Palmer (1986) defines modality as ‘semantic information associated with the speaker’s attitude or opinion about what is said.’ Language provides the means for signaling the factual status of a proposition. In addition to necessity and possibility, modality includes actuality, validity and believability of a proposition. Modality includes objective measures of factual status as well as subjective attitudes and orientations towards a proposition.

A proposition is necessarily true if there is no possibility of its being false. A proposition is possibly true if it is not necessarily false. English uses the modal auxiliaries shall, should, can, could, may, might, must and would, as well as the adverbs possibly, maybe, perhaps, necessarily, honestly, actually, and allegedly to express modality. Modal adverbs in McCawley’s (1978) and Fauconnier’s (1985) terms are world-creating—they set up a belief context or possible world for use in judging the content of a proposition. According to Chung and Timberlake (1985) modality is the way a language encodes the ‘comparison of an expressed world with a reference world’. In philosophical terms, the adverbs convey a propositional attitude–an indication by the speaker of their commitment toward the content of their statement.

In reference to the present state of actualized affairs, I am expressing the possibility of a nonactualized state of affairs, namely my going. The modal auxiliary points to an expressed world asserted by the proposition that lies outside of the reference world, normally the actual world of speech.

Where the reference world corresponds with the expressed world, we have actual modality, or realis. When the reference world does not coincide with the expressed world, we have nonactual modality, or irrealis. The modal status of a proposition depends on the extent to which the two modal deictic points diverge. This degree of coincidence translates as necessity, possibility, obligation, commitment, and so on.

Many people are confused by the terms mood and modality. Modality is a semantic feature–it captures the speaker’s attitude toward a proposition. Mood is a grammatical feature–it refers to the inflections for a subset of modal denotations. The subjunctive mood, for example, is the grammatical device a language uses to express hypotheticality or uncertainty. Mood is a structural property of verbs in certain kinds of clauses.

Logical modality examines the possible truth or necessity of a proposition according to logic. Kearns provides the following statements to illustrate logical necessity:

1a. Necessarily, the diameter of a circle passes through the center of the circle.

Logicians use ‘□’ as the symbol for logical necessity. You could translate the previous sentences into logic as:

This statement is necessarily true by the definition of the diameter of a circle.

The statement treats logical necessity as a type of quantifier. We should expect the modal operators to exhibit the same types of scopal ambiguity that we found with the other quantifiers of predicate logic. You should remember that the quantifiers of predicate logic evaluated the truth of a proposition for some set of individuals. Logical necessity evaluates the truth of a proposition for all possible circumstances.

Logicians use ‘◊’ as the symbol for logical possibility. You could translate the above statements into logic as:

We know that the proposition Napoleon won at Waterloo is false, but we can entertain the possibility that the battle could have had a different outcome. The logical expression asserts that the proposition is compatible with the laws of logic, and is not necessarily false, regardless of whether or not the proposition is in fact true.

While logical modality evaluates propositional truth according to the rules of logic, epistemic modality evaluates propositional truth according to our knowledge of the real world. Epistemic modality analyzes the necessity or possibility of a proposition’s truth in light of what we know about the reality. Consider Kearns’ example of epistemic necessity:

You could translate this sentence into pseudologic as Given what we already know, it must be the case that the dinosaurs died out suddenly. Since epistemic necessity is based on our current knowledge about the world, there is always the possibility that we will have to revise our conception of reality in light of new discoveries. We may yet discover that the dinosaurs did not die out suddenly.

Below is a copy of Hecataeus’ map of the world, originally drawn about 520 B.C. What areas correspond to modern maps? What areas diverge the most from modern maps? The second image is from the book Mundus subterraneus published by Athanasius Kircher in 1664. Kircher’s book illustrates the doctrine of signatures approach to medicine. It assumes a connection exists between each ailing part of the body and a healing plant, constellation, and planet. A line radiates from each part of the body to its corresponding planet, plant and sign of the zodiac. The third image is from the unofficial site of the Koreshan State Historical Site. This site marks a colony founded by an upstate New York alternative healer named Cyrus Teed who had a vision showing that the earth is hollow and that we really live on its inner surface.

The possibility of revision is not open to logical necessity. A proposition that is logically true will be true for all time.

All of these sentences claim the existence of intelligent life in deep space is compatible with our present knowledge. We do not know for a fact whether intelligent life exists elsewhere in the universe, but we do not know of any logical or scientific reason why such life could not exist on another planet. Epistemic possibility must follow the rules of logic so it must include logical possibility.

English used to mark the distinction between logical and epistemic possibility. May only marked epistemic modality while might could be used for either logical or epistemic modality. Consider Kearns’ examples:

a She might have fallen down the cliff—thank goodness the safety harness held.

c She might have fallen down the cliff—we’re still waiting for the rescue team’s report.

d She may have fallen down the cliff—we’re still waiting for the rescue team’s report.

The clause about the safety harness in sentences (a) and (b) suggest that the woman did not fall. The clause about the rescue team’s report in sentences (c) and (d) suggest that the woman’s fall is still being established by the rescue team. Kearns also quotes the following headline:

Deontic modality is concerned with obligation, duty, or normative action. Deontic modality is related to: orders, rights, willing, duty, exhortation, permission, requirements, and even ability. From a deictic perspective, deontic modality expresses the imposition of an expressed world on a reference world. Chung and Timberlake (1985) claim that deontic modality is the way that languages express the restriction of possible future states of affairs to a single choice–the forced convergence of the expressed world and the reference world.

Deontic necessity expresses what someone is required to do. Deontic possibility expresses what behavior is permissible. These distinctions define the two basic categories of deontic modality: obligation and permission. The following sentences illustrate these two aspects of deontic modality:

b Buildings erected after September of this year are required to comply with the Revised Building Code.

Deontic modality expresses normative action and duty as a function of the attitudinal state and judgements of the speaker. The first example of deontic necessity above indicates that the speaker has somehow judged a relation between the expressed world and the reference world. For this reason, deontics have an arguable truth value. It is possible to deny the deontic judgement by asserting a logical or epistemic claim:

Modal expressions naturally lead to alternative realities. In claiming the diameter of a circle necessarily passes through its center, you are asserting the truth of the proposition in all circumstances, not just in the current situation. The idea of possible worlds was invented to provide a concrete way of evaluating modal expressions in logic. We can then claim that the diameter of a circle passes through its center is true in all possible worlds if it is necessarily true.

We can use the concept of possible worlds to distinguish the different types of modality we have just reviewed. Logical modality uses the full set of possible worlds. A logically necessary proposition is true in all possible worlds. A logically possible proposition is true in at least one possible world. We can now use quantifiers to express logical modality:

□ (S) ≡ ∀w (S is true in w), where w ranges over the entire set of possible worlds.

◊ (S) ≡ ∃w (S is true in w), where w ranges over the entire set of possible worlds.

Epistemic modality would be evaluated over a more restricted set of possible worlds—those worlds that are compatible with the rules of logic and physics. Kearns uses the variable w_e to express this restriction. We then get the following expressions for Epistemic modality:

□_e (S) ≡ ∀w_e (S is true in w_e), where w_e ranges over epistemically possible worlds.

◊_e(S) ≡ ∃w_e (S is true in w_e), where w_e ranges over epistemically possible worlds.

Deontic modality is evaluated to the most restrictive set of possible worlds. In addition to being compatible with the rules of logic, deontically possible worlds must satisfy some code of behavior. Exactly which code applies is established by the context of speaking, i.e., pragmatics. Kearns refers to the deontically possible worlds as perfect obedience worlds or w_po. We can then write the following expressions for Deontic logic:

□_po (S) ≡ ∀w_po (S is true in w_po), where w_po ranges over perfect obedience worlds.

◊_po(S) ≡ ∃w_po (S is true in w_po), where w_po ranges over perfect obedience worlds.

The statement □A—>A is true for logical necessity, but not deontic necessity. I may fail to love my neighbor. We have to restrict ourselves to the relevant perfect obedience worlds for the deontic inference to hold.

The three types of necessity all determine states of affairs that would be anomalous if the necessarily true propositions do not hold. A system of rules of inference determines the notions of necessity. Truth in all nonanomalous states of affairs is similar to Leibniz’s characterization of necessity: true in all possible worlds.

Necessity and possibility can be paraphased by each other in combination with negation.

Kearns uses the interdefinability of the universal and existential quantifiers to motivate these identities.

∀w (S is true in w), by the interdefinability of necessity and the universal quantifier

~ ∃w ~(S is true in w), by the interdefinability of the universal and existential quantifiers

~ ◊ ~(S), by the interdefinability of the existential quantifier and possibility

∃w (S is true in w), by the interdefinability of possibility and the existential quantifier

~ ∀w ~(S is true in w), by the interdefinability of the existential and universal quantifiers

Kearns introduced conditional sentences in Chapter 2 to illustrate material implication. Conditional sentences have an antecedent clause connected to a consequent clause by material implication, e.g. If you touch me, I’ll scream.

The truth table for material implication shows that the entire proposition is true when the antecedent is false. Counterfactual conditionals are produced when we know the antecedent is false. With a false antecedent a counterfactual conditional describes a situation that is contrary to fact. Lewis (1973) discusses the analysis of counterfactual conditionals like

Logic would claim that both of these sentences are true since the truth table for material implication with a false antecedent is true. We have a problem, though, since logic doesn’t permit two contradictory statements to be true.

We could refuse to accept both statements on the grounds that kangaroos have tails. This approach would preserve the consistency of the logic at the price of giving up any analysis of sentences about hypothetical situations.

To provide a logical treatment for counterfactuals, we must find a way to handle their hypothetical character and explain why their truth value would be based on the consequent rather than on the antecedent.

An analysis based on possible worlds makes it possible to treat the antecedent as if it were a true statement. We can examine all of the possible worlds for which the antecedent holds to determine if the consequent is also true. If we imagine all the possible worlds where kangaroos are jumping about without tails, we can see if there are any in which the kangaroos topple over. If all the kangaroos remain on their feet in these worlds then the first sentence is false. If the tailless kangaroos do topple over in those worlds then the first sentence is true. This method results in a truth table for counterfactuals that only includes cases where the antecedent of the conditional is true.

One problem that remains for counterfactual sentences is specifying the correct set of possible worlds. As Kearns shows, you can imagine all kinds of possible worlds on which tailless kangaroos do not topple over. Lewis suggested picking the possible worlds which are similar to the actual world in the relevant respects, i.e., tailless kangaroos that are otherwise like the real kangaroos in the real world.

The details of specifying the relevant set of possible worlds can be complicated. We cannot simply stipulate the possible worlds will be like ours in every respect except that the kangaroos will be tailless. A world with tailless kangaroos would also be a world in which the kangaroos move differently and one without kangaroo tail soup. The absence of tails would inevitably lead to many other changes in kangaroo form and behavior, not to mention changes in their environment as well. In evaluating the truth of a counterfactual statement speakers demonstrate amazing powers of observation.

To illustrate the difficulties inherent in the possible worlds analysis of counterfactuals McCawley (1981:316-7) refers to an unpublished investigation of Dowty’s which analyzed two counterfactuals based on the sentence:

Dowty derived two possible counterfactual sentences that express an alternative outcome:

Dowty noted that in situations where (1) is true, the two counterfactuals are also generally true. In a possible worlds analysis, the first counterfactual assumes that George didn’t eat the mushrooms while the second counterfactual assumes that George didn’t die. The possible worlds in which George doesn’t die are extremely similar to those in which he doesn’t eat the mushrooms. This logical equivalence fails to account for speaker intuitions about the difference in acceptability of the two counterfactuals.

McCawley notes that while the first counterfactual obeys the English restriction that the antecedent be temporally and/or causally prior to the consequent, the second counterfactual captures the priority that (1) places on the question of how easily George could have avoided death, starting with those worlds where George doesn’t die.

We have seen that simple denotation does not provide a satisfactory theory of meaning. A title such as Mr. Muscle Beach can have multiple denotations. Negative sentences such as Orville is not flying and Orville is not eating have the same denotations. All false sentences have the same denotations. We need something more than denotations to adequately account for meaning. We have been exploring the use of possible worlds as one such addition. We have seen how possible worlds provide an explanation for expressions of modality and counterfactuals.

It would be a grave mistake to think that possible worlds provide a solution to all semantic dilemmas. One problem that an appeal to possible worlds cannot solve is the meaning of propositions that are necessarily true. Consider the following examples:

As examples of logical necessity all of these propositions are true in all possible worlds. According to possible world semantics, then, all of these sentences have the same interpretation—the set of all possible worlds. This outcome is clearly wrong. Although possible world semantics takes us a step beyond a denotational theory of meaning, it is not powerful enough to provide a complete semantic theory. The problem of assigning identical meaning to necessarily true propositions in possible world semantics has motivated the development of theories of partial knowledge of the world. Examples include situation theory (Barwise & Perry 1983) and a theory of interpreted logical forms as the object of attitude reports (Larson & Ludlow 1993).

Formal logic traditionally ignored the treatment of interrogative sentences. While declarative sentences assert the truth of a particular state of affairs, questions inquire just what state of affairs exists. Groenendijk & Stokhof (1997) and Harrah (1984) survey current approaches to the semantics of questions, while Ginzburg & Sag (2000) offer a constructionist account of questions. Groenendijk & Stokhof divide the linguistic approaches to questions into three groups: partition theories, sets of proposition theories and categorial theories. Partition theories analyze the meaning of questions as mutually exclusive subsets of possibilities which jointly exhaust the full set of possibilities (Groenendijk & Stokhof 1997). Sets of proposition theories analyze the meaning of a question as a set of propositions that answer the question (Karttunen and Peters 1980). Finally, categorial theories (Hausser 1983) also view the meaning of questions in terms of their answers, but these answers are not required to be propositions. Instead, they may be other non-sentential categories, depending on the question.

All of these approaches are extensional in nature. Groenendijk & Stokhof (1997) note that simple yes/no questions create a general problem for extensional theories. For example, the sets of proposition theories assign yes or no to the meaning of yes/no questions. It is easy to create pairs of yes/no questions in which one is the negation of the other, for example:

If we assign the value yes to one of these questions then we have to assign the value no to the other one. Materially, the two questions are equivalent, since if we know the answer to one we know the answer to the other. However, our formal theory then forces us to assign the same value to both since the equivalence of meaning should lead to identity of the semantic values of the questions. This rules out the our first intuition, since we assume yes and no are not the same entity.

Nelken & Francez (2002) offer an analysis of questions with another pair of values: resolved (r) and unresolved (ur). These values complement the truth values t and f for declarative sentences. They define the answerability conditions of an interrogative sentence as the situations in which it is assigned r, which corresponds to the truth conditions of declarative sentences. They proceed to add the interrogative operator ?, which they define as: p

In plain words, they define the meaning of the interrogative operator as resolved if the meaning of the declarative sentence p is t or f. The meaning is unresolved if the meaning of p is unknown (uk). Here, v(p) stands for the meaning of p (technically its valuation). Finally, they introduce the interrogative operators (?) or (?x) for yes/no and wh questions respectively. Questions then have the syntactic form:

A	¬ A	?A
uk	uk	ur
f	t	r
t	f	r
ur	ur	ur
r	r	r

With this system in place Nelken & Francez can show that they have solved the contradiction of Groenendijk & Stokhof. For any pair of yes/no sentences

both valuations are assigned r iff p is either true or false. Otherwise, both valuations are assigned ur.

Still, their system meets certain problems. Yes/no questions are technically answered (resolved) in their system by any disjunction of positive and negative sentences:

Nelken & Francez continue with a discussion of some classic problems in the question literature. The first deals with conjoined questions:

In an approach which analyzes questions as partitions of the set of possible worlds conjoined questions denote a simple intersection of the partitions. Disjoined questions, however, create a problem since the point-wise union of a pair of partitions does not yield a partition.

Examples with mixed coordination (Harrah 1984) also create problems for standard approaches.

Such sentences are interrogative in nature since they may be either resolved or unresolved, but it makes little sense to say they are true or false. In such contexts the coordination operators must always yield an interrogative value. Therefore neither the operators of the truth dimension

A final difficulty are wh-questions containing quantifiers or multiple wh-terms. For example,

Such questions have been analyzed traditionally as having two different readings (Chierchia 1993):

Individual Reading: which professor is such that she recommends every candidate?

The individual reading is derived directly by allowing the whterm to have higher scope over the quantified NP, however the pair-list reading is notoriously difficult to derive compositionally. Intuitively, in the pair-list reading, the universal quantifier has higher scope than the interrogative operator. In other theories, however, questions are of types that do not easily lend themselves to quantification, so it is not clear how to derive this reading.

Nelken & Francez (2002:60) give the following interpretations under their approach:

Their Pairlist interpretation employs a mixed coordinate structure. Pairlist readings are impossible in certain contexts:

Allowing the quantifier no candidate to have scope over the question would lead to a question, that requires the hearer to provide a proposition that ensures that for no candidate it is known which professor recommends her. It is hard to imagine that such an answer is well formed.

Some English verbs allow paraphrases between their own negation and the negation of their complements: so-called negative transport. Believe is one such verb; note the paraphrase relationship between the following:

Say, however, does not allow this relationship, and the following are not paraphrases:

Which of these allow the negative to be transported into the complement (that he will go) and yet are paraphrases of the originals? Which do not allow the transport of the negative? Note that certain, likely, and possible form a scale of epistemic modality, from strongest likelihood to weakest: certain > likely > possible. How could such a scale affect negative transport? Consider also the following modal scale from strong to weak: must > supposed to > might and sure of > expect > hope. Formulate a general semantic rule for this phenomenon.

Barwise, J. and Perry, J. 1983. Situations and Attitudes. Cambridge, MA: MIT Press.

Chierchia, G. 1993. Questions with quantifiers. Natural Language Semantics 1(2), 181-234.

Chung, Sandra and Timberlake, Alan. 1985. Tense, aspect, and mood. In Timothy Shopen (ed.), Language Typology and Syntactic Description, III: Grammatical Categories and the Lexicon, pp. 202-258. Cambridge: Cambridge University Press.

Ginzburg, Jonathan & Ivan A. Sag. 2000. Interrogative Investigations: The Form, Meaning, and Use of English Interrogatives. Stanford, CA: CSLI Publications.

Groenendijk, J. and Stokhof, M. 1997. Questions. In J. V. Benthem & A. T. Meulen (eds.), Handbook of Logic and Language, pp. 1055–1124. Amsterdam: Elsevier Science.

Harrah, D. 1984. The Logic of Questions. In D. Gabbay & D. Guenthner (eds.), Handbook of Philosophical Logic, Vol. 2, pp. 715–764. Dordrecht: D. Reidel.

Hausser, R. 1983. Questions and Answers. In F. Kiefer (ed.), The Syntax and Semantics of English Mood, pp. 97–158. Dordrecht: D. Reidel.

Higginbotham, James & May, Robert (1981). Questions, quantifiers, and crossing. The Linguistic Review 1. 41–80.

Karttunen, L. and Peters, S. 1980. Interrogative Quantifiers. In C. Rohrer (ed.), Time, Tense and Quantifiers, Proceedings of the Stuttgart Conference on the Logic of Tense and Quantification, pp. 181–205. Tuebingen.

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Nelken, R. & Francez, N. 2002. Bilattices and the semantics of natural language questions. Linguistics and Philosophy 25:37–64.