The second principle is usually considered as an axiom of Classical Equilibrium Thermodynamics (CET), but there are different versions of that law and the deductive relations between them are not immediately evident. That is so because the formulations were given in distinct epochs using very heterogeneous concepts, from the “mechanist" language before the stable construction of CET until the properly theoretical language of systems and states. The versions of Kelvin-Planck (KP), Clausius (C1) and Carnot (Ca) preceded the concept of entropy. That emerged tied to the formulation of the second principle given by classical entropy (SC): ΔS ≥ Σ1 ∫ dQ1/T1 (where the 1s are the systems that interchange heat with the system that varies its entropy, its heat surroundings). The law of increasing entropy (AS), that of potential entropy (PS) and the statemet of Carathéodory (Cth) are postentropic formulations of the second principle. From another point of view, the expressions KP, C1, Ca, SC and Cth can be classified as belonging to the classical developments of the theory, and the law PS as a member of the Gibbsian representations. Formulation AS could be considered as above this second clasification, since from this study it follows that it constitutes the most powerful version of the second principle, which permits the deduction equally of the Gibbsian expression (PS) as the classical ones (first SC, and on that basis the remaining four, which are more specific). The deductive tree of such diverse laws can only be realized after a systematic analysis of the basic concepts of CET, which permits a homogeneous reformulation of them all. For that task the genetic, historical and intuitive point of view should be left aside so as to adopt a purely axiomatic-deductive point of view. The results of my formulation of CET, Garrido (1983-1986), realized following the axiomatic focus designed by Bunge (1967) for factual theories, have been used in this previous conceptual analysis. On the other hand, the deductive tree of the formuIations of the second principle constructed here has as effect a modification, small but interesting, of the axiomatic base for my proposal for CET: in my previous work I considered the laws SC and AS strictly equivalent and as such interchangeable as axioms, and I had chosen precisely SC for that purpose. Now, as l argue here in detail, I have confirmed that AS is more general than SC and so should be the formulation of the second principle that appears as an axiom of the theory. Amongst the basic concepts of the domain of reference of the laws of this theory which are used here, the strangest is that of pseudoequilibrium (the state of those systems which are not in equilibrium, but whose subsystems are). In the usual statements of the theory this concept at times appears explicitly and with varying denominations (system with internal constraints, internally constrained system, etc.), and on many occasions appears in an implicit way, with no proper name (the term equilibrium is frequently used in a wide sense, covering pseudoequilibrium, but that ambiguity is a source of confusion and obscurity). Here a general and exact definition of this concept is presented and, aboye all, is systematically used, greatly easing the theoretical and homogeneous formulation of versions of the second principle, in particular that of the law of increasing entropy: (AS) For any isolated system, between two instants in which it is in a state of equilibrium or pseudoequilibrium: ΔS ≥ 0, yielding identity when in all the intermediate instants the system maintains itself in equilibrium or pseudoequilibrium. Even if this law is the most powerful expression of the second principle, it is not sufficient for the deduction of the remaining versions. It is also necessary to consider other axioma of CET, such as the first principle, the law of heat, and the postulate of isolation and equilibrium. Nevertheless, even if they are not exclusively theorems of AS, they can still be considered as (weak) formulations of the second principle, That is due to the fact that what is most peculiar about them, inequality, is deductively inherited from AS (the only axiom of the theory that contains it).
A Recursive Definition of Goodness of Space for Bridging the Concepts of Space and Place for Sustainability
