The quantum corrections related to the ideal gas model often considered are those associated to the bosonic or fermionic nature of particles. However, in this work, other kinds of corrections related to the quantum nature of phase space are highlighted. These corrections are introduced as improvements in the expression of the partition function of an ideal gas. Then corrected thermodynamics properties of the ideal gas are deduced. Both the non-relativistic quantum and relativistic quantum cases are considered. It is shown that (...) the corrections in the non-relativistic quantum case may be particularly useful to describe the deviation from the Maxwell–Boltzmann model at low temperature and/or in confined space. These corrections can be considered as including the description of quantum size and shape effects. For the relativistic quantum case, the corrections could be relevant for confined space and when the thermal energy of each particle is comparable to their rest energy. The corrections appear mainly as modifications in the thermodynamic equation of state and in the expressions of the partition function and thermodynamic functions like entropy, internal energy and free energy. Expressions corresponding to the Maxwell–Boltzmann model are shown to be asymptotic limits of the corrected expressions. (shrink)

Peter de Rivo (b. ca. 1420), argues for the existence of human freedom despite its alleged incompatibility with the truth of future contingent propositions. Rivo’s solution doesn’t follow the common medieval attempt to dissolve the alleged incompatibility, but claims that future contingent propositions aren’t determinately true. This approach troubled Rivo’s contemporaries, who thought it was incompatible with biblical infallibility, particularly the veracity of prophetic statements. Rivo tries to reconcile his solution with orthodox Christianity by grounding authentic (...) prophetic statements in God’s cognition of future events. In the end, Rivo’s attempted reconciliation fails because grounding the truth of prophetic statements in God cognition is incompatible either with his theological assumptions or his conception of free action. (shrink)

The Aristotelian commentaries by Petrus de Rivo, still unedited, represent a valuable instrument for our understanding of the major trends in the teaching of Aristotle at the fifteenth-century Faculty of Arts at Louvain. We published a preliminary survey of the manuscript material in last year’s issue of this journal, together with an account of the status quaestionis concerning Peter’s biography, works and the historical context of his thought. In the present article, we consider more closely a selection of his (...) commentaries on logic and natural philosophy. First, we offer an overview of Peter’s division of the sciences as expressed in his tractatus prohemiales, particularly those introducing his commentaries on Porphyry’s Isagoge and Aristotle’s Physics. Next, we considers Peter’s explanation of De interpretatione chapter 9, and his exposition of specific parts of Aristotle’s Physics and De anima. (shrink)

Quid enim? si Daphitae fatum fuit ex equo cadere atque ita perire, ex hocne equo, qui cum equus non esset nomen habebat alienum ? aut Philippus hasne in capulo quadrigulas vitare monebatur? quasi vero capulo sit occisus. Quid autem magnum aut naufragum illum sine nomine in rivo esse lapsum – quamquam huic quidem his scribit in aqua esse pereundum? ne hercule Icadii quidem praedonis video fatum ullum; nihil enim scribit ei praedictum: quid mirum igitur ex spelunca saxum in crura (...) eius incidisse? puto enim, etiam si Icadius tum in spelunca non fuisset, saxum tamen illud casurum fuisse, nam aut nihil est omnino fortuitum aut hoc ipsum potuit evenire fortuna. (shrink)

One concern bothering ancient and medieval philosophers is the logical worry discussed in Aristotle's De Interpretatione 9, that if future contingent propositions are true, then they are settled in a way that is incompatible with freedom. Another is if we grant God foreknowledge of future contingent events then God's foreknowledge will determine those events in a way precluding freedom. ;I begin by discussing the standard compatibilist solution to these problems as represented in Boethius's Consolation of Philosophy and then examine theories (...) that allegedly deviate from the Boethian solution. Boethius's solution to these separate problems involves showing that both problems operate on an ambiguity in the scope of the modal operator 'necessarily' present in the articulation of the problem. Once the ambiguity is removed we see that both disambiguations fail to offer a sound argument against the compatibility of free action with either God's omniscience or future contingent proposition's being true. The only difference between the solutions is that before executing the scope distinction strategy in the theological problem, Boethius reminds us that God knows future contingents rather than foreknowing them, since God is timeless. ;The rest of my discussion examines positions that allegedly deviate from the Boethian solution: positions held by Peter de Rivo, William Ockham and Plotinus. I argue that Ockham doesn't in fact deviate from the Boethian solution to the theological problem as is commonly held. Instead of offering a compatibilist position where God's omniscience includes foreknowledge, Ockham denies that God foreknows the future advocating instead a more sophisticated Boethian position. The other two philosophers, Rivo and Plotinus, deviate from Boethius, but unfortunately neither position appears philosophically plausible. Rivo's incompatibilist solution to the logical problem is inconsistent with his retention of the Boethian solution to the theological problem and is probably implausible on its own. Plotinus's compatibilist account fails not because it claims that necessity and freedom are compatible, but because the account of moral responsibility Plotinus offers to justify the compatibility fails. (shrink)

Quid enim? si Daphitae fatum fuit ex equo cadere atque ita perire, ex hocne equo, qui cum equus non esset nomen habebat alienum? aut Philippus hasne in capulo quadrigulas vitare monebatur? quasi vero capulo sit occisus. Quid autem magnum aut naufragum illum sine nomine in rivo esse lapsum – quamquam huic quidem his scribit in aqua esse pereundum? ne hercule Icadii quidem praedonis video fatum ullum; nihil enim scribit ei praedictum: quid mirum igitur ex spelunca saxum in crura eius (...) incidisse? puto enim, etiam si Icadius tum in spelunca non fuisset, saxum tamen illud casurum fuisse, nam aut nihil est omnino fortuitum aut hoc ipsum potuit evenire fortuna. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Pour une histoire de la ‘double vérité’David PichéLuca Bianchi. Pour une histoire de la ‘double vérité’. Conférences Pierre Abélard. Paris: Vrin, 2008. Pp. 192. Paper, €18.00.Since the publication of the work of the Belgian medievalist Fernand Van Steenberghen, a solid consensus seems to have emerged in the community of historians of medieval philosophy: no scholar in the Middle Ages defended the so-called “doctrine of the double truth” that (...) the Bishop of Paris, Stephen Tempier, had denounced in the prologue of his 1277 Condemnation. Moreover, how could a professional philosopher or theologian seriously contend that two contradictory statements are simultaneously true, one due to natural reason, and the other through the authority of faith? Therefore, the matter seemed settled and the file definitely closed: the double truth is nothing more than a “legend.” Yet the Italian medievalist Luca Bianchi’s in-depth critical analysis of the pseudo-historical evidence has now changed this. Without wanting to propose an entirely new interpretation of the historical ins and outs of this famous doctrine, Bianchi carries out a re-examination of the subject through a meticulous study of certain key texts that he has read in the light of a three-fold question: What impact did passages from Tempier’s decree on the “double truth” have on the medieval scholars themselves? Does the expression itself, ‘duplex veritas’, appear in some medieval documents? What were medieval thinkers’ attitudes in the face of philosophical arguments leading to conclusions contradicting the teachings of the Catholic faith?Due to space constraints, I will limit my discussion to Bianchi’s most crucial results. The answer to the first question (ch. 1) leads us in a surprising fashion to the famous Disputatio of Luther (1539). In this text, the father of the Reformation opposes the principle according to which what is true in theology must also be true in philosophy—a thesis whose paternity he attributes to Parisian theologians. Wanting to ensure the pre-eminence of faith-based teachings over all forms of human rationalization, Luther came to subscribe to a certain form of the double truth, since he claimed that what is absolutely true according to faith can be false and impossible from the perspective of philosophy. This explains his radical opposition to the “Parisian” principle that Bianchi calls “the principle of the oneness of truth.” We learn that some theologians in fifteenth- century Paris, notably Jean Gerson and Guillaume Baudin, were actually inspired by Tempier’s decree to defend some form of this principle.To respond to the second question (ch. 2), Bianchi first researched the electronic database Library of Latin Texts (CLCLT-6), and came up with “disappointing results”: no author uses the expression ‘duplex veritas’ in the relevant sense. Nevertheless, as imposing as it is, this database is not exhaustive, and Bianchi, well aware of this, extended his research by consulting other documents on paper, allowing him ultimately to reach a positive conclusion: the expression ‘duplex veritas’ appears in the course of a debate that took place around [End Page 99] 1470 on the subject of the truth-value of statements about future contingents. This debate involved a master of arts from Louvain, Pierre de Rivo, who considered it essential to distinguish between the truth of philosophers and the “popular” truth upheld by the churchmen, and a Parisian theologian whom we have already encountered, Guillaume Baudin, who invoked the “principle of the oneness of truth” to denounce the “double truth” that his adversary was promoting.From Bianchi’s multifaceted response to the third question (chs. 3–4), I can only present the groundswell. From the statutes promulgated by the Parisian Arts Faculty in 1272 up to the condemnation of Galileo in 1633, including the constitution Apostolici regiminis issued by the Fifth Lateran Council (1513), a significant evolution may be observed in the requirements academic and religious authorities imposed on professors of philosophy (as well as on scientists) in the western Christian world. Indeed, we may observe that these requirements correspond, first, to the obligation to counter, as far as possible, arguments that contradict faith, and then transform into the duty to commit to a strong... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Pour une histoire de la ‘double vérité’David PichéLuca Bianchi. Pour une histoire de la ‘double vérité’. Conférences Pierre Abélard. Paris: Vrin, 2008. Pp. 192. Paper, €18.00.Since the publication of the work of the Belgian medievalist Fernand Van Steenberghen, a solid consensus seems to have emerged in the community of historians of medieval philosophy: no scholar in the Middle Ages defended the so-called “doctrine of the double truth” that (...) the Bishop of Paris, Stephen Tempier, had denounced in the prologue of his 1277 Condemnation. Moreover, how could a professional philosopher or theologian seriously contend that two contradictory statements are simultaneously true, one due to natural reason, and the other through the authority of faith? Therefore, the matter seemed settled and the file definitely closed: the double truth is nothing more than a “legend.” Yet the Italian medievalist Luca Bianchi’s in-depth critical analysis of the pseudo-historical evidence has now changed this. Without wanting to propose an entirely new interpretation of the historical ins and outs of this famous doctrine, Bianchi carries out a re-examination of the subject through a meticulous study of certain key texts that he has read in the light of a three-fold question: What impact did passages from Tempier’s decree on the “double truth” have on the medieval scholars themselves? Does the expression itself, ‘duplex veritas’, appear in some medieval documents? What were medieval thinkers’ attitudes in the face of philosophical arguments leading to conclusions contradicting the teachings of the Catholic faith?Due to space constraints, I will limit my discussion to Bianchi’s most crucial results. The answer to the first question (ch. 1) leads us in a surprising fashion to the famous Disputatio of Luther (1539). In this text, the father of the Reformation opposes the principle according to which what is true in theology must also be true in philosophy—a thesis whose paternity he attributes to Parisian theologians. Wanting to ensure the pre-eminence of faith-based teachings over all forms of human rationalization, Luther came to subscribe to a certain form of the double truth, since he claimed that what is absolutely true according to faith can be false and impossible from the perspective of philosophy. This explains his radical opposition to the “Parisian” principle that Bianchi calls “the principle of the oneness of truth.” We learn that some theologians in fifteenth- century Paris, notably Jean Gerson and Guillaume Baudin, were actually inspired by Tempier’s decree to defend some form of this principle.To respond to the second question (ch. 2), Bianchi first researched the electronic database Library of Latin Texts (CLCLT-6), and came up with “disappointing results”: no author uses the expression ‘duplex veritas’ in the relevant sense. Nevertheless, as imposing as it is, this database is not exhaustive, and Bianchi, well aware of this, extended his research by consulting other documents on paper, allowing him ultimately to reach a positive conclusion: the expression ‘duplex veritas’ appears in the course of a debate that took place around [End Page 99] 1470 on the subject of the truth-value of statements about future contingents. This debate involved a master of arts from Louvain, Pierre de Rivo, who considered it essential to distinguish between the truth of philosophers and the “popular” truth upheld by the churchmen, and a Parisian theologian whom we have already encountered, Guillaume Baudin, who invoked the “principle of the oneness of truth” to denounce the “double truth” that his adversary was promoting.From Bianchi’s multifaceted response to the third question (chs. 3–4), I can only present the groundswell. From the statutes promulgated by the Parisian Arts Faculty in 1272 up to the condemnation of Galileo in 1633, including the constitution Apostolici regiminis issued by the Fifth Lateran Council (1513), a significant evolution may be observed in the requirements academic and religious authorities imposed on professors of philosophy (as well as on scientists) in the western Christian world. Indeed, we may observe that these requirements correspond, first, to the obligation to counter, as far as possible, arguments that contradict faith, and then transform into the duty to commit to a strong... (shrink)