Axioms and Postulates in Strategy

17 July 2011


Euclid provided the model of formal thought with his axiomatization of geometry. Legend has it that there was a sign over the door of Plato's Academy stating, 'Let no one enter here who has not studied geometry.'

Toward a Formal Strategy: Further Work in Progress

In several posts, Formal Strategy and Philosophical Logic: Work in Progress and A Comment on a Comment on a Comment among them, I have argued that a formal conception of strategy would be beneficial in clarifying our ideas of strategy. In so arguing I know that I am swimming against the tide. In his Postmetaphysical Thinking, which I briefly discussed in Post-Postmetaphysical Thinking, Habermas identified “the reversal of the classical relationship of theory to practice” (p. 7) as a theme in modern thought. Later in the same book, Habermas called the previous ascendency of theory over practice “the strong concept of theory,” which he even connects with theological motivations in the desire for “contact with the extra-ordinary,” and further claimed:

“In the modern period the concept of theory loses this link to sacred occurrences, just as it loses its elite character, which is moderated into social privilege.” (p. 33)

I think Habermas is right in his diagnosis, but this is not the whole story. While the idea of the special theological status of theory being moderated into social privilege is an interesting one, it does not do justice to the meaning of theory during its efflorescence among the Greeks. Euclid was one of the exemplars of Greek theory, and there are many extant anecdotes of Euclid’s contempt for any practical benefit of theory, the most famous one being his assertion that there is no royal road to geometry. When theoretical thought enjoyed another efflorescence during the Enlightenment, it had a somewhat more pragmatic cast to it, and this is the form that we have inherited and ultimately reacted against.

Clausewitz had a classical education and benefited from it. As paradoxical as it may sound, he attempted to bring the spirit of Enlightenment rationalism to the study of war.

I think Clausewitz would have agreed with me that a formal approach to strategy offers certain insights. Way back when, when people received what used to be called a “classical education,” they not only read literary classics, but also strategic classics like Caesar’s Commentaries as well as the classics of mathematics and logic, like Euclid’s Elements. Sometime in the twentieth century we said goodbye to all that, and with the calls for “relevance” in education the classical canon was abandoned in favor of trendy courses of study that did not tax undergraduates so much that they couldn’t also spend the better part of their time crafting revolutionary manifestos and occupying administration buildings (not that they would have remembered much of this).

If you're spending all your time protesting (or drinking, or trying to get laid) you're likely going to have a difficult time focusing on studying classic texts, which, at this age, probably doesn't seem nearly as exciting as the other pursuits.

One of the creeping intellectual calamities of our time is the belief that the facts speak for themselves. They do not. Emphatically, facts do not speak for themselves. Perhaps it would be better for formulate it like this: facts cannot be counted upon to consistently and univocally speak for themselves. In some contexts and situations, facts sometimes speak for themselves, but this is a function of the conditions under which the facts are manifested. All other things being equal — i.e., when the conditions under which a fact is manifested cannot be controlled or limited — facts do no speak for themselves. This is what Nietzsche meant when he wrote that there are no facts, only interpretations.

Friedrich Nietzsche (1844–1900)

Friedrich Nietzsche: “In opposition to Positivism, which halts at phenomena and says, 'There are only facts and nothing more,' I would say: No, facts are precisely what is lacking, all that exists consists of interpretations. We cannot establish any fact 'in itself': it may even be nonsense to desire to do such a thing. 'Everything is subjective' ye say: but that in itself is interpretation. The 'subject' is nothing given, but something superimposed by by fancy, something introduced behind. Is it necesssary to set an interpreter behind the interpretation already to hand? Even that would be fantasy, hypothesis.”

The modern quest to attain insight through accumulation of and immersion in a mass of detail is more likely to overwhelm than to enlighten. Just as the biographer selects a telling anecdote to reveal the character of his subject, so always the selection and presentation of facts in a given context shapes the meaning of these facts. The theoretical context shapes and gives meanings to facts, and it to seizes on exemplary intellectual “anecdotes” — as though playing the part of biographer of the mind — in order to make its point. These theoretical anecdotes that shape our understanding of everything else that stands in relation to them are called axioms.

Axioms are the still point in the turning world of thought. They provide our theoretical underpinnings, and are the enduring point of reference for formal thought. It was the tradition for much of intellectual history following Euclid to regard axioms as being certain, or beyond the possibility of doubt, or any other cognitive superlative you care to invoke. This attitude has fallen away in favor of what is today called hypothetico-deductivism: axioms are entertained hypothetically, without any assurance of certainty of indubitability, simply to explore their consequences and to see what follows from them.

While I regard the emergence of hypothetico-deductivism as a salutary development in formal thought, not every development in formal thought since Euclid’s time has been an unambiguous advance. While contemporary formal thought is extraordinarily subtle, there are yet distinctions incorporated into Euclid that have since fallen away since they no longer answer to the demands of contemporary logical and mathematical research, though I will suggest below that at least one such “lost” distinction may have a place in the formalization of other bodies of knowledge, and most especially in strategy.

Euclid’s axiomatization of geometry employs a distinction between axioms (also called “common notions”) and postulates. An axiom (or common notion) in Euclid is a principle that that holds good for all reasoning whatsoever, while a postulate is a principle that is specific to a particular subject matter. If we consider Euclid’s axioms, we can intuitively see how an axiom like, “Things which equal the same thing also equal one another,” is a principle of reasoning that is not specific to geometry, but can be used in any other instance of reasoning. Euclid’s postulates, however, such as, “To draw a straight line from any point to any point,” are specific to the subject matter of geometry.

There are many alternative ways to think about Euclid’s distinction. For example, in terms of Aristotle’s theory of definition, we can think of axioms as the genus and postulates as the differentia, which between the two of them define the species of geometry. A more recent way to think of the Euclidean distinction is in terms of specifying the logic of one’s argument. Twentieth century developments in logic were so rapid and so revolutionary that logicians and mathematics were forced to specify not only the axioms of their particular disciplines (which Euclid would have called postulates) but also to specify the logic by which they make their derivations (which Euclid would have called axioms), since many logics now exist side-by-side (it is in light of these developments that hypothetico-deductivism emerged). Every line of a formal proof must be justified by specifying the axiom that provides for it or the logical principle that allows for the transformation of one expression into another.

Euclid’s distinction between axioms and postulates is a perfectly good distinction in formal thought, still valid today, but contemporary logicians and mathematicians generally don’t employ it (though it is implicitly used, as noted above, which means that it is not formally invoked). Contemporary formal thought is more interested in, for example, the distinction between formation rules and transformation rules, which does not appear explicitly in Euclid, and therefore cannot be considered an aspect of Euclid’s formalization of geometry (though, again, it is implicitly used, which again means that it is not formally invoked).

I suggest that strategic though would benefit from the use of the axiom/postulate distinction, such that the axioms of strategic thought hold for all wars, or any war whatsoever, while the postulates of strategic thought would hold for particular cases of war. Moreover, a fully formal theory of war would need to specify both its axioms and its postulates in order to have an adequate theoretical context. The traditional principles of war, which I discussed and partially quoted in The Shadow of War, would constitute a good starting point for formulating the axioms of war.

It seems to me at first thought — though I readily admit I may be wrong about this; I will need to think more of this point — that bare war described by axioms of war — pure war, as it were, which would be somewhat akin of pure logic — would be unconditional war, absolute war, total war. Any further conditions superadded to the axioms of pure defining pure war would constitute limitations on the scope of war. Postulates of war added to axioms of war would be just such limitations of pure war, so that all war is, in a logical sense, limited war. And one must keep in mind that, as I noted above, any adequate theoretical framework for strategy must involve both axioms and postulates.

This last thought above — that all war is, in a logical sense, limited war — I regard as merely speculative at this point, so I only mention it here hypothetically. The most important lesson at present, regardless of whether or not I am right about limited war, is that the basic framework of war can be formally described by axioms of war (or, if you prefer, principles of war), while the fuller context of a particular war demands additional postulates that are distinct from the generality and universality of axioms, which must be added to the axioms to take account of the particular features of particular wars. For example, though all wars would equally employ the axioms of war, there would be separate postulates for conventional war, counter-terrorism, counter-insurgency operations, peacekeeping operations, and so forth.

This formulation of strategic thought in terms of axioms and postulates strikes me as a far more adequate and a far more subtle and flexible formulation than others that I have encountered — for example, an explication of strategic thought in terms of generations or gradients. In regard to generational and gradient conceptions of war, it is interesting to note Carnap’s conception of scientific concepts.

I recently quoted Carnap at some length in A Comment on a Comment on a Comment, which I gave his exposition of the difference between abstraction and concrete thinkers. This present post can even be understood as an extension of that post, and indeed as an extension of all that I have written to argue for an explicitly abstract and theoretical context for the understanding of warfare. In that post I mentioned that Carnap was very much a formal thinker at bottom (as it should be noted many scientists are not essentially formal thinkers, being more motivated by empirical considerations) and this is manifested throughout his works on the philosophy of science.

A conception of strategy that aspires to scientific status might still be essentially empirical if taken up by someone with an empirical turn of mind, but it will become a formal exercise if taken up by someone with an essentially formal turn of mind. Carnap’s approach to the philosophy of science gives as a quasi-formal theoretical framework for a scientific approach to strategy that fully embodies the spirit of methodological naturalism that defines what is distinctive about science while retaining the emphasis on formal rigor that will yield conceptual innovations for a body of knowledge that has not received any detailed formal attention.

What does this have to do with the conception of war in terms of generations and gradients? In both his Logical Foundations of Probability (which I previously quoted) and his Philosophical Foundations of Physics: An Outline of the Philosophy of Science, Carnap makes a tripartite distinction within scientific concepts between classificatory concepts, comparative concepts, and quantitative concepts. I would argue that the generational conception of war is a classificatory concept, while the gradient conception of war is partly classificatory and partly comparative.

We tend to think of scientific taxonomic schemes as exhaustive and rigorous scientific concepts, and, as far as they go, they can be considered such. But Carnap shows how, despite the detail and clarity that classificatory concepts can possess, especially in relation to the absence of any scientific concepts at all, that the progress of science often involves classificatory concepts being superseded by comparative and quantitative concepts. A systematic exposition of the classificatory concepts of warfare may yield ways in which these concepts can be gone one better by their extension to comparative and quantitative concepts derived from the initial classificatory concepts, but this will be an effort for another day.

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Grand Strategy Annex

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3 Responses to “Axioms and Postulates in Strategy”

  1. djmarsay said

    Crises often challenge one’s prejudices (assumptions), and it is important to distinguish ‘laws’ that are merely inductive from a past that may no longer be relevant (propositions), and those that are more reliable (axioms). In both world wars the UK model has been something like Russell’s type theory: there may be more than just two levels. The important level is ‘the level of the fight’. Above that everything is fixed. Below that everything has a set context, and can be fought classically. (It is clear what the ‘in order to’ is.) If one can’t win at the current level of the fight, one can increase the level, hoping at least to confuse the adversary as their propositions/assumptions are violated. Guerillas and insurgents can take this approach to extremes.

    In crisis management and perhaps wider conflict, the big mistake is to confuse what you are calling postulates and axioms. The higher levels have surely mattered.

    • geopolicraticus said

      Dear Dave,

      Honestly, I can’t fully follow you here. I would suppose, intuitively, that “the level of the fight” would be a fight of type 0, of individuals against other individuals. (In ecological terms, rather than logical terms, this would be an interaction on the level of micro-systems.) However, this interpretation won’t work because you are suggesting that there is both an “above” and a “below” to “the level of the fight.” So I conclude from this that I don’t understand you.

      If we were able to employ type theory here, we could also employ the axiom of reducibility in order to show the formal equivalent of the identity of indiscernibles, which would allow us to say that everything on other logical levels is “set” or “fixed,” but here I am only struggling with fragments because I don’t grasp your central idea.

      However, your comment did make me realize that we might say that “crisis management” enjoys a higher degree of generality than warfighting (since a war is one crisis among many), so that this is the general concept under which war falls. If this is the case, an axiomatization for strategy would actually fall under an axiomatization of crisis management. Then the axioms of crisis management would be the general principles that we employ in all crises, while warfighting would require additional postulates, and particular kinds of wars would require further postulates.

      I agree that crises can challenge premises, and often do. This is, in fact, a valuable intellectual service, and this is why we study past crises. The recent popularity of the book The Black Swan has made people aware of unknown unknowns (unanticipated developments, and perhaps unanticipatable developments). However, I suggest that if we can define a sufficiently abstract formal framework, we will not perhaps avoid unpleasant surprises, but we will always have a conceptual infrastructure in place that is prepared to deal with unanticipated contingencies.

      Given a sufficiently abstract, general, and comprehensive framework of formal thought, schematic responses to all possible crises could be formulated at least in rough outline, and while this outline may be rough indeed, it is probably better than nothing. When I read about the history of disasters, both natural and man-made, I am struck by the persistently fatal role that panic plays. When there is panic, people usually get killed. A robust intellectual framework that gives us at least a fig-leaf of what Habermas called procedural rationality may be the difference between cool heads and panic, and could therefore save lives.

      Best wishes,


  2. Nick,

    I see tactical and strategic as ‘levels’. Anecdotally, generals would often place themselves at what they thought would be the key HQs. If a force depends on maintaining constraints and achieving goals, then ‘the level of the fight’ is the place that is responsible for those constraints/goals most at risk.

    Anecdotally, the main ‘risks proper’ (in Frank Knight’s sense) are due to misplaced assumptions. In a crisis you may have more than two levels, and they may not be so obvious or rigid.

    Typically in a crisis many actors would be reasonably confident if only they could rely on the assumed context, as in Russell’s type theory. But as crises are empirical, any such reliance would be an assumption. I see setting and maintaining the key context as a typical ‘level of the fight’. Otherwise, identifying the key context or manoeuvring around what others think is the key context may be the ‘levels’. Anecdotally, it seems to me that crisis managers don’t always focus on the right type of level.

    Anecdotally, avoidable panic can be triggered by one’s attention being drawn to ‘an elephant in the room’ that is about to squash you. I am unclear about whether this calls for the sort of framework that you advocate or some psychological interventions: perhaps both.



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