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Philosophy of Meaning: A Mathematical Theory

Hassan Ajami
2020 / 9 / 15

A Mathematical Theory of Meaning

Meaning is a mathematical equation. We could successfully mathematize meaning in the following way: meaning = assertibility conditions + truth conditions (i.e. meaning is equal to assertibility conditions plus truth conditions). According to this mathematical theory of meaning, meaning is analyzed in terms of, and determined by, assertibility conditions and truth conditions. Thus, this theory is a unification of the externalist and the internalist approach of meaning. According to externalism, meanings are determined by truth conditions, while, for internalism, meanings are determined by assertibility conditions. Therefore, the mathematical theory of meaning, which says that meaning = assertibility conditions + truth conditions, reconciles between externalism and internalism. Hence, it resolves the conflict between the externalist and the internalist account of meaning. This is an essential virtue of the mathematical theory of meaning.

Basic Virtues of This Unification

Externalism maintains that sentences gain their meanings in virtue of truth conditions. And truth conditions are the conditions which should be satisfied in order for sentences to be either true´-or-false. For example, the statement “snow is white” means snow is white because the statement “snow is white” is true if and only if snow is white. For externalism, the statement “snow is white” gains its meaning in light of the conditions which are´-or-should be satisfied in order for it to be true´-or-false, namely the condition whether snow is white´-or-not (which makes the statement “snow is white” true´-or-false).

Yet internalism endorses a different account of meaning. Internalism defends the view that words and sentences acquire their meanings in virtue of their assertibility conditions. And assertibility conditions are those conditions which provide justifications for asserting certain sentences. For instance, according to internalism, the statement “snow is white” means snow is white because we are justified to assert “snow is white” in light of certain objective evidences, such as our observation that snow is white. While externalism analyzes meaning in terms of the external world, which could be presented through the truth conditions of statements, internalism analyzes meaning internally, i.e. through relying on the internal conditions of humans, such as their mental abilities. This is why it seems that there is a conflict between externalism and internalism. But, as we have seen, it is possible to successfully resolve this conflict through analyzing meaning as a mathematical relationship between assertibility conditions and truth conditions.

Since meaning = assertibility conditions + truth conditions, it mathematically follows that assertibility conditions = meaning minus truth conditions. Hence, meanings could be acquired through assertibility conditions without the actual need for truth conditions. And this is why greetings, questions and orders (such as “hello”, “how are you doing?” and “go to school”) are meaningful due to their possession of certain assertibility conditions, even though they don’t actually possess truth conditions. This shows that the mathematical theory of meaning, which says that meaning = assertibility conditions + truth conditions, successfully accounts for the fact that greetings, questions and orders are meaningful, although they have no truth conditions. And thus, the mathematical theory of meaning is plausible.

We know that greetings, questions and orders are meaningful. But they have no truth conditions. For example, there are no truth conditions which could be satisfied and enable the greeting “hello” to be true´-or-false. “Hello” is neither true nor false, and it couldn’t be true´-or-false. Hence, it has no truth conditions. Yet it is meaningful. And it is meaningful in virtue of the fact that it possesses some assertibility conditions. For example, in many circumstances, we are justified in greeting others and saying to them “hello”. The linguistic fact that greetings, questions and orders are meaningful is captured by the mathematical theory of meaning because it defines meaning in terms of assertibility conditions, and not only in terms of truth conditions.

Other statements possess their meanings in virtue of having truth conditions in addition to assertibility conditions. For instance, the statement “there are seas and trees” means that there are seas and trees because the statement “there are seas and trees” is true if and only if there are seas and trees in addition to the fact that the statement “there are seas and trees” has certain assertibility conditions, such as when we are justified in asserting that “there are seas and trees” in virtue of observing seas and trees. Here, the statement “there are seas and trees” acquires its specific meaning due to its possession of certain truth conditions and assertibility conditions. And the mathematical theory of meaning is successful in accounting for the meanings of the previous statement and similar ones in virtue of analyzing meaning in terms of truth conditions and assertibility conditions at the same time.

Further, the mathematical theory of meaning is successful in accounting for the fact that certain statements are meaningful and acquire their meanings only in light of their truth conditions. According to the mathematical theory of meaning, meaning = assertibility conditions + truth conditions. It mathematically follows from this equation that truth conditions = meaning minus assertibility conditions. Hence, some statements gain their meanings due to their possession of truth conditions, even if they don’t have assertibility conditions. This reveals that the mathematical theory of meaning successfully accounts for the fact that some statements have their meanings in light of their truth conditions, although they lack assertibility conditions.

For example, according to the mathematical theory of meaning, the statement “there are angels” is meaningful, and it means there are angels because it has its own truth conditions, even if there are no assertibility conditions which enable us to be justified in asserting that there are angels. The statement “there are angels” means there are angels because it is a true statement if and only if there are angels. So this statement acquires its meaning through its truth conditions, although scientifically and objectively we have no evidence justifying that there are angels, i.e., although the statement “there are angels” doesn’t possess assertibility conditions. All of this shows that the mathematical theory of meaning is successful in accounting for the meaningfulness and meanings of statements which don’t have assertibility conditions because it analyzes meaning in terms of truth conditions, and not only in terms of assertibility conditions. And thus, the mathematical theory of meaning is plausible.

Meaningless Sentences

Some sentences are meaningless, such as the sentence “number seven is married”. The mathematical theory of meaning accounts for the fact that meaningless sentences are actually meaningless. The sentence “number seven is married” is neither true nor false. Hence, it has no truth conditions. And it also does not possess any assertibility conditions because we aren’t justified in asserting that number seven is married, given the fact that numbers can’t be married and can’t be unmarried. Now, since the sentence “number seven is married” lacks truth conditions and assertibility conditions, and given that the mathematical theory of meaning analyzes meaning in terms of truth conditions and assertibility conditions, it follows that, from the perspective of this theory, the sentence “number seven is married” is meaningless. Therefore, the mathematical theory of meaning accounts for the fact that the sentence “number seven is married” is meaningless. All of this shows the success of the mathematical theory of meaning with respect to explaining why certain sentences are meaningless.

Other Philosophical Virtues

The mathematical theory of meaning is successfully able to distinguish between the meanings of different sentences which have the same truth conditions. This is because it does not only define meaning in terms of truth conditions, but it also defines meaning in terms of assertibility conditions. For example, the statement “this is the morning star” and the statement “this is the evening star” have different meanings, while they have the same truth conditions. The statement “this is the morning star” means this is the star which appears in the morning, while the statement “this is the evening star” means this is the star which appears in the evening. Thus, both have different meanings. But both have the same truth conditions. The statement “this is the morning star” is true if and only if the statement “this is the evening star” is true. This is because both stars are one star, which is Venus. Now, since both statements have the same truth conditions, if we only analyze meaning in terms of truth conditions, it will follow that they have the same meaning. But their meanings are different. This is why we need the assertibility conditions to distinguish between their meanings.

In fact, both statements differ in their meanings because they differ in their assertibility conditions. We are justified in asserting the statement “this is the morning star” only in the morning, while we are justified to assert the statement “this is the evening star” only in the evening. Now, the mathematical theory of meaning is successful in accounting for their different meanings because it mathematically analyzes meaning in terms of assertibility conditions as well as truth conditions. Since meaning = assertibility conditions + truth conditions, it follows, from the perspective of the mathematical theory of meaning, that the two previous statements (i.e. “this is the morning star” and “this is the evening star”) have different meanings because they have different assertibility conditions. This is how the mathematical theory of meaning gains its success in accounting for the fact that the previous statements differ in their meanings. This is another essential virtue of analyzing meaning in terms of assertibility conditions and not only in terms of truth conditions, exactly as the mathematical theory of meaning does.

Moreover, some sentences have the same assertibility conditions, while they have different meanings. This is why we need truth conditions in our mathematical analysis of meaning in order to account for the difference in meaning between these sentences. For example, we scientifically know that water is H2O. Therefore, both statements “there is water” and “there is H2O” have the same assertibility conditions. This means that whenever we are justified in asserting one of these statements, we are also justified in asserting the other, given that we know that water is H2O. Yet the meaning of “there is water” is different from the meaning “there is H2O”. This is because “there is water” means there is a colorless and tasteless liquid, while “there is H2O” means that there are hydrogen and oxygen combined in a specific chemical way. Now, the truth conditions of these two statements account for the difference in their meanings. And this is why truth conditions are essential components of the mathematical analysis of meaning.

The sentence “there is water” has truth conditions which are different from those of the sentence “there is H2O”. Hence, from the perspective of the mathematical theory of meaning which defines meaning in terms of truth conditions as well as assertibility conditions, the sentence “there is water” has a different meaning from the sentence “there is H2O”. The sentence “there is water” means that there is a colorless and tasteless liquid because the sentence “there is water” is true if and only if there is a colorless and tasteless liquid. But the sentence “there is H2O” means that there is a special chemical combination between hydrogen and oxygen because the sentence “there is H2O” is true if and only if there is a special chemical combination between hydrogen and oxygen. Thus, the difference in their truth conditions accounts for the difference in their meanings. Water might include certain different minerals in addition to H2O, and it will still be water. This is scientifically why statements mentioning water differ in their meanings and truth conditions from those mentioning H2O. And the mathematical theory of meaning accounts for the difference in the meanings between the statements mentioning water and those mentioning H2O because it doesn’t only define meaning in terms of assertibility conditions, but it also analyzes meaning in terms of truth conditions. And this success of the mathematical theory of meaning speaks for its plausibility.

Conflicting Intuitions

We have two conflicting intuitions. One intuition is leading us to believe that meaning is internally determined, i.e., meaning is determined by our internal capacities, while the other intuition is leading us to believe that meaning is externally determined, i.e., meaning is determined by the external world. The mathematical theory of meaning is successful in reconciling between both intuitions and endorsing them through expressing both of them within a single unique formula. And on the basis of this success, we are justified in holding that the mathematical theory of meaning is true. From the viewpoint of this mathematical theory, meaning = assertibility conditions + truth conditions. Therefore, meaning is determined by internal factors which are the assertibility conditions, and meaning is also determined by external factors which are the truth conditions. This is why meanings are internally related to us as humans, and they are also related to the external world. And this is how the mathematical theory of meaning accounts for both the internal and the external intuition of meaning.

Without the internal dimension of meaning, we can’t access and know what our words and sentences mean. If meanings are not determined internally through the assertibility conditions, then they are not related to us. And thus, we can’t have a privilege access to what we mean by our words and sentences. And without the external dimension of meaning, meanings can’t be successfully able to express the facts of the external world. In other words, if meanings are not determined by the external world, such as being determined by truth conditions, our language will fail to reflect the facts of the world. These are some basic reasons why we need both the internal and the external dimension of meaning. And the best way to express both dimensions is through analyzing meaning in terms of a mathematical relationship between internal conditions, such as assertibility conditions, and external conditions, such as truth conditions.




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