Logic, language, and meaning

It may be hard for many people to imagine that linguistics, a discipline usually seen as part of the humanities or the social sciences, can have much in common with mathematics and logic. In fact, I’ve often received surprised reactions from people who asked if whatever I was reading was about mathematics and were informed that it was about language. Indeed, the logical tools used by mathematicians and philosophers are also used by linguists working in the field of semantics, which aims to understand how natural languages create meaning.

Semanticists routinely make use of logical languages which involve mathematical symbols such as ‘x’, ‘∀’, and ‘<‘ to understand meaning (we’ll explore the relevance of these symbols later on). For a long time, however, linguists had thought that these kinds of logical tools were irrelevant to the study of meaning of natural language. Even the father of modern formal linguistics, Noam Chomsky, argued that the logical languages invented by logicians were too different from natural languages to be of any relevance. Moreover, because they regarded meaning as too vague and imprecise to be amenable to logical or scientific analysis, many linguists were more interested in studying syntax, i.e., the rules governing the grammatical structure of sentences.

It wasn’t until the 1960s that linguists started to notice the work of philosophers and mathematicians such as Gottlob Frege, Bertrand Russell, and Alfred Tarski, who had already been trying to use logic to analyse natural language during the earlier half of the 20^th century. When the logician Richard Montague created a logical language that could easily be integrated into what linguists already knew about the structure of sentences, linguists began to appreciate the relevance of logic to understanding natural language. In collaboration with philosophers, they started to build on the work of these earlier thinkers to create a formal approach to semantics that uses logic to understand meaning in natural language [1].

Today, semantics is firmly established as one of the main branches of linguistics. Semanticists are no longer only concerned with the meaning of individual words. Instead, they use logic as an indispensable tool to understand how larger units of meaning are created when words are combined together into sentences. They take seriously the principle introduced by the logician Gottlob Frege known as the principle of compositionality, which states that the meaning of any sentence is determined both by the meanings of its constituent words as well as the grammatical rules used to combine them.

Let’s explore what this principle means through four puzzles designed to get you started in thinking semantically and to allow you to see what a systematic approach to the study of meaning might involve. As it may take a bit of effort to understand the puzzles if you’re new to the field, I’d encourage you to go through the process of understanding and thinking through the problems before going through the solutions.

On being greater than nothing

Consider the following argument:

Premise 1: The Devil is greater than nothing.

Premise 2: Nothing is greater than God.

Conclusion: Therefore, the Devil is greater than God.

Can you explain why the conclusion doesn’t follow from the premises?

Solution:

If we were to replace the word “nothing” with some other ordinary object such as a table or an apple, the argument would be a perfectly valid one, i.e., its conclusion would logically follow from its premises (however absurd the premises might be). The problem, therefore, is that “nothing” cannot simply be treated like any other object which exists, even though “The Devil is greater than nothing” has the same superficial grammatical structure as “The Devil is greater than an apple”. Rather, “nothing” is a kind of quantifier that we use to denote the absence of an ordinary object.

The philosopher Bertrand Russell can be credited for inventing a logical language to deal with concepts like “nothing”, “something”, and “everything”. In the logical language he created known as predicate logic, these words do not refer to objects but function as quantifiers.

For example, to denote “something”, he used the existential quantifier ∃. It’s easily recognisable as the mirror image of the capital “E” which stands for “existence”. Using predicate logic, we can represent a sentence like “Unicorns exist” as:

∃x(x=unicorn)

This reads: there exists some x such that x is a unicorn. The “x” here is what logicians call a variable, which denotes an unspecified or unknown value.

To understand the sentence “Nothing is greater than God”, we can use the symbol ~ to denote negation or “not”. We can translate it as:

Proposition 1: ~∃x(x>God)

This reads: there does not exist an x such that x is greater than God. We’ve also used the standard mathematical symbol “>” to represent “greater than” [2].

If you think about it, another way of stating proposition 1 is to say that everything is not greater than God. In predicate logic, we represent this idea by using the universal quantifier , which means “for all”:

Proposition 2: ∀x ~(x>God)

This reads: for all x, it is not the case that x is greater than God. You can recognise the universal quantifier as an inverted capital “A”, which stands for “all”.

We can paraphrase proposition 2 even further to say that everything is less than God, as such:

Proposition 3: ∀x(x≤God)

“≤” is another operator borrowed from mathematics that means “less than or equal to”.

How did we paraphrase “Nothing is greater than God” into “Everything is less than or equal to God”? What we’ve done is simply to apply two general rules:

Rule 1: ~∃x A= ∀x ~A
Rule 2: ~(x>y)=x≤y

Firstly, we applied Rule 1 to express Proposition 1 as Proposition 2. Note that A represents any expression that contains x like “x > God”. We then applied Rule 2 to express Proposition 2 as Proposition 3.

If you think you’ve understood all that, try translating the statement “The Devil is greater than nothing” into predicate logic. Then, using these two rules, re-express it as a statement involving the symbols ∀ and ≤. When you’re ready, scroll down to the footnotes [3] to see the solution.

By now, it should be clear to you that the real conclusion should be that God is greater than the Devil, rather than the other way around. Evidently, mathematical symbols are especially useful as they can help us to better understand philosophical arguments by clarifying the meaning of logical statements and making natural language more precise.

On walking to school for an hour

(1) #John walked to school for an hour.

What’s wrong with sentence (1)? It doesn’t seem like there’s anything grammatically wrong with the sentence. The sentence seems odd in virtue of its meaning. A sentence like this is said to be semantically ill-formed, and we indicate this using the hash or pound symbol (#) by convention.

To understand the problem a little better, let’s compare the ill-formed sentences on the left with the well-formed ones on the right.

It seems that the difference between the two categories of sentences has something to do with the nature of the action being described. Can you figure out what the difference is?

Solution:

The fundamental difference between walking to school and walking around the park is that the former is conceived as having an endpoint to the action while the latter is not. The act of walking to school terminates upon one’s arrival at the school, while there is no sense of completion in walking around a park.

We can apply the same analysis to distinguishing the ill-formed and well-formed sentences in the table. Smashing, drinking a cup of coffee and building a house are similar in that these actions or events are presented as being complete. This sense of “completeness”, which semanticists call ‘telicity’, is absent in the idea of sleeping or watching TV. Verbs that inherently have this property are known as telic verbs, while those without it are atelic verbs.

It’s important to note that while a verb might be atelic, the entire verb phrase might be telic. For example, even though the verb “to walk” is by itself atelic, it becomes a telic verb phrase (e.g. “walked to school”) when it is combined with prepositional phrases like “to school” or “up the hill”, which convey a sense of completeness by virtue of having reached a certain location. This can be contrasted against prepositional phrases with no sense of an endpoint like “around the park” or “towards the school”:

There are a couple of other ways we can make an atelic verb like “read” or “drink” into a telic verb phrase. For one, we can add “finish” to the verb phrase, as in “John finished reading his book”, to denote the end-point of the action of reading a book. We can also add a countable noun phrase like “a cup of coffee” to the verb “drink” to create a telic verb phrase “drink a cup of coffee”. If we were to add a mass noun like “coffee” or a bare plural like “houses” instead, we would get atelic verb phrases like “drink coffee” and “build houses”.

So how can we distinguish telic from atelic events? You’ve already seen that one simple diagnostic test is to add the phrase “for (a period of time)” to the verb phrase. If the sentence seems unacceptable, it’s likely to be a telic verb phrase. This test works because “for (a period of time)” is itself atelic, which makes it incompatible with a telic verb phrase.

There’s also the opposite diagnostic for atelicity which involves the telic prepositional phrase “within (a period of time)”. As you can see from the table, only the telic verb phrases such as “walked to school” and “walked up the hill” can be combined with “within an hour” to form a well-formed sentence.

On having to play the piano

Modal verbs are verbs which are used to express ideas such as possibility, permission, ability and obligation. Some common examples of modal verbs are “can”, “must”, “should”, “might”, and “may”. Let’s consider the meaning of “can” and “must”. When I say that “I can play the piano”, I mean that it is possible for me to play the piano, whereas if I say “I must play the piano”, I mean that it is necessary for me to play the piano. In other words, “can” denotes possibility, while “must” denotes necessity.

Let’s now introduce the logical symbols that semanticists have invented to express the ideas of possibility and necessity. We will use ◊ to represent possibility and □ to represent necessity. So, if P represents the proposition “I play”, then “I can play” can be expressed as ◊P (i.e. it is possible that P). And “I must play” is □P (i.e. it is necessary that P). We’ll also use ~ to represent negation or “not”.

Here are two fundamental rules (axioms) in modal logic:

Rule 1:  ~◊~P=□P

Rule 2: ~□~P= ◊P

In other words, “not possible to not P” = “necessarily P”; “it is not the case that I can opt to not play” = “I must play”. And “not necessary to not P” = “possibly P”; “it is not the case that I must not play” = “I can play”.

I’ll mention one more rule that simply states that a double negative makes a positive:

Rule 3: ~~P=P

If you think you’ve understood those rules, here’s a puzzle for you:

Given rule 1, “I can’t not play” means “I must play”. But why does “I mustn’t not play” also mean “I must play” and not “I can play”?

Solution:

We can translate “I can’t not play” into modal logic as ~◊~P (not possibly not P). This is identical to the left-hand side of Rule 1, and accordingly, is equivalent to □P, which means “I must play”. However, “mustn’t not play” doesn’t mean ~□~P, but □~~P. According to rule 3, this is equivalent to □P.

If you’re confused, it might help to have a look at this table:

Notice that the crucial difference between “cannot play” and “must not play” lies in the order of the negation ~ and the modal operators ◊ and □. This is what linguists call the scope of the negation operator. In “cannot play”, since ~ precedes ◊, we say that the negation operator takes scope over the modal operator, while in “must not play”, we say that the modal takes scope over the negation. If we wanted the modal operator ◊ to take scope over the negation operator like in “must not”, we should say something like “can (opt to) not play”.

On even eating potatoes on Sundays

Consider a sentence like this:

(2) John even eats potatoes on Sundays.

What does the word “even” mean here? “Even” is an example of what semanticists call focus operators. These are words such as “only”, “just”, and “merely”, all of which create additional implicatures by focusing on particular parts of the sentence. An implicature is the additional meaning that is implied or suggested by a sentence.

Using sentence (2) as an example, if the focus of “even” were on “potatoes”, the implicature is:

The probability of John eating potatoes on Sundays is less than the probability of him eating anything else on Sundays.

Another way to express this is to say that potatoes are the least likely things for John to be eating on Sundays.  Notice that “even” could also take “Sundays” as its focus. If the focus were on “Sundays”, then the implicature generated would be:

The probability of John eating potatoes on Sundays is less than the probability of him eating potatoes on any other day.

Another way to express this is to say that Sundays are the least likely days for John to be eating potatoes. Given the right contexts, “even” can also select “eat potatoes” or “eat” as its focuses. I’ll allow you to work out the implicatures generated in these cases and a relevant context on your own.

But if “even” can take so many parts of the sentence as its focus, how can a listener figure out what the focus is? For one, we can use intonation and stress to mark out the focus. The listener can also infer from the preceding context. For example, in the sentence “John eats potatoes on Mondays, Tuesdays, and even eats them on Sundays”, “Sundays” is clearly identified as the focus [4].

Now here’s an additional problem. Consider this sentence:

(3) Even John eats potatoes on Sundays.

Unlike sentence (2), it’s not possible for “even” to take anything other than “John” as its focus. Why? The reason for this has to do with the grammatical structure of the sentences, which I’ve sketched out using the tree diagrams below. These diagrams tell you how the words in the sentence are grouped together. For example, since “on” and “Sundays” branch to the same node, “on Sundays” forms a group that linguists call a phrase.

Using the tree diagrams, can you figure out a rule that succinctly captures which parts of the sentence “even” can select as its focus?

Sentence (2)

Sentence (3)

Solution:

A simple way to understand this is by seeing that the sentence is made up of two groups or phrases known as the subject and the predicate. In sentence (3), only “John” can be the focus since “Even John” forms a group we call the subject. On the other hand, in sentence (2), “even eats potatoes on Sundays” forms the predicate. We can thus say that “even” can only select as its focus either words (e.g. “Sundays”) or phrases (e.g. “eats potatoes”) in the same group as itself.

There’s a more technical criterion for what “even” can select as its focus. This is a structural criterion that is known to linguists as c-command:

The node that immediately dominates “even” needs to also dominate the focus.

Sentence (2)

Sentence (3)

The nodes that immediately dominate “even” in the two sentences are circled in the tree diagrams above. Notice that only the words that these nodes dominate can be focused by “even”.

This rule also applies if we replace “even” with another focus operator like “only”. The sentence “Only John eats potatoes on Sundays” produces the implicature:

For every member of a relevant group of people, if that member eats potatoes on Sundays then that member is John. [5]

Realise that while “Only John eats potatoes on Sundays” can only select “John” as its focus, the “only” in “John only eats potatoes on Sundays” can select anything it c-commands, be it “potatoes”, “Sundays”, “eats”, or “eats potatoes”.

Conclusion

I hope these puzzles have convinced you that a systematic approach to semantics can reveal something new about how language conveys meaning. More importantly, I hope they’ve demonstrated how the meaning of sentences doesn’t only depend on the meaning of its individual words, but also how they are combined together into sentences with the help of the rules of syntax. And perhaps, every time you use words like “nothing” and “even” in the future, you’ll begin to think about how they contribute to the meaning of the entire sentence.

[1] Partee, B. (1996). The development of formal semantics in linguistic theory. In S. Lappin (Ed.), The Handbook of Contemporary Semantic Theory (pp 11-38). Oxford: Blackwell. Montague’s theory is a major contribution to semantics that is left unexplained in this article as it requires some background knowledge of linguistic syntax. Refer to http://people.umass.edu/partee/MGU_2005/MGU052.pdf to find out more if you happen to have a basic understanding of syntax but not formal semantics.

[2] The comparative “greater than” is expressed here using the mathematical operators for “more than” for simplicity. A more accurate representation could be ~( ∃t∃x(great(x,t))& x≠God & ~great(God,t)). This reads that it is not the case that there exists a threshold variable t whereby some x that is not God exceeds t while God does not. From Schwarzschild, R. (2008). The Semantics of Comparatives and Other Degree Constructions. Language and Linguistics Compass, 2(2), 308-331.

[3]Solution to Problem 1: ~∃x(Devil >x)= ∀x~(Devil>x) = ∀x(Devil≤x)

[4] Arguably, “John eats potatoes even on Sundays” is a better way to clearly select “Sundays” as the focus. Nevertheless, the sentence “John eats potatoes on Mondays, Tuesdays, and even eats them on Sundays” shows us that it’s possible for “even” to take “Sundays” as its focus although “even” does not directly precede “Sundays”.

[5] We can represent this using the predicate logic introduced earlier:  i.e. for all x, if x is a member of the relevant set of people and x eats potatoes on Sundays, then x is John.

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