Modality, scope and quantification
Modality, scope and
quantification
Modality is the term for a cluster of meanings centred on the notions of necessity and possibility, for instance;
a. You must report it. ( indicates that there is written rules)
b. You have to report it.( indicates that there is pressure)
c. You mustn’t report it. (indicates that there is a prohibition)
d. You don’t have to report it (indicates that there is no necessity to report the matter)
Relative scope is also needed for understanding quantificational meanings. Quantifiers are words such as all, some and most. s and modality: what ‘must be’ is expected under all circumstances, and if a situation is possible in some circumstances, then it ‘may be’.
Modality
There are example from modality:
a. You must apologise.
b. You can come in now.
c. She’s not able to see you until Tuesday.
d. Acting like that, he must be a Martian.
e. With an Open sign on the door, there ought to be someone inside.
f. Martians could be green.
The main carriers of modality are a set of auxiliary verbs called modals: will, would, can, could, may, might, shall, should, must and ought to. Modality is encoded in various other expressions too, such as possibly, probably, have (got) to, need to and be able to.
Modal verbs and tense
The modal verbs would, could, might and should are past tense forms, but the examples in (7.4) show that past forms of modal verbs often do not mark past time.
The requests in (7.4a–7.4c) are more tentative and polite than those in (7.4d), but none of them is semantically about the past. Also, the (7.4a) sentences have almost the same meaning as those in (7.4d) where the modals are not past tense forms.
(7.4) a. Would/Could you help me tomorrow?
b. Might you be free to help me tomorrow?
c. Should you have the time tomorrow, please help me then.
d. Will/Can you help me tomorrow?
Past tense forms of the modals, particularly would and could, do sometimes have reference to past time, as in (7.5a, b).
(7.5) a. Previously we would meet every New Year, but not anymore.
b. Two years ago she could swim fifty lengths, but not anymore.
Deontic and epistemic modality
Epistemic interpretations have to do with knowledge and understanding. Markers of epistemic modality are understood as qualifications proffered by speakers or writers (or from someone they are reporting) regarding the level of certainty of a proposition’s truth. For example:
(7.6) a. The whole hillside is slipping down into the valley. ( Epistemic)
b. The whole hillside could be slipping down into the valley.
(7.7) a. They meet in the centre court final tomorrow. (Epistemic)
b. They may meet in the centre court final tomorrow.
(7.8) a. Jessica went by motorbike. (Epistemic)
b. Jessica probably went by motorbike.
(7.9) a. The car was travelling very fast, so it came unstuck at the bend. (Epistemic)
b. The car must have been travelling very fast because it came unstuck at the bend.
Modality comes in different strengths. A gradient from weak to strong
can be seen in the modally marked (b) examples (7.6b–7.9b). Because of could, someone who produces (7.6b) is likely to be understood as conceding that the possibility of
the hillside slipping down into the valley is not ruled out by available evidence.
The presence of may in (7.7b) is likely to convey that a meeting between the players in question, in the centre court final the next day, is compatible with some available information.
Use of probably in (7.8b) signals that the speaker or writer regards the available evidence as not just compatible with Jessica having gone by motorbike, but that the balance of evidence points towards this having been her mode of travel.
Must (7.9b) is a mark of strong modality: a speaker or writer who says this is vouching that all the available evidence leads to the conclusion that the car was going very fast.
Deontic interpretations of modality relate to constraints grounded in society: duty, morality, laws, rules. Deontic modality lets language user express their attitudes (or relay the attitudes of others) as to whether a proposition relates to an obligatory situation or permissible one, or somewhere in between. For examples:
(7.10) a. You can ride my bike anytime you like. ( premission)
b. The consul could have been more helpful. ( can be or no)
c. You should send him an email. ( must)
d. Tax forms must be submitted by the end of September. (must)
Explanation (7.10a) using can (or may), the utterer offers no objection to the addressee riding the bike. Could contributes to a presumption behind (7.10b): that the consul was not very helpful. Additionally – and this is the deontic modality part of the meaning – could conveys a judgement that it would have been preferable if the consul had been more helpful. Should makes (7.10c) a statement that the desirable course of action is for an email to be sent to ‘him’. must, (7.10d) conveys an obligation regarding tax returns.
The expressions of modality are in italics.
(7.11) a. Might you have put the ticket in your jacket pocket? (Epistemic)
b. Might I have another piece of cake, please? (Deontic)
(7.12) a. It may be dark by the time we’ve finished. (Epistemic)
b. OK, we’ll permit it: you may copy these two diagrams. (Deontic)
(7.13) a. Prime numbers can be adjacent: 1, 2, 3. (Epistemic)
b. The pigeons can have this bread. (Deontic)
(7.14) a. The tide should be turning now; I looked up the times before we came here. (Epistemic)
b. You should try harder.
(7.15) a. The tide ought to be turning now; I looked up the times earlier today. (Epistemic)
b. You ought to try harder. (Deontic)
(7.16) a. Warmer summers must be a sign of global warming. (Epistemic)
b. The treaty says carbon dioxide emissions must be reduced. (Deontic)
(7.17) a. At 87 metres this has (got) to be one of tallest trees in the world. (Epistemic)
b. He has (got) to be more careful or he’ll break the crockery. (Deontic)
Core modal meanings
(7.18) a. She expected the coffee to be strong; she’d tasted that blend before. (Epistemic)
b. She told the waiter that she expected the coffee to be strong and not accept it otherwise. (Deontic)
Explanation (7.18a), which is epistemic, a moderately strong expression of conviction abouthow reality would turn out, and (7.18b), which deontic, a moderately strong demand about how she wanted it to be.
P stands for any proposition and the double-headed equivalence arrow represents paraphrase (sameness of sentence meaning), The double-headed arrow is a reminder that entailment between paraphrases goes in both directions.
(7.19) a. P is necessarily true ⇔ It is not possible for P to be untrue.
b. necessarily P ⇔ not possibly not P
(7.20) a. It is possible that P is true ⇔ It is not necessarily so that P is untrue.
b. possibly P ⇔ not necessarily not P
(7.21) a. It is impossible for P to be true ⇔ P is necessarily untrue
b. not possibly P ⇔ necessarily not P
Epistemic interpretations arise when the presuppositions are propositions assumed to be facts: common knowledge, or propositions that have recently been accepted in the conversation or that are made obvious by sights, sounds and so on available to be experienced in the context of utterance.
Deontic interpretations arise when preferences, wishes, requirements or recommendations form the contextual presuppositions.
Relative scope
Part of the issue was whether un- has as its scope – which is to say the material that it applies to – the verb lock or the adjective lockable. Questions of relative scope arise when there are two operators – items that have scope – in the same expression. Interesting things happen when a marker of modality interacts with negation. Consider deontic interpretations of the sentences in (7.25).
(7.25) a. You mustn’t provide a receipt. (demand)
b. You don’t have to provide a receipt. (Advice)
c. You must provide a receipt.
d. You have to provide a receipt.
A conventional notation that is helpful is used in (7.26), and is explained immediately
below the example.
(7.26) a. necessarily (not (you provide a receipt)) ( must)
b. not (necessarily (you provide a receipt)) ( do not have to)
c. necessarily (you provide a receipt) ( must and have to)
The scope relations are informally indicated for the first three sentences in (7.28).
(7.28) a. It is not possible that they received the invitation.
not (possibly (they received the invitation))
b. It is possible that they have not received the invitation.
possibly (not (they received the invitation))
c. They may not have received the invitation.
possibly (not (they received the invitation))
d. not possibly P ⇔ necessarily not P
Quantification
The letter I, for intersection, has been written into the region of overlap. If there are any vegetarian corgis, then they belong in this intersection. (An intersection of sets is itself a set.) Some possible answers to the question of whether any corgis are
vegetarian are shown in (7.29). The quantifiers are in italics.
(7.29) a. No corgis are vegetarian. | C n V | =0
b. Several corgis are vegetarian. 2 < | C n V | <10
c. At least three corgis are vegetarian. 2 < | C n V |
Figure Corgis and vegetarians. I labels the intersection of the two sets, C n V
d. Some corgis are vegetarian. 1 < | C n V|
e. At least one corgi is a vegetarian. 0 < | C n V|
The notation ‘C n V’ stands for the intersection of C and V, the set of corgi vegetarians or vegetarian corgis (if there are any such). Enclosing the label of a set within a pair of vertical lines is a way of representing the number of elements in the set, its cardinality; for example, with C standing for the set of all corgis, |C| is the total number of corgis that there are.
Proportional quantifiers
The sentences in (7.30) have what are called proportional quantifiers (italicised). These do not exhibit the symmetry found with cardinal quantifiers. s. Most meat eaters are corgis is clearly different in meaning from the sentence in (7.30a). while (7.30b) is probably false, reversing the nouns – Less than half the world’s meat eaters are corgis – yields a sentence that is probably true. Switching (7.30c) to Few vegetarians are corgis does not state the case nearly strongly enough: hardly any of them are corgis!
(7.30) a. Most corgis are meat eaters. | C n M | > | C – M | (more 500)
b. Less than half the world’s corgis are meat eaters. | C n M | < | C – M |( less 500)
c. Few corgis are vegetarian. | C n V | << | C – V |
Figure 7.2 Corgis and meat eaters. M labels a subset of corgis that are not meat eaters, C – M
M labels the set of all meat eaters and, as before, C is the set of all corgis. The bulge is the set of corgis minus the (large) subset of meat eaters amongst them: C – M. The meat-eating corgis, as in Figure 7.1, are in the intersection of corgis and meat eaters: C n M. The specification given in (7.30a) for Most corgis are meat eaters is simply that the number of meat-eating corgis | C n M | is greater than the number of corgis who do not eat meat | C – M |. Another way of putting this is to say that more than half of the members of the corgi set are meat eaters
The sentence in (7.30b) is probably false. It would be true if, and only if, the balance went the other way and there were fewer corgis in the intersection with meat eaters than | C – M |,With few (7.30c, and you might find it useful to look back to Figure 7.1) truth requires the intersection to be quite a lot smaller (doubled ‘less than’ sign: <<) than | C – M |.
The truth of the sentences in (7.30) depends on how the totality of corgis is split between an intersection and a remainder, hence the name
proportional quantifier.
Though the same set-theoretical specification is given for all and every on the right in (7.31), these two quantifiers are not identical in meaning. Every is a distributive quantifier, so that, for instance, Every corgi at the dog show was worth more than £1,000, would mean that if there were ten of them, the total was over £10,000. All, however, is ambiguous between a collective and distributive reading: if it is true that All the corgis at the show were worth more than £1,000, then that figure could be the value per dog or could be the total for all of them. Each is another distributive quantifier. Like every and all, it is specified in terms of a subset relationship, C ⊆ M.
quantification
Modality is the term for a cluster of meanings centred on the notions of necessity and possibility, for instance;
a. You must report it. ( indicates that there is written rules)
b. You have to report it.( indicates that there is pressure)
c. You mustn’t report it. (indicates that there is a prohibition)
d. You don’t have to report it (indicates that there is no necessity to report the matter)
Relative scope is also needed for understanding quantificational meanings. Quantifiers are words such as all, some and most. s and modality: what ‘must be’ is expected under all circumstances, and if a situation is possible in some circumstances, then it ‘may be’.
Modality
There are example from modality:
a. You must apologise.
b. You can come in now.
c. She’s not able to see you until Tuesday.
d. Acting like that, he must be a Martian.
e. With an Open sign on the door, there ought to be someone inside.
f. Martians could be green.
The main carriers of modality are a set of auxiliary verbs called modals: will, would, can, could, may, might, shall, should, must and ought to. Modality is encoded in various other expressions too, such as possibly, probably, have (got) to, need to and be able to.
Modal verbs and tense
The modal verbs would, could, might and should are past tense forms, but the examples in (7.4) show that past forms of modal verbs often do not mark past time.
The requests in (7.4a–7.4c) are more tentative and polite than those in (7.4d), but none of them is semantically about the past. Also, the (7.4a) sentences have almost the same meaning as those in (7.4d) where the modals are not past tense forms.
(7.4) a. Would/Could you help me tomorrow?
b. Might you be free to help me tomorrow?
c. Should you have the time tomorrow, please help me then.
d. Will/Can you help me tomorrow?
Past tense forms of the modals, particularly would and could, do sometimes have reference to past time, as in (7.5a, b).
(7.5) a. Previously we would meet every New Year, but not anymore.
b. Two years ago she could swim fifty lengths, but not anymore.
Deontic and epistemic modality
Epistemic interpretations have to do with knowledge and understanding. Markers of epistemic modality are understood as qualifications proffered by speakers or writers (or from someone they are reporting) regarding the level of certainty of a proposition’s truth. For example:
(7.6) a. The whole hillside is slipping down into the valley. ( Epistemic)
b. The whole hillside could be slipping down into the valley.
(7.7) a. They meet in the centre court final tomorrow. (Epistemic)
b. They may meet in the centre court final tomorrow.
(7.8) a. Jessica went by motorbike. (Epistemic)
b. Jessica probably went by motorbike.
(7.9) a. The car was travelling very fast, so it came unstuck at the bend. (Epistemic)
b. The car must have been travelling very fast because it came unstuck at the bend.
Modality comes in different strengths. A gradient from weak to strong
can be seen in the modally marked (b) examples (7.6b–7.9b). Because of could, someone who produces (7.6b) is likely to be understood as conceding that the possibility of
the hillside slipping down into the valley is not ruled out by available evidence.
The presence of may in (7.7b) is likely to convey that a meeting between the players in question, in the centre court final the next day, is compatible with some available information.
Use of probably in (7.8b) signals that the speaker or writer regards the available evidence as not just compatible with Jessica having gone by motorbike, but that the balance of evidence points towards this having been her mode of travel.
Must (7.9b) is a mark of strong modality: a speaker or writer who says this is vouching that all the available evidence leads to the conclusion that the car was going very fast.
Deontic interpretations of modality relate to constraints grounded in society: duty, morality, laws, rules. Deontic modality lets language user express their attitudes (or relay the attitudes of others) as to whether a proposition relates to an obligatory situation or permissible one, or somewhere in between. For examples:
(7.10) a. You can ride my bike anytime you like. ( premission)
b. The consul could have been more helpful. ( can be or no)
c. You should send him an email. ( must)
d. Tax forms must be submitted by the end of September. (must)
Explanation (7.10a) using can (or may), the utterer offers no objection to the addressee riding the bike. Could contributes to a presumption behind (7.10b): that the consul was not very helpful. Additionally – and this is the deontic modality part of the meaning – could conveys a judgement that it would have been preferable if the consul had been more helpful. Should makes (7.10c) a statement that the desirable course of action is for an email to be sent to ‘him’. must, (7.10d) conveys an obligation regarding tax returns.
The expressions of modality are in italics.
(7.11) a. Might you have put the ticket in your jacket pocket? (Epistemic)
b. Might I have another piece of cake, please? (Deontic)
(7.12) a. It may be dark by the time we’ve finished. (Epistemic)
b. OK, we’ll permit it: you may copy these two diagrams. (Deontic)
(7.13) a. Prime numbers can be adjacent: 1, 2, 3. (Epistemic)
b. The pigeons can have this bread. (Deontic)
(7.14) a. The tide should be turning now; I looked up the times before we came here. (Epistemic)
b. You should try harder.
(7.15) a. The tide ought to be turning now; I looked up the times earlier today. (Epistemic)
b. You ought to try harder. (Deontic)
(7.16) a. Warmer summers must be a sign of global warming. (Epistemic)
b. The treaty says carbon dioxide emissions must be reduced. (Deontic)
(7.17) a. At 87 metres this has (got) to be one of tallest trees in the world. (Epistemic)
b. He has (got) to be more careful or he’ll break the crockery. (Deontic)
Core modal meanings
(7.18) a. She expected the coffee to be strong; she’d tasted that blend before. (Epistemic)
b. She told the waiter that she expected the coffee to be strong and not accept it otherwise. (Deontic)
Explanation (7.18a), which is epistemic, a moderately strong expression of conviction abouthow reality would turn out, and (7.18b), which deontic, a moderately strong demand about how she wanted it to be.
P stands for any proposition and the double-headed equivalence arrow represents paraphrase (sameness of sentence meaning), The double-headed arrow is a reminder that entailment between paraphrases goes in both directions.
(7.19) a. P is necessarily true ⇔ It is not possible for P to be untrue.
b. necessarily P ⇔ not possibly not P
(7.20) a. It is possible that P is true ⇔ It is not necessarily so that P is untrue.
b. possibly P ⇔ not necessarily not P
(7.21) a. It is impossible for P to be true ⇔ P is necessarily untrue
b. not possibly P ⇔ necessarily not P
Epistemic interpretations arise when the presuppositions are propositions assumed to be facts: common knowledge, or propositions that have recently been accepted in the conversation or that are made obvious by sights, sounds and so on available to be experienced in the context of utterance.
Deontic interpretations arise when preferences, wishes, requirements or recommendations form the contextual presuppositions.
Relative scope
Part of the issue was whether un- has as its scope – which is to say the material that it applies to – the verb lock or the adjective lockable. Questions of relative scope arise when there are two operators – items that have scope – in the same expression. Interesting things happen when a marker of modality interacts with negation. Consider deontic interpretations of the sentences in (7.25).
(7.25) a. You mustn’t provide a receipt. (demand)
b. You don’t have to provide a receipt. (Advice)
c. You must provide a receipt.
d. You have to provide a receipt.
A conventional notation that is helpful is used in (7.26), and is explained immediately
below the example.
(7.26) a. necessarily (not (you provide a receipt)) ( must)
b. not (necessarily (you provide a receipt)) ( do not have to)
c. necessarily (you provide a receipt) ( must and have to)
The scope relations are informally indicated for the first three sentences in (7.28).
(7.28) a. It is not possible that they received the invitation.
not (possibly (they received the invitation))
b. It is possible that they have not received the invitation.
possibly (not (they received the invitation))
c. They may not have received the invitation.
possibly (not (they received the invitation))
d. not possibly P ⇔ necessarily not P
Quantification
The letter I, for intersection, has been written into the region of overlap. If there are any vegetarian corgis, then they belong in this intersection. (An intersection of sets is itself a set.) Some possible answers to the question of whether any corgis are
vegetarian are shown in (7.29). The quantifiers are in italics.
(7.29) a. No corgis are vegetarian. | C n V | =0
b. Several corgis are vegetarian. 2 < | C n V | <10
c. At least three corgis are vegetarian. 2 < | C n V |
Figure Corgis and vegetarians. I labels the intersection of the two sets, C n V
d. Some corgis are vegetarian. 1 < | C n V|
e. At least one corgi is a vegetarian. 0 < | C n V|
The notation ‘C n V’ stands for the intersection of C and V, the set of corgi vegetarians or vegetarian corgis (if there are any such). Enclosing the label of a set within a pair of vertical lines is a way of representing the number of elements in the set, its cardinality; for example, with C standing for the set of all corgis, |C| is the total number of corgis that there are.
Proportional quantifiers
The sentences in (7.30) have what are called proportional quantifiers (italicised). These do not exhibit the symmetry found with cardinal quantifiers. s. Most meat eaters are corgis is clearly different in meaning from the sentence in (7.30a). while (7.30b) is probably false, reversing the nouns – Less than half the world’s meat eaters are corgis – yields a sentence that is probably true. Switching (7.30c) to Few vegetarians are corgis does not state the case nearly strongly enough: hardly any of them are corgis!
(7.30) a. Most corgis are meat eaters. | C n M | > | C – M | (more 500)
b. Less than half the world’s corgis are meat eaters. | C n M | < | C – M |( less 500)
c. Few corgis are vegetarian. | C n V | << | C – V |
Figure 7.2 Corgis and meat eaters. M labels a subset of corgis that are not meat eaters, C – M
M labels the set of all meat eaters and, as before, C is the set of all corgis. The bulge is the set of corgis minus the (large) subset of meat eaters amongst them: C – M. The meat-eating corgis, as in Figure 7.1, are in the intersection of corgis and meat eaters: C n M. The specification given in (7.30a) for Most corgis are meat eaters is simply that the number of meat-eating corgis | C n M | is greater than the number of corgis who do not eat meat | C – M |. Another way of putting this is to say that more than half of the members of the corgi set are meat eaters
The sentence in (7.30b) is probably false. It would be true if, and only if, the balance went the other way and there were fewer corgis in the intersection with meat eaters than | C – M |,With few (7.30c, and you might find it useful to look back to Figure 7.1) truth requires the intersection to be quite a lot smaller (doubled ‘less than’ sign: <<) than | C – M |.
The truth of the sentences in (7.30) depends on how the totality of corgis is split between an intersection and a remainder, hence the name
proportional quantifier.
Though the same set-theoretical specification is given for all and every on the right in (7.31), these two quantifiers are not identical in meaning. Every is a distributive quantifier, so that, for instance, Every corgi at the dog show was worth more than £1,000, would mean that if there were ten of them, the total was over £10,000. All, however, is ambiguous between a collective and distributive reading: if it is true that All the corgis at the show were worth more than £1,000, then that figure could be the value per dog or could be the total for all of them. Each is another distributive quantifier. Like every and all, it is specified in terms of a subset relationship, C ⊆ M.
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