Is Axiomatic Deduction Knowledge? (Also, how the categorical syllogism works!)

To answer the question  the definitions of axiom, knowledge, deduction, and induction must first be understood.

According to the Oxford English Dictionary Online:

an axiom is “A statement or proposition which is regarded as being established, accepted, or self-evidently true;”

knowledge is defined as “True, justified belief; certain understanding, as opposed to opinion;”

deduction is “The inference of particular instances by reference to a general law or principle [an axiom];” and induction is ” The inference of a general law from particular instances.” 

In science the general rule for what is scientific knowledge is whether a theory is experimentally testable and falsifiable. However, not all knowledge is scientific, nor should it be. Under an experimental testability definition of (scientific) knowledge axiomatic deduction is not (scientific) knowledge, because axioms are not strictly testable.   However, under this confining definition of (scientific) knowledge, in which only inductive, experimentally testable things are knowledge, pure mathematics is not (scientific) knowledge. Why?

Under the induction-only definition of knowledge mathematics should be rejected as not (scientific) knowledge. Here mathematics means pure mathematics (abstract, ex. calculus) as opposed to applied mathematics (applied, ex. engineering). Mathematics is AXIOMATIC and based on DEDUCTION. For example, the fundamental assumption (axiom) on which all mathematics is based is A = A, or the identity property. This is an axiom because it is not testable or learnt through observation/ experience. A = A is a given (axiom) that allows one to do something as simple as 1 – 1 or something as complex as e^ π√-1. If it weren’t for the A=A axiom neither would be possible, id est if 1 did not equal 1 then 1-1 wouldn’t equal zero; similarly if e did not equal e, π did not equal π, and √-1 did not equal √-1 [1], the equation would not equal -1. Take another fundamental axiom of mathematics, is the commutative property, that is A + B = B + A.  Again this is not testable or derived from observation. and not higher math would be possible without it. There are many more examples, of mathematics axiomatic deductive nature.

So, perhaps the inductive only definition of knowledge is not a useful definition. It may serve as the basis for good scientific knowledge. Induction (experience based knowledge) is the only type of knowledge supported by the traditional scientific method, hypothesize – test – observe – collect – report. The traditional scientific method is all well and good but it cannot (should not) be applied to many fields. Image a study of theology based on the scientific method. It is basically impossible, which is probably why some many modern scientists are atheists. Linguistics based purely on the scientific (inductive) method is being done, but couldn’t deductive axiomatic linguistics also work. For example, axiom: All languages have grammar. Syllogism based on axiom: Lipan is a language, therefore Lipan has a grammar. Certainly it is not much of an insight but it is at least a useful bit of information. A language learning example might go: axiom: All vocabulary is learnable. Axiomatic syllogism: the Spanish for hand is ‘mano’ this is a bit of vocabulary. Therefore, the Spanish word ‘mano’ is learnable. Again not much, but something. Now, try to image an axiomatic science (ignore that this would be a violation of the scientific method). Take Physics for an example. Axiom: all matter has mass. Axiomatic syllogism: the Earth consists of matter, therefore, the Earth has mass. Another example based on the same axiom: atoms are matter, therefore, atoms have mass. Deductive science is possible! However, “absolutely” verified science, as modern scientism seem to want is necessarily inductive.

Rejecting deductive knowledge because it is not “discover” through empirical observation is a tragedy and a betrayal against the advancement of human understanding. Almost all philosophy is deductive, as is pure mathematics, history, and political science. All of these are being assaulted by the practitioners of scientism, who want everything to be study through using the scientific method. If they had their way morals, ethics, pure mathematics, and history would be studied by the scientific method, a near impossibility.  Try for a moment to image history as study by the scientific method. Example, if William McKinley was a U.S. President, then he would be assassination. There is a hypothesis, now come up with a scientific test (an appeal to the historical record does not work as a true test of the hypothesis). This is not to say induction and the scientific method should be rejected. In the opinion of this writer, a pluralistic system of knowledge is the best system. Induction and deduction, a priori and a posteriori,   axiomatic and non-axiomatic study, researcher, thinking, and seeking of knowledge should all be excepted. Each has its place, though attempt to use one method where the other is more common are and should be done.

Axiomatic deduction is knowledge. LET THERE BE THINKING (both inductive and deductive)!

{If you don’t agree with the views expressed here, have counterexamples, think this writer has missed something, or see a mistake, please leave a comment below!}

[1]: As an interesting note √-1 is defined as i (imaginary number), which allows one to perform mathematical operations requiring taking the root of a negative.

The Simple Categorical Syllogism: The categorical syllogism consists of three categorical propositions (a truth-telling sentence made up a subject- a form of to be in the present tense – and an object). Two premises and one conclusion.  There are 3 constituent rules of the syllogism and six inferential rules. The constituent rules are:

1. 3 propositions, 2 premises, 1 conclusion;

2. 3 terms:

3. no term appears twice in the same propositions.

The six inferential rules are made of up of 2 rules of quality, 2 rules of quantity, and 2 rules of distribution, each set is broken down into one categorical rule and one hypothetical rule. The six inferential rules are:

1. Must be at least one universal premise;

2. If there is a particular premise there must be a particular conclusion;

3. Must be at least one affirmative premise;

4. If there is a negative premise there must be a negative conclusion, and vice versa;

5. The middle term (the term shared by both premise, but not the conclusion) must be distributed at least once;

6. If a term is distributed in the conclusion it must be distributed in the premise in which appears.

For a syllogism to be a syllogism it must met the first three, constituent rules. For the syllogism to be logically valid it must meet all six inferential rules.

An example of a valid syllogism:

All dogs are animals (universal, affirmative) {dog extended}

This thing is a dog (particular) {neither term extended}

Therefore, this thing is an animal (particular) {neither term extended}

(Middle term: dog is extended at least once).


Aristotle, McKeon, R. (ed.) (2001). Posterior Analytics, The Basic Works of Aristotle. New York: Random House, Inc.

von Gottenberg, M. (2012). Another Attempted Rescue of Praxeology. Retrieved from

Oxford English Dictionary Staff (2015). Oxford English Dictionary Online. Retrieved from

Science Buddies Staff (2015). Steps of the Scientific Method. Retrieved from


4 thoughts on “Is Axiomatic Deduction Knowledge? (Also, how the categorical syllogism works!)

  1. Excellent article. But I may add, there is a further recent form of deduction which appeared in my own self-published work. I call it the categorical deduction or coherent categorical deduction. It is a form of non-causal form of inference, meaning that it relies on abstractions as the content for an inductive argument. It then uses abstract relationships as the basis for determining that the result is not probablistic, but instead, absolute.

    The results take the form AB:CD and AD:CB in the case of quadra, in which opposites always relate along the diagonal (makes sense, because it is the longest distance), and only non-opposite terms are directly compared (resulting in a cyclical operation).

    As far as I can tell, I make fewer assumptions than in Aristotle’s reasoning. Perhaps the most burdensome aspect is the reliance on the existence of all opposite terms, but only RELATIVELY. Because they are relativized, I find the assumptions acceptable.

    Here are the assumptions for categorical deduction, in my method:

    1. All opposite terms relatively exist.
    2. Relatively, an opposite can be found for any term that is used.
    3. Opposites relate along the diagonal, since diagonal is a longer distance.
    4. Since opposites destroy one another, opposites are never compared directly, resulting in a cyclical order in which opposites are in opposite positions.

    It has the advantages that it’s coherent and analogous to math.

    I hope this method catches on and I get cited for it, as I feel it is a valuable contribution to, if not the creator of, objective knowledge.

      • My text is available on Amazon, Barnes & Noble, and other major websites. If you’re okay with an e-book edition, I can give you a free PDF in exchange for a review. Just e-mail contact [at] In my opinion the physical text is much more satisfying. But if you’re going to buy the PDF anyway, this is a good way to get it for free.

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