Apodeixis
from 350 BCE Posterior Analytics
Not just a valid argument but one whose premises are prior to, better known than and explanatory of the conclusion — proof that shows why, not merely that.
- Active
- from 350 BCE
- Where
- Athens
- Also
- apodeixis
- Status
- In use
- Tradition
- Classical procedures
What it is
Apodeixis is Aristotle's account of what a scientific proof must be. Validity is not enough. The premises must be true, primary, better known than the conclusion, and — the crucial condition — they must be the cause of the conclusion rather than merely correlated with it.
The distinction he draws is between demonstration *that* something is so and demonstration *why*. Both can be valid; only the second is knowledge.
How it works
Identify what is to be shown. Find premises that are true, that are more evident than the conclusion, and that state why the conclusion holds. Derive the conclusion deductively from them.
The chain must terminate. Since the premises must be better known than what they prove, and this cannot go on forever, demonstration rests finally on first principles that are grasped directly and not demonstrated at all.
The form of the inference
- The premises are true and better known than the conclusion
- The premises state the cause of the conclusion, not merely evidence for it
- The conclusion follows deductively
- Therefore the conclusion is known, and known in the sense of being understood
The argument
Proposition This beam will carry the load.
- The demonstration that it will is that an identical beam carried the same load last year without deflecting. This is true and it is evidence, but it does not say why.
- The demonstration why begins from the material's modulus and the section's second moment of area, both established independently of this beam and both more securely known than any claim about it.
- From those, with the span and the load, the deflection under load follows by calculation.
- The calculated deflection is within the allowable limit, and the calculation states the cause: the beam carries the load because of what it is made of and how it is shaped.
Therefore The beam will carry the load, and the argument shows why it will, so it also tells you what would have to change for it to stop being true.
The difference between the two arguments is the whole point of the method. The first would collapse silently if the load changed; the second tells you exactly how much change it can take.
What it is good for
Any body of knowledge that is meant to be systematic rather than a list of findings. Mathematics, engineering and the parts of science that have been axiomatised all aspire to this shape.
Where it fails
The requirement that first principles be grasped directly is where it breaks. Aristotle needs a faculty, nous, that apprehends them without demonstration, and this has satisfied almost nobody since — it looks like a name for the problem rather than a solution to it.
It is also a poor description of how science actually proceeds. Real inquiry runs from observations upward toward principles, which is the opposite direction, and only reorganises itself into demonstrative order afterwards. The Posterior Analytics describes a finished science and not the making of one.
Common misconceptions
That it is simply deduction. Every apodeixis is a deduction; almost no deduction is an apodeixis, because the causal and epistemic conditions on the premises are far stronger than validity requires.
Where it came from
The Posterior Analytics, which is the first theory of scientific knowledge in the Western tradition and the model for every axiomatic system after it.
What became of it
Realised most fully in Euclid, aspired to by Newton and Spinoza, and eventually abandoned as a model of scientific method while surviving as the model of mathematical exposition.
Lasting influence
Euclid's Elements is its purest execution, and through Euclid it shaped what a proof looks like for two thousand years.
Why it still matters
Alive in mathematics, and in the engineering distinction between a component that has been tested and one that has been calculated.
Put another way
Knowing the tide table rather than having watched the water rise.
Texts
Aristotle, Posterior Analytics
Book I sets the conditions on demonstration; Book II confronts the problem of how first principles are known, and does not obviously solve it.
Where to start
Compare any proposition in Euclid with a modern engineering calculation. They are the same shape, twenty-three centuries apart.
The surprising bit
Aristotle's model of science, which requires proof from causes, made almost no contribution to the science that eventually worked, which proceeded by exactly the direction of inquiry he treated as preliminary.
In their own words
We think we understand a thing when we know its cause, and that it could not be otherwise
Who held it
Euclid
Connections
- Relies on
- Deduction, Categorical syllogism
- Set against
- Endoxa
- Influenced
- Formalisation, Grounding
- Parallel to
- The infinite regress
The verdict
The right account of what a finished body of knowledge should look like, and the wrong account of how to get one.