It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the$\Pi ^1_1$reflection strength order. We prove that there are no descending sequences of$\Pi ^1_1$sound extensions of$\mathsf {ACA}_0$in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any$\Pi (...) ^1_1$sound extension of$\mathsf {ACA}_0$. We prove that for any$\Pi ^1_1$sound theoryTextending$\mathsf {ACA}_0^+$, the reflection rank ofTequals the$\Pi ^1_1$proof-theoretic ordinal ofT. We also prove that the$\Pi ^1_1$proof-theoretic ordinal of$\alpha $iterated$\Pi ^1_1$reflection is$\varepsilon _\alpha $. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles. (shrink)

Introduction The editors of this volume hope that it will prove useful for the study of philosophy in the Middle Ages by virtue of the comprehensiveness of ...

Why are natural theories pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. This approach was inspired by Martin's Conjecture, one of the most prominent conjectures in recursion theory. Fixing a reasonable subsystem $T$ of arithmetic, the goal was to classify the recursive functions that are monotone with respect to the Lindenbaum algebra of $T$. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate $\mathsf{Con}_T^\alpha$ of the consistency operator ``in (...) the limit'' within the ultrafilter of sentences that are true in the standard model. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as $\mathsf{Con}_T$ in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as $\mathsf{Con}_T$ in the limit nor as strong as $\mathsf{Con}_T^2$ in the limit. In fact, for every $\alpha$, we produce a function that is cofinally equivalent to $\mathsf{Con}^\alpha_T$ yet cofinally equivalent to $\mathsf{Con}_T$. (shrink)

It is a well known empirical observation that natural axiomatic theories are pre-well-ordered by consistency strength. For any natural theory $T$, the next strongest natural theory is $T+\mathsf{Con}_T$. We formulate and prove a statement to the effect that the consistency operator is the weakest natural way to uniformly extend axiomatic theories.

This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 : \mathcal{O}^A \leq_H B\}$ is well-founded. We provide an alternative proof of this fact that uses G\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, (...) we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any $A\in\mathbb{R}$, if the rank of $A$ is $\alpha$, then $\omega_1^A$ is the $(1 + \alpha)^{\text{th}}$ admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of $X$ is $\omega_1^X$. (shrink)

A critical discussion of Aristotle's thoughts on moral weakness, or Akrasia, with a look at the contributions of other philosophers, such as, Socrates and Plato on this subject.

We examine recursive monotonic functions on the Lindenbaum algebra of EA. We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and (φ∧Con(φ)). We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function f, if there is an iterate of Con that bounds f everywhere, then f must be somewhere equal to an iterate of Con.

What will it take for business to improve lives? Many think we need a theoretically sound meta-narrative to articulate the proper place for business in our lives. Important as that is, this meta-narrative will only come to life when everyone articulates his and her personal narrative, shares it with others, and ultimately fine-tunes it into a personal theory-in-use, one that guides everyday decision-making. Hoping that the Humanistic Management Association willsoon find room on its webpage for those of us in business (...) and business education to share our values and beliefs with each other, I thought I would start usoff by sharing mine here. (shrink)

We present an analogue of Gödel’s second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are $\Pi ^0_1$ -sound and $\Sigma ^0_1$ -definable do not prove their own $\Pi ^0_1$ -soundness, we prove that sufficiently strong theories that are $\Pi ^1_1$ -sound and $\Sigma ^1_1$ -definable do not prove their own $\Pi ^1_1$ -soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

Ordinal analysis is a research program wherein recursive ordinals are assigned to axiomatic theories. According to conventional wisdom, ordinal analysis measures the strength of theories. Yet what is the attendant notion of strength? In this paper we present abstract characterizations of ordinal analysis that address this question. -/- First, we characterize ordinal analysis as a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories, namely, the partition whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence (...) relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same $\Pi^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true $\Sigma^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make. -/- Second, we characterize ordinal analysis as an ordering on arithmetically-definable and $\Pi^1_1$-sound theories, namely, the ordering wherein $T\leq U$ if the proof-theoretic ordinal of $T$ is less than or equal to the proof-theoretic ordinal of $U$. The standard ways of measuring the strength of theories are consistency strength and inclusion of $\Pi^0_1$ theorems. We introduce analogues of these notions---$\Pi^1_1$-reflection strength and inclusion of $\Pi^1_1$ theorems---in the presence of an oracle for $\Sigma^1_1$ truths, and prove that they coincide with the ordering induced by ordinal analysis. (shrink)

On the nature of Aristotle's Ethics, by R. A. Gauthier.--Reason, happiness, and goodness, by F. Siegler.--The nature of aims, by J. Dewey.--Thought and action in Aristotle, by G. E. M. Anscombe.--On forgetting the difference between right and wrong, by G. Ryle.--Aristotle and the punishment of psychopaths, by V. Haksar.--Suggested further readings (p. 121-123).

We fix a gap in a proof in our paper Reducing ω-model reflection to iterated syntactic reflection.

Journal of Mathematical Logic, Volume 23, Issue 02, August 2023. In mathematical logic there are two seemingly distinct kinds of principles called “reflection principles.” Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic reflection principles assert that every provable sentence from some complexity class is true. In this paper, we study connections between these two kinds of reflection principles in the setting of second-order arithmetic. We prove that, for (...) a large swathe of theories, [math]-model reflection is equivalent to the claim that arbitrary iterations of uniform [math] reflection along countable well-orderings are [math]-sound. This result yields uniform ordinal analyzes of theories with strength between [math] and [math]. The main technical novelty of our analysis is the introduction of the notion of the proof-theoretic dilator of a theory [math], which is the operator on countable ordinals that maps the order-type of [math] to the proof-theoretic ordinal of [math]. We obtain precise results about the growth of proof-theoretic dilators as a function of provable [math]-model reflection. This approach enables us to simultaneously obtain not only [math], [math] and [math] ordinals but also reverse-mathematical theorems for well-ordering principles. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:LettersJames L. Walsh, Moira M. McQueen, Kevin O'Rourke, and Jean deBloisEarly Delivery of the Anencephalic InfantMadam:In the March 1994 issue of the Kennedy Institute of Ethics Journal, Kevin O'Rourke and Jean deBlois have replied to an article of ours (KIEJ, December 1993) on the early induction of the anencephalic fetus. They agree with our conclusion that such early delivery may be morally acceptable, but argue that our justification is (...) flawed. Exchanges on the subject probably could continue ad nauseam; here we simply state some of the elements of their article that leave us perplexed.1. The phrase "prolonging the life of" (p. 49) has one meaning in the case of artificial life supports, which may be unwarranted interventions, but it has a different meaning when it refers to not interfering with a person's enjoyment of necessary natural means of preserving his or her life. Inducing labor prior to viability, which denies the natural means of life support without a possible substitute, has more in common with smothering a dying patient than the nonemployment of or withdrawing of artificial means of life support. To refer to the death of the fetus here as an effect is rather like saying that I suffocated you and the effect of the suffocation is your death. This is quite different from induction after viability where the induction can have different effects.2. If the authors were correct that "the prohibition on inducing delivery before viability is predicated upon the assumption that the fetus in utero is a human being with the potential for further development" (p. 49) and the anencephalic has none, why is there need to employ the principle of double effect? They say that there is no moral obligation to prolong the life of the anencephalic fetus. Just as there is no need for the principle in cases of withdrawing unwarranted artificial life support, so there would be no need for it here. Because the authors seem to perceive a complete disjunction between direct and indirect killing (allowing to die being in the category of indirect killing), it seems that they are unable to let their argument rest here, but have to justify the death that occurs.3. The example concerning smothering and the nonemployment of aggressive interventions (p. 49) explains well the difference between killing and allowing to die. But allowing to die is not indirectly killing since it occurs as a result of the disease and not from the causality of the agent.4. How can we cause something and not intend it? There are instances in which we simply "permit" evil events, but where we are not the cause. For example, evil events may be attributable to the bad will of others. Legislators make just laws knowing that some people of bad will will break them for their own advantage. Here the legislators really do permit evil. Neither are we the cause of death when we allow a person to die by not imposing unwarranted means of life support; we permit his or her death. But can we so readily designate by "permit" [End Page 184] what we knowingly choose to bring about? O'Rourke and deBlois use interchangeably "indirectly intended" and "unintended" for what is caused by our actions; "indirectly intended" is a better designation than "unintended." The whole package, including all foreseen effects, is intended. Evil "happening" (p. 51) as a result of an action that I place can refer either to evil as caused by my action or to evil as arising from a cause apart from my action. Only in this latter sense do I permit evil and not intend it, either directly or indirectly.5. The fourth condition of the principle of double effect is given as follows: "There must be a proportionately grave reason for placing the act. The greater the evil, the more cogent the reason must be for performing the act" (p. 50). How the authors can abstract from balancing goods and evils here is beyond us.6. We cannot follow the authors' use of the term "morally good" with respect to the induced delivery (p. 50). The problem is concerned with the morally right... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:LettersJames L. Walsh, Moira M. McQueen, Kevin O'Rourke, and Jean deBloisEarly Delivery of the Anencephalic InfantMadam:In the March 1994 issue of the Kennedy Institute of Ethics Journal, Kevin O'Rourke and Jean deBlois have replied to an article of ours (KIEJ, December 1993) on the early induction of the anencephalic fetus. They agree with our conclusion that such early delivery may be morally acceptable, but argue that our justification is (...) flawed. Exchanges on the subject probably could continue ad nauseam; here we simply state some of the elements of their article that leave us perplexed.1. The phrase "prolonging the life of" (p. 49) has one meaning in the case of artificial life supports, which may be unwarranted interventions, but it has a different meaning when it refers to not interfering with a person's enjoyment of necessary natural means of preserving his or her life. Inducing labor prior to viability, which denies the natural means of life support without a possible substitute, has more in common with smothering a dying patient than the nonemployment of or withdrawing of artificial means of life support. To refer to the death of the fetus here as an effect is rather like saying that I suffocated you and the effect of the suffocation is your death. This is quite different from induction after viability where the induction can have different effects.2. If the authors were correct that "the prohibition on inducing delivery before viability is predicated upon the assumption that the fetus in utero is a human being with the potential for further development" (p. 49) and the anencephalic has none, why is there need to employ the principle of double effect? They say that there is no moral obligation to prolong the life of the anencephalic fetus. Just as there is no need for the principle in cases of withdrawing unwarranted artificial life support, so there would be no need for it here. Because the authors seem to perceive a complete disjunction between direct and indirect killing (allowing to die being in the category of indirect killing), it seems that they are unable to let their argument rest here, but have to justify the death that occurs.3. The example concerning smothering and the nonemployment of aggressive interventions (p. 49) explains well the difference between killing and allowing to die. But allowing to die is not indirectly killing since it occurs as a result of the disease and not from the causality of the agent.4. How can we cause something and not intend it? There are instances in which we simply "permit" evil events, but where we are not the cause. For example, evil events may be attributable to the bad will of others. Legislators make just laws knowing that some people of bad will will break them for their own advantage. Here the legislators really do permit evil. Neither are we the cause of death when we allow a person to die by not imposing unwarranted means of life support; we permit his or her death. But can we so readily designate by "permit" [End Page 184] what we knowingly choose to bring about? O'Rourke and deBlois use interchangeably "indirectly intended" and "unintended" for what is caused by our actions; "indirectly intended" is a better designation than "unintended." The whole package, including all foreseen effects, is intended. Evil "happening" (p. 51) as a result of an action that I place can refer either to evil as caused by my action or to evil as arising from a cause apart from my action. Only in this latter sense do I permit evil and not intend it, either directly or indirectly.5. The fourth condition of the principle of double effect is given as follows: "There must be a proportionately grave reason for placing the act. The greater the evil, the more cogent the reason must be for performing the act" (p. 50). How the authors can abstract from balancing goods and evils here is beyond us.6. We cannot follow the authors' use of the term "morally good" with respect to the induced delivery (p. 50). The problem is concerned with the morally right... (shrink)

In situations where anencephaly is diagnosed and where the mother's life or health is threatened Roman Catholic hospitals are faced with the dilemma of waiting until viability before inducing the fetus, thus potentially putting the mother at further risk. According to most Roman Catholic ethicists, induced delivery before viability is contrary to the Church's prohibition of direct killing of the innocent. The authors propose for discussion a reconsideration of this position in the case of the anencephalic fetus and conclude that (...) taking the life of such a fetus does not constitute an attack on its personal dignity and therefore is morally permissible. (shrink)