On the analytic-synthetic distinction

Already Immanuel Kant, investigating the foundations of knowledge in his 1781 Critique of Pure Reason, distinguished between analytic and synthetic truths, for him the latter being the most interesting. While in later days ‘analytic’ became almost a synonym for a priori – before experience – Kant argued that some a priori truths, such as 7 + 5 = 12 are synthetic because they produce new knowledge.

In the heydays of logical empiricism, the first half of the previous century, the analytic-synthetic distinction was severely dichotomized. Logic and mathematics were meant as tools to process the results of observations and experiments, but tools and subject matter were to be kept strictly distinct. In the 1950s, this view came under pressure, mostly due to the work of Willard van Orman Quine.

The original distinction

“Snow is white” and “Seven plus five is twelve” are propositions of different nature. The first we state from observation, and therefore it is empirical, also known as a posteriori, after experience. The second is logical, supposedly meaningless with respect to material reality. It is also known as a priori: before experience. Empirical statements are called synthetic because they produce new knowledge, for example by connecting a predicate to a substance – “white” to “snow” – or by expressing relations: “The cat is on the mat”. But besides observation, some logic is required here as well; sense experience and logic are one process, stimulated by our competences and our memories of past experiences. These also enable us to linguistically shape our newly acquired knowledge. The separation of sense experience from logic is a less fruitful offspring of Cartesian dualism and its assumption of a metaphysically independent mind, nowadays supported by a very small minority of philosophers.

The so-called mind-body problem is a soul-body problem in disguise. Both ‘mind’ and ‘soul’ are metaphysical concepts that do not exist materially. Reality is one and essentially physical, and so is thought., see Truth and the mind.

Thought is private and inscrutable, but that is no reason to deny that it is material.

But what to do with “Mammals breastfeed”? It denotes a real-life property of mammals: their breastfeeding. It may also be called a tautology: we call mammals ‘mammals’ because they breastfeed. Quine, in his 1951 Two Dogmas of Empiricism, used “All bachelors are unmarried men” to prove that the truth value of this proposition relies on synonymity. Since synonymity cannot be theoretically justified, tautologies cannot be distinguished from synthetic statements, although their truth value completely relies on the meanings of their terms. After all, if ‘bachelor’ and ‘unmarried man’ are not synonyms, the proposition becomes synthetic, combining the substance ‘bachelor’ with the predicate ‘unmarried’.

Maybe Kant still would have objected to this, because he thought that to be synthetic, a proposition should not contain terms that contain each others’ meaning. Such a distinction is problematic. Let us consider “Tim is a bachelor” and “Tim is unmarried.” These sentences may mean the same, but they just as well may not. Suppose somebody remarks: “Tim’s house is not very tidy.” Someone else may either reply “Well, he is a bachelor” or “Well, he’s unmarried”. The first connotes a supposed general pattern of behavior of bachelors, the second the possible influence of partners on that behavior, or even the assumed traditional duty of women to clean up after their husbands.

It can, on the other hand, not be denied that the names ‘bachelor’ and ‘unmarried man’ share concrete referents, a group of people to whom both terms properly apply. But that is not the same as the categories of ‘bachelors’ and ‘unmarried men’ referring to each other. We call Tim a bachelor because he is unmarried. But we do not call Tim unmarried, because he is a bachelor. Along Kant’s lines, Tim will drop between categories. His two possible predicates largely contain each other, yet they do not seem to be equivalent.

The analytic nature of ‘truth’

But now for: “ ‘Snow is white’ is true .“ In The Metaphysics of truth, we already saw that ‘is true’ adds nothing material to the proposition. This gives reason to say that “p is true” is an analytic proposition. And here we stumble across the question if a priori truths exist at all. If we accept a priori truths, they do not need material proof. The key questions are why “Snow is white” and “Seven plus five is twelve” are thought of as true.

For “Snow is white”, the simple answer is that if we show snow to somebody who has never heard of it and ask her to name its color, the answer will be: “Snow is white”. But this presupposes that this person not only has knowledge of “white” but is also able to link that knowledge logically to her new experience of snow. If we ask somebody: “How much is seven plus five?” she needs to know arithmetic. Both cases need logical procedures, but only the second needs formal knowledge.

Gottlob Frege, the founder of modern logic, wrote that in “p is true”, ‘true’ is redundant. It is not a predicate expressing a material property such as ‘white’. Of truth there simply exists no universal account. Up into the 20th century, substantive truth theories were dominant, such as the identity theory, the correspondence theory, the coherence theory, and the pragmatic theory of truth, see Knowledge and truth. In the course of the century, the deflationary theory of truth became ever more popular, seeing truth not as an existent, but at the most as any proposition bearing the ‘true’ predicate.

An opposite view is that ‘truth’ is an ‘abstract object’, just like a number or a mathematical shape. As opposed to concrete objects, ‘abstract objects’ are causally inert, but there is discussion about this distinction. A more natural and parsimonious view is to deny abstract objects and speak of intersubjective notions. These may, as Frege suggested, constitute a ‘third realm’ between concrete objects and individual consciousness.

‘True’ is one of a realm of predicates with other members such as ‘correct’, ‘valid’, ‘just’, ‘righteous’, ‘good’, ‘beautiful’ and all of their opposites, synonyms and gradations. All of these are subjective and used to express opinions, not facts. In natural language, they have several informal meanings. Philosophers, however, use to formalize these expressions, as is shown by their tormenting work on precise definitions. This practice conflicts with interpreting the above expressions by what they mean in ordinary language, see also Alfred Tarskli’s position in Knowledge and facts. And for some thinkers, even defining them is not enough. Predicates are elevated to the level of substances and called ‘abstract objects’, suggesting, of course, their objective existence. But they do not exist mind-independently: they are intersubjectively adopted notions that became ever more general across history.

One of the reasons for the emphasis on abstract objects was a long tradition of seeing mathematics as a coherent, mind-independent system of truth. Thales of Milete, the godfather of western philosophy, proved that all triangles inscribed in a semicircle are rectangular. This and similar miraculous discoveries led to the idea that mathematics showed a transcendent truth. From 13th-century philosopher William of Ockham onward, however, first of all the empiricists started to see mathematical shapes not as existents, but as formalized models. Triangles are seen in nature, but none of those has the perfectly calculable surface of base times half the height.

Formal and informal propositions

There can be little doubt that formalization of arithmetic, geometry and logic – and philosophical vocabulary! – serves heuristic purposes. Formalization enables us to improve their shareability and general usefulness. The same happens when we state propositions. However, we do this at the cost of dismissing a lot of detail, both at the beginning of the epistemological track – observation and the like – and at its end – the expression of knowledge in language.

With for instance field observations, we look for what we want to know. Since also detail is part of the material reality that we observe, the propositions we derive from our observations are a selection, and therefore, if knowledge is to mirror reality, it is necessarily incomplete. However, this does not mean that these propositions can never be true because of something missing from them. It only means that the truth value of propositions can neither be based on correspondence – because we ourselves choose our examples from reality – nor on their relation to the whole of reality – since that whole is unknown to us.

At the end of the track, expressing knowledge in language, we see a conflict between the versatile, dynamic and informal semantics of expressions in ordinary language and the formalized, restricted use of these very same expressions by philosophers and scientists. Besides the discomfort of having to read their findings with some private list of terms on the side, we should be wary of the logical consequence of this practice, which is that “p is true if and only if {n₁…n_n} and {r₁…r_n} necessarily consist of ordered pairs.” This means that the constituent names n₁…n_n of a proposition p should match their referents r₁…r_n one to one. Even if we do not take into consideration the syntactic structure of p, this is impossible, because Frege taught us that different names – e.g. Morning Star and Evening Star – may have the same referent, – e.g. the planet Venus, just like single referents may be known under different names, such as the author of Huckleberry Finn being both Samuel Clemens and Mark Twain.

The linguistic mechanisms governing conversations have been well analyzed ever since the 1950’s work of John Langshaw Austen and Ludwig Wittgenstein, the former explaining the contextual nature of expressions and therefore of truth, the latter introducing the famous concept of language games. Our general conclusion is that natural language and logic are at odds with each other, to put it mildly.

Truth values as formal values

We should base both the analytic-synthetic distinction and truth values on formal criteria only. First of all, we need to move away from seeing knowledge as resting in the minds of individuals. Knowledge is a collective asset of human culture, and since ‘true’ is a desirable quality of knowledge, ‘truth’ is a collective issue. In the unlikely case that I ever were to accept abstract objects, knowledge would be my first candidate, and it to be materially true is a most worthwhile claim for any proposition. In fact, if we exclude all personal and socio-political motives interfering with knowledge, it is the only sensible claim. However, human knowledge is a dynamic whole. If we take materially true to be an essential criterion, we shoot ourselves in the foot, since this would mean that what is knowledge today might become false tomorrow and the other way around. The best we can do is focus on logic; focusing on material truth is a dead-end, see Truth-makers, truth-bearers, truth-breakers.

Just like mathematics, also logical truths are mind-dependent. In logic, we distinguish between valid arguments and sound arguments. Arguments are valid if the premises of an argument lead to its conclusion by following a formal system of rules. A valid argument only is sound if its premises are materially true, implying that its conclusion is true also. However, since there are various types of logic each with their own rules, the validity and therefore the soundness of arguments may differ according to the logical system we use. Even if we assume that the premises of these arguments are materially true, arguments in different logical systems may still produce contradictions.  But since we can never be sure about material truth, logical soundness remains an ideal and arguments can only be tested on formal grounds, that is, for validity.

Redefining the analytic

Returning to the analytic-synthetic distinction, the way to define ‘analytic’ is a formal one and it is neither metaphysical – does it refer only to what is in the mind or also to the outside world? –, nor epistemic – where is the border between logic and experience? Keep in mind that before Quine, the distinction was made by only defining analytic statements and calling all other statements synthetic.  

Sadly enough, both ‘analytic’ and ‘synthetic’ can only be correctly understood by their formal philosophical meanings. In ordinary language, we may just as well turn them upside down, by stating that first, we analyze reality – and make selections from it – to then synthesize our analysis with the rest of knowledge using a formal system of logic, which is indeed reminiscent of both functionalist accounts in the philosophy of mind and accounts of innate competences such as Noam Chomsky’s. Therefore, the formal-informal distinction should be preferred over the analytic-synthetic. Spontaneous observations, either perceptional or mental, can only become knowledge by formalizing them following standard rules.

Back to Kant

Above I wrote that Kant suggested the possibility of a priori synthesis and that he wrote that 7 + 5 = 12 is essentially producing new knowledge. There is discussion about if Kant was right. Let us end that and consider Tim’s less bachelor-apt brother Tom, who, after visiting his new girlfriend Phyllis’ parents for the first time, tells his mother that “In front of the house there are seven trees” (p₁). “Behind the house, there are five trees”, Phyllis adds (p₂). Tim’s mother concludes to “Around Phyllis’ house there is a total of twelve trees” (p₃), or simply p_{1 &}  p₂ -> p₃. Obviously, under the condition that numbers represent similar concrete objects, Kant was right – please note the relationship between resemblance and countability: the objects that Tom and Phyllis take into account need to resemble trees: shrubs, even if they are of the same species as one of the trees, are not counted, so here resemblance precedes physical qualities.

Only if numbers would represent abstract objects, Kant would have been wrong. From the history of mathematics, we know that in its beginnings calculations were made using pebbles – calculi – that were very concrete. Only later, numbers became abstract, followed by the abstraction of elementary geometrical shapes and derivations such as irrational numbers, to not speak of functions, sets and the entire rest. But originally, mathematics was very foundationalist.

An interesting question is if 7 + 5 = 12, when adapted to concrete objects such as trees or pebbles is still to be considered a priori knowledge. It looks as if also the a priori – a posteriori distinction is artificial. Arithmetics has to be learned, and as every math teacher knows, is best explained by concrete examples, that is, examples referring to a posteriori experience. No new knowledge will come of ‘truths’ as long as we do not know what it is we call true. ‘Truth’ as such is blind. But sensible propositions at least claim to represent material reality. The essence of knowledge is rather the formalization of observations through a well-tried system with the intention to share them with the rest of the world.

To combine the ‘truth’-criterion with informal knowledge is gastronomic cooking without recipes.