A Priori-A Posteriori, Analytic-Synthetic, and Necessary-Contingent Distinctions

Understanding: the A Priori-A Posteriori Distinction, the Analytic-Synthetic Distinction, the Necessary-Contingent Distinction, and Other Terms of Propositional Logic

We explain the a priori-a posteriori distinction, analytic-synthetic distinction, necessary-contingent distinction and other logic-based terms. In doing this we will define Kant’s analytic a posteriori, synthetic a posteriori, analytic a priori, and synthetic a priori from his Critique of Pure Reason (in which he defines many terms and rules of propositional logic; that is, terms and rules pertaining to the validity of statements and arguments).^[1][2][3][4]

Understanding Kant and the the a Priori-a Posteriori Distinction

The above terms may sound intimidating at first, but the gist is simple. They are all terms used by Immanuel Kant that speak to whether a statement (a judgement or proposition) is based on empirical data (facts based on experience), rationalized ideas (facts based on ideas), or a mix of the two.

This concept can be illustrated in a number of ways, including by placing terms into two distinct categories (rational and empirical) like we do below.

What do a priori and a posteriori mean? a priori means “prior to experience” (“pure” “formal” imagination and reason; rationalization not based on experience), and a posteriori means “after experience” (concepts we get from observation via our senses; based on empirical experience). All our terms speak to one of these two categories or a mix, as ultimately everything we conceptualize is either the observed properties of an object, an imagined idea, or a mix.

Rationalism Vs Empiricism. This video works as a quick overview of our subject. What it lacks in detail it makes up for by offering a quick and easy visual of our overarching subject.

“Hume’s” Two Pronged Fork of Rationalism and Empiricism

As noted, the above terms are all essentially describing the same “two pronged fork” (called “Hume’s fork“). The prongs are:

1. Rational (facts based on ideas), Relations of Ideas – Statements about ideas. These are analytic, necessary statements (tautology) that are knowable a priori. They are known through reason (rationalism). Things we know through thought alone. Ex. “1+2=3,”“no apples are blue,” “all bachelors are unmarried.”
2. Empirical (facts based on experience), Relations of Facts – Statements about the world. These are synthetic, contingent, and knowable a posteriori. They are gained through empirical evidence (empiricism). Things we know through the senses. Ex. “An apple tastes sweet,” “the man is sitting in a chair.”

Of course, with that two prong distinction in mind, we can note that each set of terms and each term itself has its own important and meaningful definition (as do combinations of those terms).

TIP: The trick to understanding Kant is understanding what these terms mean in isolation and how they relate to each other and to the study of human knowledge. If you get the two-pronged fork, and you get the distinctions below, you get the basis of Kant’s theory of epistemology.

Introducing A Priori and A Posteriori, Analytic and Synthetic, and Necessary and Contingent

Each of the terms in the above fork can actually be considered as a pair where we consider the A Priori and A Posteriori, the Analytic and Synthetic, and the Necessary and Contingent.

Each pair speaks to different aspects of “what we can know” about statements (AKA logical judgements or propositions): The necessary and contingent speak to reality itself, the a priori and a posteriori speak to human knowledge and what we can know, and the analytic and synthetic speak to the language we use.

When we consider the nature of a statement/claim/judgement/proposition, we must consider “reality as it is,” “what we can know about reality as humans and consider knowledge,” and “the language we use to express knowledge about reality.” After-all, there is a difference between reality itself, how we conceptualize it, and how we communicate what we know about it.

Considering all three aspects of “what we can know” at once allows us to examine all aspects of a proposition, and shows us how we can have useful rational knowledge about the empirical world.

With that covered, it’ll help to give specific definitions to each set of terms. So let’s do that now.

TIP: A proposition is a statement containing at least two terms rational and/or empirical terms conjoined by qualifier like “and,” “or,” “if…then,” or, “not.” Humans conceptualize reality and rational ideas, and then use propositions (or in common language statements) to speak about that. Thus, Kant’s focus is on dealing with human knowledge in the proposition form, and then relating that back to reality itself using logic and reason.

A Priori vs. A Posteriori, Analytic vs. Synthetic, Necessary vs. Contingent Explained

With the above in mind, we can define the a Priori-a Posteriori Distinction, the Analytic-Synthetic Distinction, the Necessary-Contingent Distinction as follows:

• The analytic – synthetic distinction: Analytic statements can be proven true by analyzing their terms (they are tautological), meanwhile synthetic statements cannot be proven true by analyzing their terms.
• The necessary – contingent distinction: Necessary statements are necessarily true in all cases, meanwhile contingent statements depend on more information (they are conditional).
• The a priori – a posteriori distinction: A priori statements do not rely upon direct experience (they are rationalized), meanwhile a posteriori statements do rely on direct experience (they are empirical).

Then combinations of the above terms speak to different types of rational, empirical, or mixed propositions (for example, we can consider a contingent synthetic a priori, an a priori statement that does not rely on experience, that is synthetic and thus can’t be shown true based on its terms alone, and is contingent and thus depends on more information to determine its truth-value).

A Priori-A Posteriori, Analytic-Synthetic, and Necessary-Contingent Table

Placed in a simple table that shows the a priori-a posteriori, analytic-synthetic, necessary-contingent distinctions, their relations, and some examples look like this:

Reality vs. Knowledge vs. Language: As noted above, all the definitions on this page speak to the relations of terms in propositions (the relations of subjects and predicates in statements). The point is that they can help us to better understand both the statement (the validity of the statement) and the truth behind a statement (the reality as it is, not just how we refer to it). For example, “all men are mortal” is a analytic a priori statement that tells us about the mortality of all men (where mortality is necessarily a property of any man; a tautological thing to state). The sentence is an analytic a priori, but there is no widget in reality called an analytic a priori and there is no widget called mortality. Instead, mortality is a quality of mortal beings and a priori is a logical category that helps us understand reality by understanding statements and language. When we speak, we necessarily speak in the language form, but despite this we are almost always referring to reality as understood by the human mind. These definitions help us to better understand reality, by examining the language form, to arrive at human knowledge as it relates to conception and understanding.

Definitions Related to Kant’s Terms

Below are important definitions related to Kant’s terms to help the above make more sense.

First, here are some underlying terms to help frame the general concept:

• Concepts and Terms: To make a judgement, one must compare terms. To have a term, one must conceptualize something. To conceptualize something, one must have observed the attributes/properties of an object or imagined something. In general there are two types of concepts/terms: empirical (observed) and rational (imagined).
• Proposition: A logical judgement (or simply “a statement”) about two or more terms (a subject and a predicate; ex. “a bachelor is sitting in the chair” is a proposition or judgement about the subject, “a bachelor,” who is “sitting in the chair,” the predicate). In other words a proposition is a proposed logical judgement about at least two terms.
• Category: A class of things that share properties. TIP: When we say a priori we could be referring to a term/concept, a judgement/proposition, or a class of things. It depends on context. Space and time are a priori terms, judgements about space and time are a priori judgements, all terms and judgements about space and time are of the category a priori.
• Premisses and Conclusions: Two types of propositions; where a premise is a proposition that leads to a conclusion (another proposition).
• Empiricism: Knowledge through empirical evidence (information from the senses). Facts about the world. What we observe. We observe something and form a concept by observing its attributes. All real objects and real attributes and the real relations of objects are empirical.
• Rationalism: Knowledge through ideas (information originating in our minds). Facts about ideas. Everything that isn’t material, and is therefore formal, is rational. All argument involves rationalizing about rational and empirical concepts.
• Deduction: Deductive reasoning where one compares certain truths to draw out other certain truths.
• Induction: Inductive reasoning where one compares specifics facts and probable truths to draw out likely truths.
• Objectivity: That which is confirmable as true. The state or quality of being true even outside of a subject’s individual biases, perspectives, interpretations, feelings, imaginings, and/or opinions. Ex. “Water is wet” or “1+1=2.”
• Subjectivity: That which we perceive. Knowledge based on individual biases, perspective, interpretations, feelings, imaginings, and/or opinions. Ex. “The water feels cold to me.”
• Skepticism: Generally, the art of questioning and contradicting. In this case, being skeptical that rationalism (pure reason) can result in true knowledge about the world. Can be interpreted broadly as skepticism about both empirical and rational knowledge. For instance, Kant suggests fusing the two styles as, “our senses themselves could be tricking us.”

The three basic distinctions we are working with (as noted above) are:

• The analytic – synthetic distinction: Analytic statements can be proven true by analyzing their terms (they are tautological), meanwhile synthetic statements cannot be proven true by analyzing their terms.
• The necessary – contingent distinction: Necessary statements are necessarily true in all cases, meanwhile contingent statements depend on more information (they are conditional).
• The a priori – a posteriori distinction: A priori statements do not rely upon direct experience (they are rationalized), meanwhile a posteriori statements do rely on direct experience (they are empirical).

The terms used in those distinctions can be defined in terms of propositions (logical statements) like this:

• Analytic proposition: A statement that is true by definition (can be proven true by analyzing the terms).  Ex. “All bachelors are unmarried.” The bachelor is unmarried because he is a bachelor. “Being unmarried” is an attribute of “bachelor,” thus the statement is necessarily true by definition.
• Synthetic proposition: A statement that is not true by definition and requires observation or more information (cannot be proven true by analyzing the terms alone). Ex. “The man is sitting in a chair.” Nothing about sitting in a chair makes one a man, but we can look to see a man is sitting in the chair. The statement is not necessarily true by definition, it is contingent on more information (such as observing to see if it is true in this case).
• a priori proposition: a proposition whose justification does not rely upon experience. Ex. “1 + 2 = 3,” or “all bachelors are unmarried.” It stands to reason all bachelors are unmarried, but I can’t meet every bachelor to confirm this empirically (I have to confirm it rationally). Likewise, we know 1 +2 = 3 rationally, but numbers aren’t tangible material things we can confirm with our senses (we can’t look at see a 1, 2, or 3 directly).
• a posteriori proposition: a proposition whose justification does rely upon experience. Ex. “The man is sitting in a chair.” I can confirm the man is in the chair empirically, via my senses, by looking.

This gives us four possibilities (four mixes of the analytic-synthetic and a priori-a posteriori) of which:

• Analytic a posteriori proportions: experience based propositions that can be shown to be true by their terms alone. This produces a contradiction and can be ignored. There are no Analytic a posteriori statements.
• Synthetic a posteriori proportions: experience based propositions that can’t be shown to be true by their terms alone. Ex. “The man is sitting in a chair.” I can confirm the man is sitting in the chair by looking.
• Analytic a priori proportions: propositions not based on experience that can be shown to be true by their terms alone. Ex. “All bachelors are unmarried.” By their nature, all bachelors are unmarried, although we can’t confirm it via direct experience.
• Synthetic a priori proportions: propositions not based on experience that can’t be shown to be true by their terms alone. Ex. “F=ma.” F=ma is necessarily true and not tautological, yet only indirect evidence can prove it (we can’t observe force, mass, and acceleration directly).

TIP: Kant “proves” that synthetic a priori judgements are possible early on in his Critique, pointing to mathematics (ex. “7 + 5 =12”), geometry (“a straight line between two points is the shortest”), physics (“F=ma”), and metaphysics (“God gave men free-will”). The main question he then seeks to answer is, “how are a priori synthetic judgements possible?” To be clear, Kant doesn’t explicitly give all those examples, but they do fit the bill. Since metaphysics, in its dealing with freedom, God, and the will, deals with the unknowable a priori, the key to figuring out the limits of our knowledge and the usefulness of rationalism, are found in mathematics (including geometry) and physics. Here one should note that which Kant eludes to, that the physical, logical, ethical (metaphysics as it relates to human action or conduct), and metaphysical are all classes of phenomena with different properties (the physical, we can know with things like physics and observation, the logical with things like mathematics and logic, the ethical with things like social science and the law, the metaphysical with things like individual experience and imagination). To learn about the world, we need to consider classes rooted in the physical world, so the physics (which explains natural things) and mathematics (which can be used to explain natural things indirectly) are good places to look (as ethics involves free-will and metaphysics involves “that which we cannot know”).

Terms of these four categories of propositions can then be of the following types:

• A necessary proposition (necessarily true): Any proposition which is necessarily true or necessarily false.  Ex. “The white cat is white; or, the white cat is not black.” The white cat is by definition necessarily white.
• A contingent proposition (dependent on more information): Any proposition in which the truth of the proposition depends on more information. They are propositions that are neither “true under every possible valuation (i.e. tautologies)”, nor “false under every possible valuation (i.e. contradictions)”. Ex. “a black cat is sitting in the chair.” That statement is only true when a black cat is sitting in the chair, otherwise it is false. It is contingent on what the case is in this instance.
• Tautological proposition (necessarily true but redundant): That which must be true (or could be true) no matter what the circumstances are. Ex. “the black cat is black.” It is redundant (tautological) to say the black cat is black. TIP: It may be redundant to state a fact about a property of a system (such as the blackness of a black cat), but it is still a useful trick of analysis. So it isn’t useless, it is just tautological.
• Contradictions (necessarily not true as it contradicts itself): That which must necessarily be untrue, no matter what the circumstances are or could be. Ex. “the man is in a chair and not in a chair.”
• “Possible” proposition (is true under certain circumstances): Are true or could have been true given certain circumstances. Ex. “x + y = 4.” That could be true, it depends on the values of x and y.

With that in mind, let’s put the basics together in a table like we did above, but this time with more detail to better illustrate all this.

Phenomena and noumena: Kant also considers other terms like phenomena and noumena. Phenomena are the appearances and properties of things; that which constitutes what we can experience and sense. Meanwhile, noumena are posited objects or events that exist without sense or perception (that which constitutes reality). In other words, the properties and effects of a thing that we can sense directly are phenomena, and the rest is noumena. All synthetic a priori judgements that tell us about the world are rationalizations about phenomena (like F=ma which describes the phenomena of force, mass, and acceleration). Understood loosely, 1. noumena is of the rational and phenomena is of the empirical, and 2.noumena is the thing-in-itself and phenomena is the effects (the manifestations of those things that can be perceived via the physical senses). TIP: See Plato’s theory of the forms (a theory of a noumenal world; as a metaphor at least) for more on different ways to understand noumena. NOTE: Empirically speaking, an object is a collection of properties (ex. a photon isn’t a widget with properties as far as we know; the only way to describe a photon is to describe its properties, its phenomena). From this perspective there is only phenomena in the physical world and noumena is just a metaphysical idea (at best describing a collection of properties; directly observable or not). With that said, loosely speaking, it helps to understand that we can have useful knowledge of an object beyond what we can sense about an object directly. Still, the takeaway is “the noumenal world may exist, but it is completely unknowable through human sensation… and therefore it is a purely metaphysical concept.”^[5][6]

What does “transcendental” mean in Kantian terms? An important but complex concept of Kant is the “transcendental.” Essentially each part of our discussion gets a transcendental, which generally describes where one category (like a priori) transcends into another (like a posteriori). Important for our conversation is the Transcendental Aesthetic, which describes the a priori of empirical things (like space, time, geometry). Here it describes not the metaphysical aspects of space and time, but the useful physic concepts used to predict behaviors of physical bodies that transcends the limits of pure rationalization and becomes useful knowledge about the world. Meanwhile, to flesh out the picture, Transcendental Logic describes the aspect of logic that relates to the empirical (like the categorizing of relations between objects). A synthetic a priori is of the transcendental aesthetic and we have categorized it using transcendental logic. Learn more Kant’s Transcendental.

TIP: As you can see a from the above, some terms are very similar, this is because all these terms speak to different aspects of “what we can know.” All of logic is a bit like that, sometimes we are talking about the process of thought, sometimes about the product. Sometimes about a judgement, sometimes about a term. A justification that relies on experience (a posteriori), and a statement that is true on observation (synthetic) can use some of the same exact examples (as they are both speaking about an empirical judgement). Likewise, we can consider synthetic a priori terms, judgements, and categories (not just judgements/propositions/statements). Despite this, each term speaks to a different aspect of thought and has a slightly different meaning. In other words, many terms are similar, but they have specific meaning, and need to be considered on their own merit.

A Table Showing the Analytic a posteriori, Synthetic a posteriori, Analytic a priori, Synthetic a priori, and the Necessary/Contingent Distinction in More Detail

Below is a table that illustrates the above terms as used by Immanuel Kant in his Critique of Pure Reason (his examination of the validity of using rationalized formal logic only to find useful truths about the world; as opposed to Hume’s idea that only direct empirical observations of the world detected via our senses produced useful truths; HINT: Kant ends up concluding Pure Reason is useful).

In other words, Kant’s goal was to prove Hume’s idea that pure rationalization tells us nothing about the world wrong. He does this by proving the existence of a synthetic a priori (a statement not based on experience that can’t be shown to be true by its terms alone).

The table below helps show what a synthetic a priori is, by showing how the analytic-synthetic and a priori-a posteriori relate.

TIP: Speaking loosely, rationalism is related to deduction and empiricism is related to induction. Learn more about these reasoning types.

Conclusions and Creating a Synthetic a Priori

Above we illustrated the basics of what you’ll learn from Kant directly if you read his Critique of Pure Reason from a modern perspective.

To summarize what that means:

• The subject tells us about the works of the skeptic David Hume and the rationalist Immanuel Kant.
• This whole subject speaks to fundamental aspects of logic and reason as they apply to epistemology (specifically in terms of judgements/statements/propositions); so these terms don’t just help you understand Kant, they help you understand logic and reason as applied to any modern field.
• The underlying duality is between rationalization and empiricism. Since everything is an abstraction of that concept, what Kant and Hume are doing is essentially helping to guide our thinking through from that concept to the logic conclusions we can make based on that.
• The general take away is the synthetic a priori (a judgement that “crosses forks” successfully).
• The goal of “crossing” these forks is to show that pure rationalization can tell us something useful about the world, and that we should not, like Hume suggests, go throwing all our books on pure reason on the fire (although to be fair to Hume, I suspect he would have revised his theory or offered a counter-theory had he not passed away before Kant’s rebuttal was written; see the story of Kant and Hume).

Using a Synthetic a Priori to Cross Forks

In his Critique of Pure Reason, Kant generally points to mathematics (ex. “7 + 5 =12”), geometry (“a straight line between two points is the shortest”), physics (“F=ma”), and metaphysics (“God gave men free-will”) to show “synthetic propositions a priori” possible.

Specifically, he tells us we should focus on mathematics (including geometry) and physics. Kant then zeroes in on the a priori concepts/terms of space and time to justify his ideas about “synthetic propositions a priori.”

Of course space and time are complex concepts (terms), and not simple judgements using terms like “the man is on the chair,” and thus they are a little harder to explain (especially considering Kant’s sometimes unclear and dense writing).

To simplify Kant’s ideas into my own words.

“Synthetic propositions a priori” are a category of things (concepts and judgements) that are pure imagination, pure rationalization, and having no physical form, but also exist as truths that tell us about the world.

A straight line doesn’t exist in real life, nor does a perfect circle, but yet geometry does. Is geometry just reasoning by analogy? Perhaps. But the result is none-the-less useful knowledge about the world.

Likewise, time and space aren’t any more material objects than a prefect circle (although they do have different qualities), but they are none-the-less real.

Here we can note that judgements that use terms from this category of synthetic a priori (for example, judgements about the world that use terms related to geometry or space and time) are synthetic a priori judgements.

Any Synthetic a priori judgement that is valid then is also an example of “crossing forks” (the synthetic from the empirical fork is mixed with the a priori from the rational fork; see Hume’s fork for the metaphor).

Synthetic a priori judgements include statements like “all phenomena in general, that is, all objects of the senses, are in time and stand necessarily in relations of time”  and equations like Newton’s F=ma or Einstein’s E=mc² are examples of synthetic a priori judgements.

All these judgements are Pure Reason (Pure Logic; a Priori), despite being both necessarily true (valid statements / very strong theories), and they are not tautological (not purely analytic and redundant).

These synthetic a priori judgements tell us about the real world… in theory at least.

To confirm these tell us about the real world it helps to understand that we can use judgements like this to make predictions about phenomena.

In this respect, we can’t confirm that synthetic a priori judgements tell us anything about the world until we test and confirm them via experiment and actually physically “cross forks” (we have to not only create a “Synthetic a priori,” but prove it is true empirically via testing to show there is merit in all this rationalizing about synthetic a priori).

And of course, equations like F=ma do just that.

Even though we can’t reach out and touch their forms directly, we confirm spacetime, geometry, the equations of physics, and other valid synthetic a priori judgements “are true,” in that they can help us to predict what we will observe empirically with perfect accuracy (and thus we can treat them as scientific theories and facts).

Thus, examples like these are good examples of a “synthetic a priori.” The complex part is dealing with “Synthetic a priori” that can’t be proven indirectly with empirical testing, such as is the case with Moral Philosophy…

…and to end, this explains why the necessary-contingent distinction is so important.

Synthetic a priori judgements include pure metaphysical statements like “all men have free-will” and physics judgements like “F=ma.” Both are contingent on data, but where “F=ma” can be proven with testing (and can then be considered a necessarily true rule), “all men have free-will” is at best only partially unobscured via social science and metaphysical thinking.

Kant helps us to see that while we can have useful a priori knowledge about the world, the class of things that is metaphysics is destined to remain at least partly unknowable. However, Kant also helps us to see that anything that speaks to a phenomena in the real world can be better understood through rationalism.

On the example of F=ma as a synthetic a priori: To clarify and qualify the above. F=ma is used as an example of a synthetic a priori judgement on this page. This is because: 1. it fits the bill of a necessary synthetic a priori judgement (a statement, not based on experience, that can’t be shown to be true based on its terms alone, but which is necessarily true), 2. it is a nod to Kant’s main examples of space and time as a priori with which synthetic judgements can be made (F=ma loosely speaks to gravity as an effect of spacetime curvature where the mass and acceleration of the earth relate to the force of Earth’s local “gravity”), 3. because Kant specifically discusses “the doctrine of motion” as synthetic a priori (and F=ma is essentially the equation that describes Newton’s second law of motion), and 4. because Kant spends more time fleshing out his concept than he does offering us an example of the perfect objective synthetic necessary a priori statement. The thing to get here is that space and time are pure a priori (they aren’t tangible things), but yet they can tell us useful things about the empirical a posteriori world (in this vein, other statements that contain objective synthetic a priori knowledge include “mass and energy are equivalent” and “time is relative to frame of reference;” both of these statements are examples that concern what Kant calls the “transcendental aesthetic”). Consider Kant’s own words below:

“Thus our conception of time explains the possibility of so much synthetical knowledge a priori, as is exhibited in the general doctrine of motion, which is not a little fruitful.”

“Time and space are, therefore, two sources of knowledge, from which, a priori, various synthetical cognitions can be drawn. Of this we find a striking example in the cognitions of space and its relations, which form the foundation of pure mathematics. They are the two pure forms of all intuitions, and thereby make synthetical propositions a priori possible.”

“We have now completely before us one part of the solution of the grand general problem of transcendental philosophy, namely, the question: “How are synthetical propositions a priori possible?” That is to say, we have shown that we are in possession of pure a priori intuitions, namely, space and time, in which we find, when in a judgement a priori we pass out beyond the given conception, something which is not discoverable in that conception, but is certainly found a priori in the intuition which corresponds to the conception, and can be united synthetically with it. But the judgements which these pure intuitions enable us to make, never reach farther than to objects of the senses, and are valid only for objects of possible experience.”

– Kant on SECTION II. Of Time.

TIP: Just to phrase the bit on Hume’s fork one last time so it is clear: Kant successfully synthesizes Hume’s ideas with his own in his masterwork a Critique of Pure Reason, thus “crossing Hume’s fork,” by saying (paraphrasing), “although all knowledge begins with the senses, we can use our experiences to inform our reason, and vice versa; We can’t rely on our senses alone, but nor can we rely on pure rationalization.” Thus we can say, Kant “crosses Hume’s fork” by proving that we can create a confirmable [via testing] “synthetic” “a priori,” a proposition that is “necessarily” true and not dependent on itself, yet can’t be proven via direct empirical evidence (it can only be proven indirectly).

1. David Hume’s Fork and Immanuel Kant’s Synthetic A Priori. “Hume’s Fork” is a concept laid out by Kant to refute Hume and labeled by later philosophers.

1. A Priori and A Posteriori
2. Analytic–synthetic distinction
3. Critique of Pure Reason. Wikipedia.
4. READ: Critique of Pure Reason
5. Kant: Experience and Reality
6. Noumenon