strength is grammatically governed by the tense structure of its propositional argument (Abu Helal, 2023;Bahloul, 2008;Fassi, 2012).When the qad particle selects for a past tense proposition, it denotes "certainty".For example, the qad sentences in (6) all mean that the eventuality described by the qad's propositional complement is certain.PRT MODAL be-PAST-3SG.M arrived "He had indeed arrived."(Fassi, 2012 p. 8 & 105) Similarly, when qad takes a present tense proposition, it denotes likelihood or possibility.In the following example, the qad sentences in (7) mean that the eventuality described by the qad's propositional complement is possible or likely. 9 (7) QAD y-as ʕ il-u l-yaam-a PRT MODAL arrive-PRESENT-3SG.M the-today "He may arrive now."(Fassi Fehri, 2004 "The boy might play today/tomorrow."(Bahloul, 2008, pp. 123-125) Like modals in the English-like languages, modal qad has fixed quantificational force which is indifferent to context.For example, the English modals in (8'.a) and (8'.b) have universal and existential forces of strength, respectively, regardless of context of use.
(8') a. John must be the culprit.(Epistemic necessity: in view of the available evidence) b. John may be the culprit.(Epistemic possibility: in view of the available evidence) (Abu Helal, 2023, p. 37) Unlike English-like modals, the force of modal qad is not strictly lexicalized in the semantics of the modal particle, but it is determined by the tense of its propositional argument.This fact is re-stated in the following descriptive generalization: (9) When qad selects for a tense-denoting TP, it gives rise to a modality meaning as follows: while qad taking a past tense is a certainty-denoting modal, qad taking a present tense is a likelihood-denoting modal.For ease of use, we will refer to the former as qad Φ past and to the latter as qad Φ present (Fassi, 2012) (Fassi Fehri, 2012: p. 8) The second puzzling fact has to do with qad's scalar structure.Both qad Φ past and qad Φ present are roughly equivalent to the gradable modal predicates muʔakkd "certain" and muħtamal "possible", respectively as evidenced by the valid bi-directional implications in ( 10) and ( 11).
( The fact that the modal predicate muʔakkd "certain" can be modified by the maximizer tamam-n "completely" indicates that muʔakkd "certain" is a maximum-standard predicate, meaning that it is a gradable predicate that involves an upper-closed scale structure.The semantically equivalent qad Φ past , on the other hand, never admits modification by maximizers.This indicates that qad Φ past is a non-gradable, scalar operator with a threshold that cannot be manipulated by maximizers. Consider now the pattern in ( 16).Therefore, it is slightly probable that he will win as Ocampos mentioned.'

b. wa mɪn l-muħtamal qaalil-n ann-hu fi l-sanwaat l-muqbilh,
and from the-likely-GEN slightly that-3SG in the-years the-coming, an najid-a l-maħkamh t-t ʕ awir-u haað-a l-firiʕ min iħtimalit l-qanuun to find the-court improve this branch from likelihood the-law "And it is slightly likely in the coming years to find that the court improves this branch of the probability of law." 11 As shown in ( 17), the modal predicate l-muħtamal "likely" can be modified by the minimizer qaaliln "slightly".This indicates that l-muħtamal "likely" is a minimum-standard predicate, with a lowerclosed scale structure.The semantically equivalent qad Φ present , on the other hand, never admits modification by minimizers as shown in ( 17).This indicates that qad Φ past is a non-gradable, scalar operator with a threshold that cannot be manipulated by minimizers. ( PRT MODAL (*slightly) arrive-PRESENT-3SG.M the-today "He may arrive now." The puzzling behavior of modal qad invites a semantic theory with the following features.First, the theory provides a unified compositional analysis to modal qad which is compatible with the semantics of the gradable epistemic modals of certainty and likelihood.Second, it explains the pattern of qad strength that depends on the tense specification of its propositional complements.At this point, let us review two dominant theories that explain GEMs in natural language and show how the one that is built around probability (Lassiter, 2010, subsequent work) appears to be superior to the one that is based on comparative possibility (Kratzer, 1991(Kratzer, , 2012)).Kratzer (1977Kratzer ( , 1981Kratzer ( , 1991Kratzer ( , 2012) ) proposed a version of the quantificational theory of modality with many empirical advantages over the classical modal logic.One advantage is its ability to give a unified analysis to the English-like modal auxiliaries which have a variety of interpretations such as epistemic or deontic based on context as exemplified in ( 18) and ( 19).Under Kratzer's (1977, Kratzer, 1981, 1991), the necessity epistemic modal sentence (18.a) asserts that for each possible world compatible with the speaker's knowledge in the actual world, John is the murderer in that world.Its existential counterpart (18.b) asserts that there is at least one world compatible with the speaker's knowledge in the actual world and in which John is the murderer in that world.Similarly, the necessity deontic sentence (19.a) asserts that for each world in which John obeys the house's actual world rules, John goes to bed at 8 pm in that world.The possibility necessity deontic sentence (19.b) asserts that there is at least one world in which John obeys the house's actual world rules and in which John watches TV in that world.

Comparative possibility vs. probabilistic semantics
Kratzer derived these truth conditions using a contextual parameter which she called conversational background.The parameter is interpreted by means of two contextually determined free parameters which are functions from words to sets of propositions. 12The first function is the modal base f which maps for each world w a set of propositions that describe consistent facts about the modal use underlying its flavor (e.g., epistemic, deontic, circumstantial, etc.).For example, the epistemic modal base characterizing (20) can be formalized as follows.
(20) a. f(w) = {p s.t.p is a proposition that describes an established piece of knowledge known by the speaker in w} (i.e., where ∩ f(w) = {w s.t.all propositions p ∈ f(w) which describe an established piece of knowledge known by the speaker in w hold true}) 13   The second function g is an ordering source which is a function from worlds obtained from f to sets of propositions that describe ideals or morals.This function is used to order the worlds given by f through the relation ≽ g(w) using the propositions given by g which is defined as follows. ( One way to incorporate these functions into semantic composition is to assume that the interaction of f and g yields the best worlds of quantification using the following function BEST, which takes as its arguments f , g and w and returns the best set of worlds on the basis of the ordering ≽ g(w) as defined in (22). 15 The epistemic and deontic readings can be derived using the following unified semantics of the universal and existential modal by varying the particular choices of the conversational background based on f and relative to context.
Another big advantage of the Kratzerian semantics, which is relevant to the current paper, lies in its ability to generate non-trivial truth conditions for intermediate and comparative modals including Gradable Epistemic adjectives (GEA) such as possible, likely and certain.To handle this case of intermediate modals, Kratzer (1991Kratzer ( , 2012) ) proposed a binary relation ≽ g(w) on propositions similar to the binary relation ≽ g(w) on worlds on the basis of the notion of comparative possibility defined as follows.
(24) For each α and β ∊ D ρ(w) , α is as good a possibility as β with respect to < f, g, w> (Kratzer, 1991, p. 644). 16  Using comparative possibility of (24), we can derive the truth conditions of the following two examples of intermediate and comparative possibility in (25) as follows. 17 (25) a. ⟦ probably P ⟧f,g,w is TRUE iff P is a better possibility than ¬ P b. ⟦ P is more probably Q ⟧f,g,w is TRUE iff P is a better possibility than Q Although the comparative possibility in (24) correctly predicts the truth conditions of intermediate and comparative modalities, it was criticized to be non-compositional (Lassiter, 2017, pp. 69-71).Lassiter, however, showed that it is not difficult to convert the non-compositional semantics of (25) into a compositional analysis for both intermediate and comparative modalities.He discussed a proposal which is due Portner (2009) in which a gradable modal adjective denotes a measure function from propositions to degrees which reside in admissible scales relative to the quantitative model < P(W), ≽ g(w) > derived from a particular modelling of modal base f and ordering source g.Consider, for example, the lexical entry of the following gradable epistemic modal in English.On the assumption that certain and likely are measure functions μ mapping proposition to different quantitative scales of likelihood and certainty, respectively, we end up with the following lexical entries of the two adjectives.
(27) a. α is at least as likely as β b. α is at least as likely as γ c. ∴ α is at least as likely as β ∨ γ To see how the inference is invalid, consider the following scenario (Lassiter, 2017, pp. 4-5).
(28) "A lottery with one million tickets.Only one ticket will be chosen as the winner so it is fair.Everyone buys no more two tickets including three siblings-Mar, Sam and Sue-who buy 2 tickets each." From this scenario, it follows that Sam is as likely to win the lottery as anyone else is.Given the supposed-to-be valid inference in ( 27), the following sentence ends up an intuitively false statement.
(29) Sam is as likely to win the lottery as Mary or Sue is.
The sentence in ( 29) is a false reading relative to the context (28).Mary and Sue have four tickets and it cannot be the case that Sam who has only two tickets is likely to win the lottery as one of them.To make this point clearer, suppose that A = {Participant 1 , . . . .Participant 999999 } to be the list of the participants other than Sam (i.e., the other 999,999 participants).Given the supposed-to-be valid inference in ( 27), again, we have the following.
(31) Sam is likely to win the lottery as Sam doesn't win.
But (31) is clearly false.Since Sam has only 2 tickets which is chosen from one million tickets, Sam is much more likely not to win the lottery than he is to win.
As Lassiter (2014) argued, the puzzle of disjunctive inference appears to be an Achilles Heel for Kratzer's comparative possibility.To see this, consider the following reasoning which validates the disjunctive inference under Kratzer's comparative possibility.
(32) a. ∀v: d. Since w is arbitrary, then for every w ∊ (β ∨ γ), there is w' ∊ α such that w' ≽ g(w) w α ≽ g(w) β ∨ γ holds Building on insights from Yalcin (2007Yalcin ( , 2010)), Lassiter (2010Lassiter ( , 2014) ) proposed a scalar semantics based on probability for gradable epistemic adjectives.One theoretical advantage of this approach is that it avoids the disjunctive inference puzzle (i.e., it doesn't validate the inference in ( 27)) without complicating the picture with further semantic machinery beyond what's needed to predicate the semantics of gradable adjectives (Kennedy, 2007).Another advantage is that it provides standard quantitative comparisons of likelihood to account for the degree modification in (33) in which quantitative comparisons of likelihood hold between two propositions or between propositions and maxima or minima of a relevant scale.
(33) a. α is twice as likely as β b.It is half certain that α c.It is 95% certain that α (Lassiter, 2010:210) Lassiter's (2010) theory is based on the following probability calculus.
(34) A probability space is a pair <W, μ> where W is a set of possible worlds and μ is a function from the subset of P(W) to real numbers 0 and 1 which satisfies the following conditions.
This approach also provides a straightforward mechanism of deriving standard quantitative comparisons of likelihood of ratio degree modification.It therefore generates the correct truth conditions of the modal sentences in (36).
(36) a. α is twice as likely as With this theoretical background in mind, we can proceed to solve the question of modal qad using assumptions of probabilistic semantic along the lines of Yalcin (2010) and Lassiter (2011).

Proposal: Deriving the two epistemic occurrences of modal QAD
With this background on gradable modals, we want to explain the phenomenon of modal qad by addressing the two puzzling facts pertinent to its meaning: first, the graded strength of qad is regulated by the tense of its propositional complement with the certainty reading being associated with the past tense and the possibility reading with the present tense as exemplified in the following pattern repeated from (10) and ( 11 Second, as evidenced by its incompatibility with degree modification as in ( 39) and ( 40), modal qad is non-gradable.The fact that the modal qad cannot be modified by other degree operators which serve as threshold manipulators such as the maximizer tamam-n "completely" and the minimizer qaalil-n "slightly" indicates that modal qad in ( 39) and ( 40) involves a kind of sales with a fixed or for all threshold values, which cannot be grammatically manipulated, but determined relative to the tense of qad's complement.

as mentioned Ocampos
The facts in (39) and (40) suggest that modal qad is a non-gradable epistemic modal predicate with a scalar structure that is not lexically inherent but derived in relation to the time scale of its propositional complement.The following sub-section presents and analyzes this derived scale structure.

A Scale of Probability based on a Time-Modal Framework
This section presents and analyzes a model theoretical framework used for modal qad on the basis of Thomason's (1984) world-time branching model. 18Assume an abstract time line Τ with the following characteristics.
(41) a. Τ is a set of times.
b. Τ includes a deictic center which is the speech time t 0 ∊ D i (von Stechow, 2010).
c. Τ includes the reference time t* ∊ D i (i.e., past, speech time) which is interpreted as a contextually determined pronominal free variable vis the assignment function K as follows. 19  i. ⟦ PAST t* t0 ⟧ K is defined iff K(t*) and K(t 0 ) ∈ D i and K(t*) ≺ K(t 0 ).When defined, ii. ⟦ PRESNT t* t0 ⟧ K is defined iff K(t*) and K(t 0 ) ∈ D i and K(t*) ≽ K(t 0 ), when defined, (Where ≺ and ≽ stand for temporal precedence and successiveness along T) For example, the equivalence class of worlds Γ (t 2 ) (w 0 ) = {w s.t.w ≃ t w 0 }, which is the set of worlds: {w 2 , w 0, w 5 }.
(44) a. Ρ ∊ D <it> is a set of propositions.b. w ∊ W is a set of worlds c. t ∊ T is a set of times d.Π is a function that takes as its arguments the two elements of propositions Ρ and worlds W and it returns the predicate of words describing the equivalence class of words which branches from the reference time of P. i.
e. μ is a probability measure function from sets of worlds Π (P)(w) to the real numbers between [0-1].

A Compositional Analysis: computing the meaning of the two epistemic occurrences of QAD
The following inferential pattern of equivalence between qad Φ past and qad Φ present , on one hand, and the gradable epistemic modal predicates l-muʔakkd-i "certain" and l-muħtamal "likely", on the other hand, indicates that qad Φ past and qad Φ present have the same truth conditions as those of the two epistemic predicates of certainty and likelihood, respectively.
(  (Lassiter, 2010 and2014), respectively, and which therefore involve an inherent scale that makes use of maximum/minimum threshold values, the modal qad structures in (45.a) and (46.b) are non-gradable modal sentences with no inherent but derived maximum/minimum intermediate scale that appears to be supplied by the past and present tense time of its propositional complement.
Given these facts, we want the semantics of modal qad to derive the correct truth conditions of its certainty/likelihood meaning based on the tense its propositional complement and to account for its non-gradability and derived scalarity.Following the mainstream literature on graded modality (mainly Lassiter, 2010;Portner, 2009;Yalcin, 2007Yalcin, , 2010)), we propose that the qad operator is a measure function that takes as its argument propositions of type <it> and returns degree denoting objects of type d which is represented by the probability measure function μ.This operator is relativized with respect to the speech time of evaluation t 0 with a branching equivalence class of worlds as represented by ( 47).ii.Lexical entries iii.Composition a. ⟦ PAST t 0 t * he arrived yesterday ⟧ K =: he arrived yesterday at K(i*) b. ⟦ Π PASTt 0 t * he arrived yesterday ⟧ K =: We argue that μ(Π PASTt 0 t * P) has two properties.First, it is a downward monotonic interval: for every t 0 ≻ t 2 ≻ t 1 , it follows that μ (Π PAST t 0 t * P) < μ(Π PAST t 2 t* P) < μ(Π PAST t 1 t* P).Second, μ(Π PASTt 0 t * P) is a closed interval with the upper-closed point t 0 as represented as ( 49). ( To see that, suppose TO THE CONTRARY that T =: (μ t*t*, μ t 0 t*] in ( 48) is an open interval.By this assumption, the following in (49) hold true of T.
(51) ⟦ μ (Π PASTt 0 t * he arrive yesterday) (where MAX is a function that takes T as its argument and returns the maximal elements in T's range.) Let us now look at the truth conditions of the qad Φ present sentence.Our semantics derives the likelihood reading of qad Φ present as follows.
( ii.Lexical entries iii.Composition To see how it is upward monotonic, assume that for every t 0 < t 1 < t 2 , it follows then that μ (Π PRESENT t 0 t * P) < μ(Π PRESENT t 1 t* P) < μ(Π PRESENT t 2 t* P).The interval also meets the conditions of open intervals as shown in (54).
(54) i. Conditions of Open Intervals  53) necessarily defines an open interval with the minimum lower element μ t 0 t* and no maximum element.Because the operator QAD returns the minimal probability degree element μ t 0 t* when it operates on the past tense T in (53), the likelihood meaning of qad Φ present automatically arises.
(55) ⟦ μ (Π PRESENTt 0 t * he arrives now) (where MIN is a function that takes T as its argument and returns the minimal element in T's range.)

Theoretical Implications
This section discusses two theoretical implications of modal qad for the two notions of gradability and measurement in natural language.Let us discuss them in turn.
First, under the standard theory of Kennedy and McNally (2005) and Kennedy (2007), gradable adjectives denote measure functions from individuals to points on abstract scales.Kennedy and McNally (2005) proposed that gradable adjectives have one of two forms: an absolute and relative types.The absolute gradable adjective can be modified by an overt explicit degree modifier that takes measure functions as its argument and returns properties of individuals with maximum or minimum standard of comparison as determined by the lexical semantics of the adjective.Maximum gradable adjectives pick maximum elements on closed intervals. 20An example of maximum predicates is muʔakkd "certain" which pertains to a scale with maximum element as shown in (56).Minimum gradable adjectives, on the other hand, pick minimum elements on closed intervals. 21An example of minimum predicates is muħtamal "likely" which pertains to a scale with minimum element as in (57).Relative gradable adjectives such as high have neither maximum nor minimum interpretations.Such a type of gradable adjectives is modified by the null morpheme pos, which takes measure functions as its argument and returns properties of individuals with contextually determined standard of comparison SP.The standard represents the lowest degree on the relevant scale that stands out on that scale properties  Kennedy (2007), the different maximum/minimum interpretations of gradable adjectives are governed by a last resort condition in grammar which he dubbed "Interpretive Economy"(IE).It says that the "standout" notion of the relevant degree of the standard of comparison in gradable adjectives is determined by the conventional meaning of the gradable adjective, mainly the scale structure given by the gradable adjective's lexical semantics.That is, adjectives with the same scales have the same positive meanings.In ( 56) and ( 57), the lexical semantics of muʔakkd "certain" and muħtamal "likely" which incorporate different scale structures give rise to a maximum and minimum readings, respectively.In the absence of this meaning in the lexical semantics of gradable adjectives as in the case of the positive relative gradable adjectives in (58), which is taken to lexicalize an open scale structure, the interlocutors make a last resort strategy to context in order to determine the semantic value of the "stand out" notion represented by the covert morpheme pos.
The semantic behavior of modal qad seems to contradict IE in the following way: the certainty and likelihood meanings of qad don't seem to follow from qad's conventional meaning based on two distinct scale structures.The scalar epistemic meaning of modal qad is grammatically derived by means of qad's association with the tense specification of its propositional argument.At this point, we have two positions: one view meets the IE.Accordingly, we postulate a lexical ambiguity between a certaintydenoting qad Φ past and a likelihood-denoting qad Φ present .This view needs to motivate and justify the assumption that the certainty and likelihood meanings of qad relate to different degree scales represented by the two-time intervals of the past and present.It is far from clear how it can be the case.
The other view maintains a unified semantics for qad which derives the two interpretations of certainty/likelihood relative to the tense of qad's propositional argument.This view can only be supported if we assume a unified lexical semantics of qad based on a common probability scale derived from a unified time-world model.In this way we extend Lassiter's (2011Lassiter, 2017) proposal that certainty/likelihood involve a common probabilistic scale (contra to Klecha (2012)).
Since modal qad is neither gradable (i.e., as evidence by qad's incompatibility with degree modification) and nor lexically scalar (i.e., as qad's scalarity is determined by the tense specification of its propositional argument), the common probabilistic scale it has is neither ratio nor fully closed.It is then grammatically determined by the tense specification of its propositional complements.This explains an important fact about modal qad with respect to graded modality: qad cannot be intermediate and comparative in the same way as its lexical synonyms muʔakkd "certain" and muħtamal "likely".It is a kind of extreme scalar modal that involves a threshold that cannot be manipulated grammatically by degree modifies.Its threshold is fixed once and for all with the maximum meaning of certainty and minimum meaning of likelihood in the extremes.It follows then that the type of probabilistic scale associated with qad is not additive as proposed by Lassiter (2011) for the lexical gradable epistemic adjectives (e.g., certain or likely) but an intermediary one (so 44.f follows).This explains the following pattern about certainty and likelihood meanings of qad Φ past and qad Φ present .Both qad Φ past sentences in (59) express the same maximum probability degree of extreme certainty extreme regardless of how far the past event precedes the speech time (yesterday vs. last week).Similarly, the two Φ present sentences denote the same minimum probability degree of extreme likelihood regardless of how far the present event follows the speech time.This indicates that the type of the probability scale derived from the time scale is of intermediary with fixed for once and all maximum and minimum points, unlike the other lexical GEMs such as certain or likely.
Second, it has been generalized that all open degree monotonic properties never have maximum elements; they cannot be maximized using relevant maximal operators (Fox andHackl 2006, Sharvit 2014). 22It follows that an upward monotonic property may not have a top maximum element and similarly a downward monotonic property may not have a bottom maximum element.This generalization is the consequence of the universal density of measurement in which all measurements involved in natural language semantics are mappings into dense scales.
Given this generalization, our analysis straightforwardly accounts for the fact that qad Φ past denotes certainty and qad Φ present has the likelihood meaning.If as Lassiter (2011Lassiter, 2017) suggested that certainty denotes the maximum degree of probability, qad Φ past only selects μ(Π PASTt 0 t * Φ) which represents the maximum degree point in a downward monotonic time interval as shown in figure (48).Given the generalization on open intervals, it is clear why qad Φ past cannot select any other probability degree that is lower than μ(Π PASTt 0 t * Φ): by density of time interval, for any μ(Π PASTt ' t * Φ) lower than μ(Π PASTt 0 t * Φ), there is another probability degree that is lower than μ(Π PASTt 0 t * Φ) and greater than μ(Π PASTt ' t * Φ).In other words, any other probability degree lower than μ(Π PASTt 0 t * Φ) is undefined by virtue of the ban of maximizing open intervals given the universal density of measurement.Therefore, the particle qad selecting for a past tense has no choice but selecting the maximum degree of probability and hence it automatically gives rise to the certainty meaning.
By similar reasoning, qad Φ present has a default likelihood meaning.Assume with Lassiter (2011) that likelihood denotes the minimum degree of probability, qad Φ present can only select μ(Π PASTt 0 t * Φ) which represents the minimum degree point in an upward monotonic time interval as in figure (52).Given the generalization on open intervals, it is clear why qad Φ present cannot select any other possibility degree that is higher than μ(Π PASTt 0 t * Φ): Again by density of time interval, for any μ(Π PASTt ' t * Φ) greater than μ(Π PASTt 0 t * Φ), there is another probability degree that is greater than μ(Π PASTt 0 t * Φ) and lower than μ(Π PASTt ' t * Φ).Therefore, qad Φ present can only select the minimum degree of probability in the present tense axis with the likelihood meaning arising by default.

Conclusion
This squib has presented a well-known tradition puzzle involving the semantics of modal qad in Standard Arabic.It has proposed a solution to the puzzle of qad's modality based on a probabilistic semantics which makes use of time-world structures of the relevant tense expressions of qad's propositional complement.The key part of the proposal is re-defining the two time intervals of the past and present in terms of time-world branching structures with different monotonic orientation.Modal qad is analyzed as a probabilistic measure function that operates on the predicates of worlds; the world classes obtained from the time-world branching structures.With the past being as a downward monotonic interval maximized at speech time, qad Φ past automatically returns the maximum degree of probability which is interpreted as certainty.The present, being an upward monotonic interval, minimized at speech time, qad Φ present returns the minimum probability interpreted as likelihood.The current study has the following empirical and theoretical implication for the theory of gradable epistemic modality: epistemic modal-like qad is a kind of extreme scalar expression whose scalar meaning is grammatically encoded in the semantic computation with two extreme probabilistic values: a maximum degree of likelihood that triggers certainty and a minimum likelihood degree that gives rise to possibility.(Blackburn & Bos, 2005)Blackburn 20.Be it fully closed (i.e., [0,1]) or depending on the monotonicity of the relevant property: an upward monotonic property may associate with an upper closed scale (0,1] where it picks the maximum upper end element.A downward monotonic property may associate with a lower closed scale [0,1) where it picks the maximum lower end element.21.Unlike maximum gradable predicates, the minimum type either involves fully closed scales or an upper/lower closed one providing the following monotonicity orientation: an upward monotonic property may associate with a lower closed scale [1,0) where it picks the minimum lower end element.A downward monotonic property may associate with an upper closed scale (0,1] where it picks the minimum upper end element.22.For Fox and Hackl (2006), a maximal operator takes an interval and returns the most informative element in that interval as defined in (i).
(i) MAX inf (P <α,st> ) (w) = the unique x s.t.if P(x)(w) = 1 & ∀y(P(y)(w) = 1 then P(x) a-symmetrically entails P(y) Fox andHackl (2006: 559) Fox andHackl (2006) tested this condition on open intervals against several degree-based phenomena involving maximal operators such as the define article, the question how many/much and exhautifier only.The generalization was shown to be at work in all degree constructions, hence universal.

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We can define a world-time model uses W × T frame Γ in which each time point t branches into an equivalence class of worlds defined as follows.(OurpresentationcloselyfollowsthatofMari (2017:  197-198)).(42)a.Γ is an algebra of equivalence classes of worlds in which for all t ∊ D i and w ∊ D s , Γ (t) (w) = {w' s.t.w' ≃ t w} which are closed with respect to the Boolean operations of conjunction, disjunction, and negation.b.An equivalence class of worlds ≃ t is defined as a 3-place relation on W × W × T for each time point t iff ∀〈w, w'〉 ∊ D s ∀ 〈t, t'〉 ∊ D i : w' ≃ t' w and t ≺ t' → w' ≃ t w.To illustrate this model, consider the following time line T.