In this very thing, which the dialecticians teach among the elements of their art, how one ought to judge the truth or falsehood of a hypothetical judgement like “If day has dawned, it is light”, how great a contest there is; Diodorus has one opinion, Philo another, Chrysippus a third. — Cicero, 45 BC, complaining about conditionals

The search for an answer to the problem of conditionals has a very long history — the philosophers of Cicero’s time were neither the first nor the last to long for a satisfying solution. In this article, we will develop a framework of conditionals and modals, both of which we will find to be intimately linked.

First, what is meant by conditional? In simple terms, a conditional is a sentence of the form “If A, (then) B” or a variation thereof. The left side is called the antecedent, the right side is called the consequent. The question, then, is how one ought to interpret sentences of this form.

It may be tempting to think that conditionals correspond to material implication (→). As it turns out, however, material implication is hopelessly inadequate for modeling the semantics of conditionals in natural languages.

One property of material implication A → B is that it is truth-functionally equivalent to ¬A ∨ B. This has many well-known “paradoxical” consequences[1]. One is that any material conditional statement with a false antecedent is true. Thus, the sentences

“If the moon is made of cheese, then the world will come to an end tomorrow.”

“If 2 is odd, then 2 is even.”

are true if understood as a material conditionals.

Such an understanding goes against our truth-value intuitions of conditionals. The consequent is not convincing given the antecedent, so the sentences would more likely be judged as anything between false and meaningless, but not as true.

Another example of how counter-intuitive the material conditional is can be observed in the following example[2]: A doctor says to the nurse

“If the patient is still alive in the morning, change the dressing”.

Assuming, for the sake of the argument, that this is taken as a material conditional under a command[3], what would be an appropriate course of action for the nurse? The nurse, knowing that A → B is equivalent to ¬A ∨ B acts according to the doctor’s wish… and puts a pillow over the patient’s face and kills them. The nurse “made it the case that either the patient is not alive in the morning, or they change the dressing”. Surely, this is not what the Doctor meant.

The previous two examples show ways in which material implication is implausible for reasons of not matching our intuitions. The next example will reveal a different problem.

“If it is sunny, we often go outside.”

The reading where the adverb is under the scope of the conditional can be discarded immediately. Let us rewrite this as:

“Usually, if it is sunny, we go outside.”

Can we interpret this as an adverb scoping over a material conditional? The analysis would look something like this:

(A) Most of the time, if it is sunny, we go outside.

(B) [For most events e] (e is an event where it is sunny) → (e is an event where we go outside)

Assuming that (B) is the logical form of (A), and that our domain of quantification is made up of a thousand individual days, then (A) would come out true if spoken by someone living in the cloudiest city in the world, where 10 days per year are sunny, and, out of those, only 2 days are times where they go outside. The problem is that (A) is intuitively false under those conditions, because then most of the sunny days are not days on which we go outside. But (B) comes out true, since most of the 1000 days in our domain are not sunny days to begin with.[4]

This is a problem. In fact, no truth-functional conditional connective can predict the correct truth conditions of a sentence like (A) in any compositional way. The failure of material implication is final. However, its loss is our gain. In looking for the correct analysis of our sentence, we will discover the key to a promising semantics of conditionals.

The problem can be solved if the if-clause “if it is sunny” is instead interpreted as restricting the domain of the adverbial quantifier. So, instead of thinking of (A) as the combination of an adverbial operator scoping over a conditional operator, we should understand that there is only one operator (the adverb) and that if itself does not express any conditional operator on its own.[5]

For our example, this would mean:

(A) Most of the time, if it is sunny, we go outside.

(C) [For most events e : e is an event where it is sunny] (e is an event where we go outside)

“Most events which are events where it is sunny are events where I go outside.”

And this does in fact give the right meaning (and correctly predicts the truth-value in the cloudy city hypothetical).

It is worth noting that it has the same form as normal quantified sentences:

[ Qx : F(x) ] G(x) quantifier restriction scope "Q x which are F are G."

This will become useful in a minute.

We just saw that there are conditionals that cannot be analyzed as material conditionals and that the solution in the case of adverbs of quantification is to treat if-clauses as restricting the domain of the quantifier expressed by the adverb.

The next logical step, then, is to generalize this pattern to all if-conditionals.[6]

So what kinds of conditionals are there? There are two main kinds, exemplified by this minimal pair of sentences[7]:

(1) “If Oswald didn’t kill Kennedy, someone else did.”

(2) “If Oswald hadn’t killed Kennedy, someone else would have.”

Conditionals of the first kind are called indicative conditionals, and those of the second kind are called subjunctive (also: counterfactual) conditionals.[8]

Anyone who knows that Kennedy was in fact assassinated will judge (1) true. Maybe it wasn’t Oswald who killed Kennedy, but if it wasn’t Oswald, then someone must have done it. In (2), we take the fact that Oswald did kill Kennedy for granted, and make a claim about what would have happened if things had gone differently.

Now, coming back to the idea that if is a restrictor, and looking at (1) and (2), neither sentence seems to contain an adverb, so if if is to restrict something, what is there to restrict? The key idea here is that if there is no explicit operator, then we must assume an implicit operator[9]. In English, this is usually an (epistemic) necessity modal or something like a generic frequency adverb. The necessity modal works quite well for (1):

(1b) “If Oswald didn’t kill Kennedy, someone else must have.”

And again, we follow the same line of thinking as we did in the adverb example we reach:

(1c) [Necessarily : if Oswald didn’t kill Kennedy] someone else did.

Before we can interpret the meaning of “necessarily”, we need to roughly define some basic modal operators:

Logically necessary modal (□) claims are true iff they are true in all possible worlds.

□P

= [∀w : w ∈ W] P in w

= “For all worlds w in of all the possible worlds W, P is true in w.”Logically possible modal (◇) claims are true iff they are true in some possible world.

◇P

= [∃w : w ∈ W] P in w

= “There exists some world w in all the possible worlds W such that P is true in w.”

The modal operator thus has a quantifier, a restriction, and a scope. The pieces of the puzzle are starting to come together. We can now return to (1c) and apply our definition of the necessity modal:

(1d) [In all words in which Oswald didn’t Kill Kennedy] someone else did.

The if-clause has become the restrictor of the quantifier expressed by the modal operator. We can now give meaning to sentences like (1), although we should really refine our definitions a bit, since we don’t want to quantify over literally all possible worlds but only over contextually accessible or relevant worlds.

How do we interpret subjunctive conditionals like (2)?

We evaluate subjunctive conditionals by starting from the actual world, adding the antecedent to our information state and looking for contextually relevant worlds that differ minimally from the actual world (i.e., keeping fixed all actual facts that are not strictly tied to the contextually relevant case) while making the antecedent true, and then evaluating the consequent in those worlds.

For (2), then, our truth-value judgement depends on what we believe the circumstances of Kennedy’s assassination to have been. If we believe that there was some kind of conspiracy and that sooner or later, someone would have killed him, we will judge it true. If we believe that the assassination attempt was a one-off event, we will deem it false.

The following humorous example[10] shows why, in our semantics, we should only include those contextually relevant worlds in the domain of the modals that are as similar as possible to the current world:

(3) “If kangaroos had no tails, they would topple over.”

(4) “If kangaroos had no tails but used crutches, they would topple over.”

We can’t really make the inference from (3) to (4). We can accept (3) but reject the move to (4).

For the inference to be valid, the following pattern would need to hold:

A → C entails A ∧ B → C

This move, known as strengthening the antecedent, is generally valid under material implication. It would also be valid in our example if our domain included not only the most similar worlds but all the worlds in which the antecedent is true, i.e., all worlds in which kangaroos have no tails. The worlds in which they have no tails but use crutches are a subset of the ones where they have no tails. This gives us incorrect results; the worlds where kangaroos have no tails but are otherwise as similar as possible to the current world are not worlds where kangaroos use crutches, and so we would not include them in our interpretation of (3).

We should keep in mind then, that subjunctive conditionals take us to minimally different worlds relevant to the current context.

We are finally ready to look at the way modal operators and conditionals are implemented in Toaq.

The four basic modal operators in Toaq

Indicative Subjunctive Necessity she ao Possibility daı ea

Each modality type (necessity vs possibility) comes in two forms: indicative and subjunctive.

Since they are based on dual quantifiers, she (necessity) and daı (possibility) are duals as well, as is standardly the case in common systems of modality:

bu daı bu = she

bu she bu = daı

The general place structure pattern of modal predicates is as follows:

/1: ___ is [modal-ly] the case. /2: ___ is [modal-ly] the case in world(s) where ___ is the case.

Thanks to their place structure, Toaq’s modals can be used to express bare (unrestricted) modal claims and subjunctives by using the unary meaning via an adverbial construction (7th tone), or to express a (restricted) conditional statement by using the binary meaning via a prepositional construction (6th tone).

We will now look at examples of each of the four basic modal predicates.

Indicative

With indicative predicates we make claims about the actual (current) world. We increment our stock of knowledge hypothetically with the antecedent and then evaluate on that basis the consequent.

Necessity:

Shè tî jí mí Pảrī bı tỉ jí Fárāqsēgūa.

“If I’m in Paris, then I am in France.”

Possibility:

Dàı tî súq roı jí ní dỏaq bı chẻo gẻq súqjī.

“If you and I are in this city, we can meet (it is possible that we meet).”

Subjunctive

The subjunctive predicates are counterfactual, i.e., the antecedent is contrary to the facts of the current world. They make claims about alternative universes, ones in which the antecedent holds.

Necessity:

Ào tî hó ní bı hỉaı hó.

“If they were here, they would laugh.” (but they are not here)

Possibility:

Èa tî súq ní bı sỏa súq jí.

“If you were here, you could help me.” (but you are not here)

Due to their place structure, every modal operator can also be used without the restrictor phrase. Usually, this will mean using it with the adverbial tone or as the head of the main predicate phrase. An unrestricted ao corresponds to a bare if-less would-sentence in English, for example:

Ảo pủa súq.

Pủa ão súq.

“You would have a good time.” (but you are doing something else)

Adverbs of quantification

Adverbs of quantification (such as “usually”, “sometimes”, “always”, “often”) follow the same place structure pattern as modals.

/1: ___ is [adverb-ly] the case. /2: ___ is [adverb-ly] the case in world(s) where ___ is the case.

As such, they, too, appear either restricted or unrestricted:

Dàqfāı jîobūı jí bı kảqsī jí séoq.

“Often if I’m outside, I look at the sky.”

Dảqfāı kảqsī jí séoq. /

Kảqsī dãqfāı jí séoq. /

Dãqfāı bı kảqsī jí séoq.

“I often look at the sky.”

Since Toaq derives adverbs and prepositions productively from its verb forms, modal operators, adverbs of quantification and so on form an open class in Toaq, which means that as many kinds of operators as needed can be created easily. This is convenient, because there is a huge number of potentially useful modals.

To exemplify this, let us consider deontic modality. Deontic modality is a large group of modalities that indicate how the world ought to be according to certain norms. One important subkind is formed by the pair legal necessity and legal possibilty.

These can be considered subtypes of the logical kind with additional restrictions built into the possible worlds. More specifically, we can define legal necessity and legal possibility via logical necessity and logical possibility by restricting the possible worlds to those worlds where everybody acts according to the law:

Legal necessity claims are true iff they are true in all legally perfect worlds.

= [∀w : w ∈ L] P in w

= “For all worlds w in of all the legally perfect worlds L, P is true in w.”

(and likewise for legal possibility)

Thus:

Bủ jủaodāı mủaqtūa súq jí.

“You cannot kill me” (meaning you killing me would be against the law)

“In no relevant possible world where everyone acts according to the law do you kill me.”

Jùaoshē bûa hó ní jỉo bı tẻq hó.

“If they live in this building, they must pay.” (not paying would be violate a law)

“Every relevant possible world where they live in this building and where everyone acts according to the law is a world where they pay.”

There are countless other varieties of necessity and possibility, and compiling an exhaustive list is way outside the scope of this article[11]. There is comfort in knowing that as many subtypes as needed can be derived via the usual compounding machinery.

In this article we saw how Toaq unifies conditionals and modality into one coherent system. We will wrap up with another list of examples.

Ảo pủ jẻa jí máq da. /

Pủ jẻa ão jí máq da.

“I would have bought it.” (but the circumstances were different)

Ào pû nủaımīa jí bı pủ jẻa jí máq da.

“If I had been rich, I would have bought it.” (but I wasn’t rich)

Ẻa pủ tủa dủa jí súq hóq.

“I could have told you that.” (but the circumstances were different)

Shè bû pủ kủo ékū bı bảo máq.

“If the knight was not black, it was white.” (spoken of a chess piece)

Dàı jîa dảqshēı súq bı jỉa fả súq pátī.

“If you have time (then), you will be able to go to the party.”

Sa pủı bı ảo tâo jí púı.

“There are many things I would do.” (if a certain thing were the case, which it isn’t)

Ảo tâo jí sa pủı.

“I would do many things.” (if a certain thing were the case, which it isn’t)

Sıa rảı bı dàı tâo súq ráı bı shảı mảı jí súq.

“There is nothing you can that could make me stop loving you.”

Sòqdāq rûqshūa bı kảı jí.

“Usually when it rains, I write.”

Èa pû shỉe hó bı pủ kảqgāı hó súq.

“If they had been awake, they could have seen you.” (but they weren’t awake)

Coming back to Cicero, one might wonder…

Mả ào mîe mí Kỉkērō nãı bı jảı hó moq.

“If Cicero were alive today, would he be happy?” (but he is not alive today)

And with that, we reach the end of this article.

Even the west wind itself is whispering that it is time for us to move on, and also I have said enough; so I ought to round off. — Cicero, De natura deorum, Book II

Footnotes

[1]: https://en.wikipedia.org/wiki/Paradoxes_of_material_implication

[2]: Example taken from Edgington 2008.

[3]: Such an analysis of conditional commands is not convincing, as the example demonstrates. One popular approach nowadays is to treat conditional commands as introducing a modal operator that is restricted by the if-clause. Another area where the restrictor view of conditionals seems to work much better than material implication.

[4]: I adapted this explanation from Kratzer 2012.

[5]: This insight goes back to Lewis, whose 1975 paper on adverbial quantification set in motion a radical rethinking of conditional semantics, though Lewis did not generalize the restrictor analysis of if-clauses in adverbial quantification to if-clauses found elsewhere.

[6]: Kratzer took this step. In Kratzer 1986, they write: “The history of the conditional is the story of a syntactic mistake. There is no two-place if … then connective in the logical forms for natural languages. If-clauses are devices for restricting the domains of various operators.”

[7]: Due to Adams 1970.

[8]: This terminology is due to the fact that, in English, indicative and subjunctive conditionals differ in their mood marking (subjunctives tend to carry an additional layer of past tense morphology). We will adopt this terminology for Toaq even though it does not have special morphology or syntax for marking mood.

[9]: A genius move by Kratzer.

[10]: Example from Lewis 1973.

[11]: https://plato.stanford.edu/entries/modality-varieties/ has some useful examples.