Frege’s Contradiction

In philosophy and mathematics, the use of words and linguistic expressions is equivalent to the use of the tongue in the process of producing a coherent utterance: neither philosophy nor speech would be possible without those respective features. Words are of such importance that philosophical texts can be accompanied by personalized dictionaries that clarify the nature of specific words which are to be found throughout a text; the vessel for the message is as important as the message. And yet, many scholars have little, if any, grasp on that quite elusive feature associated with linguistic expressions: namely, their meaning.

For some time now, scholars have attempted to establish a coherent theory of meaning; a theory which will grant to those who grasp its conceptual roots an ability to reliably interpret the same word in the same way at different times and in different contexts. Their deliberations have, most certainly, lead to a vast forest littered more with books and essays rather than trees and flowers: a library; the philosophy of language has seen the rise and fall of more intellectuals than newborns have daybreaks and evening stars. As a consequent, more than a few theorists who regard semantics as their domain of investigation have put forth various theories of meaning. One such semantic theorist, Gottlob Frege, had proposed a quite unique, non-traditional theory of meaning which has impacted both mathematics and the philosophy of language; of him and his theory, the remainder of this essay will concern.

Frege’s approach to meaning establishes two features as being central for the interpretation of linguistic phenomenon: sense and reference. Frege developed these two notions while considering mathematical problems, like, whether the equal sign is a sign of identity or equivalence (Lycan, 2000). He, contrary to his peers, argued that (2 x 5 = 10) is both identical and different by means of sense and reference. By reference, 2 x 5 expresses the same referent as 10, and so are identical in that regard; but the modes of expressions each have their own cognitive significance and so differ in their sense.

For Frege, a sense is, “the connection of each sign with the same nominatum,”. In more simplistic language, that means that a sense is the meaning for a given expression or symbol, and that the meaning (sense) is what connects two distinct symbols or expressions to the same object (nominatium). So, the sense determines the reference. For example, the utterances, “that large cup looks cool,” and, “that is probably a well-brewed cup of coffee,” express different senses but relate to the same nominatium: the cup of coffee. Frege would express that in the following manner:

That is probably a well-brewed cup of coffee = The cup of coffee being drank by Jane Doe

That large cup looks cool = The cup of coffee being drank by Jane Doe

When one is thinking about the magnitude of the cup, they are having a distinct cognition from the person who is thinking about the contents inside the cup; however, the two thinkers have as their reference a specific cup of coffee. It also is the case that some senses have no reference; for instance, “mickey mouse” and “the cartoon mouse” do not have a reference, according to Frege, but they do have a sense. This allows Frege to appear more pragmatic than those who would suggest that “mickey mouse” is not meaningful when used in sentences: I.e. mickey mouse jumped over the wall.

Reference, for Frege, is the correspondence between a word and an object, though he allows for there to be a correspondence between thought and an object as well. When the correspondence is between the word and the object, the words are in their customary sense, and words in their customary sense have specific objects as their reference; words in their customary sense are expressions understood by those that are sufficiently familiar with the language of choice according to Frege. Moreover, when the correspondence is between a thought and an object, it is so by virtue of the accompanied expression; which is to say, A and B from “A = B” have the same referent but different cognitive significance.

Frege makes another distinction between indirect sense and reference. When an expression is uttered in an indirect context, words drop their direct references and adopt a sense instead; consider some examples:

Plato is bald

Frege believes that Plato was bald

In these examples, “Plato” has a reference in the initial sentence but then loses its reference in the second sentence. This is because the second sentence is about Frege’s beliefs rather than Plato, and so “Plato” has some sense but lacks a direct reference.

Thus, the sense of, “Plato was bald,” in, “Frege believes that Plato was bald” relies on Frege’s belief for reference; that is, Frege’s belief has the referent . Comparatively, the initial “Plato was bald” is about Plato, which means it has reference to Plato.

From this short exposition on Frege, we see that Frege was attempting to account for thought in his theory of meaning; indeed, Frege believed that there was a relationship between thoughts, words, and objects. However, in his attempt to account for thought, Frege makes some claims that force him into a contradiction. On one hand, Frege is not a linguistic determinist: words do not determine thoughts for Frege; rather, he believed that the same cognition can be expressed in multiple ways. But on the other hand, Frege is a linguistic determinist:

“The words ‘the heavenly body which has the greatest distance from the earth’ have a sense; but is very doubtful as to whether they have a nominatum. Therefore, the grasping of a sense does not with certainty warrant a corresponding nominatum”

Here it seems that Frege is suggesting that certain words deterministically pick out objects and others do not; that is, some words when used in their customary manner pick out their nominata. Thus, it logically follows that when a speaker is using words in their customary sense the speaker’s thoughts must be about the nominata. Therefore, Frege is both a linguistic determinist and not a linguistic determinist.

In addition, there are other oddities which stick out of his theory like sore thumbs. Consider that, in accordance with Frege’s theory, one can use multiple senses for one object but not multiple objects for one sense, especially if it is a customary sense. So, “behaviour” in the customary sense is restricted; it must be restricted because in the customary sense it would correspond to a nominata. But, if that is the case, then the customary sense for “behaviour” must be ontologically specific, all while the image schema for “behaviour” is ontologically relative. To elaborate, “behaviour” is a linguistic vessel that relies upon an image schema[1] for semanticity; I must caveat, the linguistic vessel could have also been “tree” because the image schema and the linguistic vessel are not directly linked in a deterministic fashion (Pinker, 2015). Put more explicitly, words do not determine the nature of thought. Whatever word one chooses to pair with the image schema, as far as I can tell, matters very little. So, because the image schema that, through Pavlovian conditioning, corresponds to “behaviour” can be applied to multiple ontologies, it follows that the word is inherently polysemous; thus, it will always lack a customary sense since its relative to the activation of an image schema. That said, the image schemas physical form could be argued to be less relative given that many human brains interpret movement quite similarly, though that would be radically different from what Frege was doing.

More can be said about Frege, though to do so at this point would go beyond the scope of the essay. So, to conclude, we can see that, within the first few paragraphs of Frege’s essay, there are contradictions; the contradiction stems from his notion of customary sense. His customary sense produces the contradiction because it entails that some words have designated objects, which entails that some thoughts inherently designate objects; therefore, word and thought are in a deterministic relationship where in which the word pre-determines the thought. If this were not the case, then there could never be a customary sense, and so the entire theory of reference collapses. If one holds the position that there are customary senses, then they are committed to a contradiction. The way to circumnavigate this contradiction, I believe, is to abandon customary senses and reference.

[1] The image schema for “behaviour” is determined by the recognition of patterns within the environment, and so can be applied to many ontologies. For instance, atoms behave, molecules behave, humans behave, and so forth.


Lycan, W. G. (2000). Philosophy of language: A contemporary introduction. London: Routledge

Matrinich, A. (2008). The philosophy of language. New York: Oxford University Press.

Pinker, S. (2015). The language instinct: How the mind creates language.


Ideasinhat is a business development analyst and longtime reader of academic literature. He writes books and essays on science and philosophy, and posts them to this website. The essays, as with the books, cover topics from psychology, philosophy, and cognitive science to economics, politics, and law.

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