Definitions of provability noun capability of being demonstrated or logically proved synonyms: demonstrability see more Cite this entry Style: MLA "Provability. When the 1st Theorem talks about "arithmetical statements that are true but unprovable", "true" means "true in the standard model". provability 意味, 定義, provability は何か: 1. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much. Instead of emphasizing truth, as in classical logic, or proof, as in intuitionistic logic, linear logic emphasizes the role of formulas as resources. Provability is our term for demonstrating competence. It can be used to make pictures of the scrotum and testicles. There is sometimes a divide between. Provability logic is a modal logic that interprets the modal operator of K as provability and an additional axiom derived from Löb's theorem. Arguments. Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. We study interpolation properties for Shavrukov's bimodal logic GR of usual and Rosser provability predicates. The only legitimate reason. ə. provability (countable and uncountable, plural provabilities) The condition of being provable; Translations [edit] condition of being provable. Provability logic. 1. Fewer examples v t e Part of a series on statistics Probability theory Probability Axioms Determinism System Indeterminism Randomness Probability space Sample space Event Collectively exhaustive events Elementary event Mutual exclusivity Outcome Singleton Experiment Bernoulli trial Probability distribution Bernoulli distribution Binomial distribution provability. Right Bound (Rb): Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. The Provability Gap episodes: Introduction; Part I: The Victim; Part II: The Police; Part III: The Prosecutor; Part IV: The Public; If you or anyone you know needs help following a sexual assault, call the 24-hour SAFEline in Austin at 512-267-SAFE (7233) or the National Sexual Assault Hotline at 800-656-HOPE (4673). At last, we investigate the hierarchy of. The box as Bew (x) 4. Two representatives of this tradition are Gödel himself and the logician Emil Post. of Provability, each targeting its own set of applications (cf. “On notions of provability in provability logic. As a modal logic, provability logic. To achieve this focus, linear logic does not allow the usual structural rules of contraction and weakening to apply to all formulas but. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. 10. a. 00032. 0008 = 0. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. 5. This fun game allows students to earn tokens to the fair by demonstrating their understanding of probability. As a modal logic, provability logic. 98. Rosser provability predicates were essentially introduced by Rosser [] to improve Gödel’s first incompleteness theorem. The plural form of possibility is possibilities. Instead, nonprobability sampling involves the intentional selection of. v. . There are two possibilities: So 2 + 6 and 6 + 2 are different. Peano arithmetic 3. گودل یک افلاطونی بود و بنابراین هیچ مشکلی با اثبات درستی و نادرستی گزاره ها مستقل از اثبات پذیری آنها نداشت. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. Assume that the probability that a de. the level of possibility of something happening or being true: 2. Transparency and Provability: Because crypto tokens live on the blockchain, users can easily trace their provenance and transaction history in a way that is cryptographically verifiable. Verifiable Random Functions Silvio Micali y Michael Rabin Salil Vadhan z Abstract We ef ficiently combine unpredictability and verifiability by extending the Goldreich–Goldwasser–Micali constructionStandard Normal Distribution Tables STANDARD NORMAL DISTRIBUTION: Table Values Re resent AREA to the LEFT of the Z score. The inclination to settle problems in this way, I argue, reflects an implicit adherence to the axiom of solvability. Roll two dice. The only. "?) Yes, "provability of provability" makes total sense: as you indicate, we do this relative to some kind of formal system with inference rules: using the rules, the system can create formal proofs and thus "prove" statements. provability algebras [12], aims to establish links and find applica-tions of provability logic in the mainstream proof theory tradition. Reflection principles and provability algebras in formal arithmetic. Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. Impossible events have a probability of 0, and events that are certain to happen have a probability of 1. First published Wed Aug 27, 2003; substantive revision Fri Nov 2, 2018. 3 examples: 1. Artemov. We give several sequential calculi for GL, compare their merits, and use one calculus to syntactically derive several metamathematical results about GL. AI. Thus, the probability that they both occur is calculated as: P (A∩B) = (1/30) * (1/32) = 1/960 = . ) That last point is in stark. The theorems are widely, but not universally, interpreted as. German: Beweisbarkeit. Adams@NASA. 0. Finally, another reason to believe that what Ethical Relativism holds is indeed true is the undeniable fact of moral dispute occurring between and among groups as well as individuals. Then, write the probability of drawing certain colored marbles from a bag. As a modal logic, provability logic. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. In words, a model is one of total provability if the set of events an agent can prove coincides with the set of events she knows. We. 2. The truth provability logic of T is the set of all A such that A T * is true in the standard model, for all realizations *. Given that a student passes the examination, the probability that the student gets above 90% marks is ____. The word possibility is derived from the Latin word possibilitas which means able to be done. AC2 (a) is a necessity condition. provability (countable and uncountable, plural provabilities) The condition of being provable; Translations [edit] condition of being provable. Share. V́ıtězslav Švejdar ON PROVABILITY LOGIC∗. One may proceed to derive the provability predicate Prf from this by existentially quantifying the. Yes, it has its weird ways and sometimes its really shady but all in all, the mods and helpers are super nice, especially cyborg and joseph, big kudos to them and really the whole team behind it, its a really enjoyable experience with a daily reward system that is unrivaled!GST applies to logical systems with a notion of provability that is effective. 2 Propositions and Normal Mathematical Processes. We are interested in the probability of an event — the likelihood of the event occurring. t̬i / Add to word list the fact of being able to be proved: Unlike religion, science depends on provability. vəˈbɪl. As a modal logic, provability logic. 1) PMF, Mean, & Variance. The added assumptions are: you can only add if the two events are disjointPaperback. Provable prime, an integer that has been calculated to be prime. The meaning of PROVABILITY is the quality or state of being provable. Gödel’s two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. The two that I am familiar with look quite different to me but there is a long tradition on stats. The syntactic consequence of provability should be contrasted to semantic consequence, denoted by the double turnstile symbol . The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel–Löb provability logic. We study reflection principles in fragments of Peano arithmetic and their applications to the questions of comparison and classification of arithmetical theories. proved , proved or prov·en , prov·ing , proves v. Whenever we’re unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. It is therefore very strange that the Assumable Provability Theorem argument declares that Suspicious-FairBot will Cooperate with itself, while Naive-FairBot will Defect against itself. –3– 2 Law of Total Probability. Probability also describes something that might occur, the chance that something might happen. The language of the modal system will be different from the language of the system T under study. To Summarize So Far. First published Thu Mar 7, 2013; substantive revision Thu Aug 17, 2023. As a modal logic, provability logic. There is no such thing as objective truth. Beklemishev. Unit 7 Probability. 1. 1289–1291. Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. ‘‘’ for provability using universal generalization and modus ponens; within this chapter, ‘‘’ will always refer to provability within PA. For these strategies, we recommend you take profits at 3-5% or sooner – and set stop losses at 1-2% losses. During an ultrasound you lie on your back with your legs spread. In the first, is fast. Truth is a notion that depends on interpretation (i. Rule 1: The probability of an impossible event is zero; the probability of a certain event is one. The analysis of events governed by probability is called statistics. As with most automated theorem proving systems, HOL is a computer-aided proof tool: it proves simple theorems and assists in proving more complicated. This article is a philosophical study of mathematical proof and provability. How to pronounce provability. Listen to the audio pronunciation in the Cambridge English Dictionary. rocen_pro on October 10, 2023: "- BUSINESS 101 : NEURO 易⚡️ - What level we talkin’? - ️Talk only goes so high wit. Therefore, for any event A, the range of possible probabilities is: 0 ≤ P (A) ≤ 1. Learn more. provability pronunciation. Assume thatSML has become the formal methods workhorse because of its strong typing and provability features. One can transform a probabilistic signature scheme, such as the scheme in [GMR88], into a deterministic one if the signer uses a GGM pseudorandom oracle to replace the. Add the numbers together to calculate the number of total outcomes. There are a number of provability logics, some of which are covered in the literature mentioned in. We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. Whenever we’re unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. This system has played an important role in some applications of provability algebras in proof theory, and has been extensively studied since the late 1980s. Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic. vəˈbɪl. The task is to determine whether the conjecture is true or false. There are many more ways to earn even more Bitcoins on this site. Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. there is a tendency to focus attention only on the most difficult moral issues. Traducciones de inglés a español [PRO] Matemáticas y estadística / math terms. A proof. Indeed, Kurahashi [9] proved that for any natural. 1980d, Review of Raymond M. Provability is relative to a deductive system, whereas truth is relative to an interpretation (/structure/model). Add to word list. The course covers sample space, random variables, expectations, transforms, Bernoulli and Poisson processes, finite Markov chains, and limit theorems. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. 5 = 0. We usually write this as Δ ⊢ R φ, using the provability operator ⊢ (which is sometimes called single turnstile). Merriam-Webster unabridged. Definition. Solovay [100] who showed that the modal logic L4 axiomatized all propositional properties of the formal provability, and by Artemov [4] and Vardanyan [112] who demonstrated that the first order logic of formal provability was not axiomatizable. 5 Proof Systems. ( extsf{GLP}) is a provability logic in a propositional language augmented with a possibly transfinite sequence of modal operators; our case of interest is that in which the sequence is wellfounded. Hypothesis Essential #2: Testability (Provability) A statement must be testable to qualify as a research hypothesis. Moreover, we introduce three extensions , , and of and investigate the arithmetical semantics of these logics. Not able to produce an actual contradiction (if we are consistent) but with a proof that there is a counterexample and unable to produce such a counterexample. provability – philosophical and historical remarks. In this paper it is proved that the four infinite series of provability logics known at the present time - , , , and (, , finite) - include all such logics. 7-Probability Theory and Statistics amounts of data or characteristics of that data are also called statistics. Now the SEP shows that it's possible to derive Gödel's 2nd incompleteness theorem from this setup. Some key information on P-P plots: Interpretation of the points on the plot: assuming we have two distributions (f and g) and a point of evaluation z (any value), the point on the plot indicates what percentage of data lies at or below z in. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. What is the probability that a blue marble gets picked? Number of ways it can happen: 4 (there are 4 blues). The problem actually arose with the birth of set theory; indeed, in many respects it stimulated the birth of set theory. 05. Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. Synonyms for Provability. In 1994 Jech gave a model-theoretic proof of Gödel’s second incompleteness theorem for Zermelo–Fraenkel set theory in the following form:As nouns the difference between truth and provability is that truth is the state or quality of being true to someone or something while provability is the condition of being provable. 0. The modal logic GL has been found by Solovay [13] to formalize the provable propositional properties of the provability-predicate for Peano's Arithmetic PA (cf. 04 and 36. Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. A health care provider puts a clear gel on the scrotum. We would like them to be the same; that is, we should only be able to prove things that are true, and if they are true, we should be able to prove them. Continue this process until you have calculated the individual probabilities for each independent event. 42. Provability and interpretability logics (1985-1998) While a student, introduced polymodal provability logic GLP, and proved its arithmetical completeness. Collapse 10 Absolute Provability and Safe Knowledge of Axioms 10. Answer is: 1 4 1 4. , 2j!j is valid on any topological space. Single Event Probability Calculator. – Arturo Magidin. Let's learn all about probability in a practical way! Using a coin, dice and deck of cards. Question 2. However, what is its connection with Gödel's 1st incompleteness theorem? It seems unlikely that this. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form. Example 1. The mathematic probability is a Number between 0 and 1. Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. We provide two salient instances of the theorem. 8 -3. Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. Proof. We will prove: Soundness: if something is provable, it is valid. For your first example the order is important so the number of choices are 11*10*9*8*7, for the second it assumes that the team members are not ordered so you have count all arrangements the same by dividing number of arrangements of the choices. e. the fact of being able to be proved: . “On notions of provability in provability logic. The Provably Fair algorithm involves three major components: Server seed (comes from the casino side); Client seed (generated through the player’s browser); Nonce (a number that increases with every subsequent bet). Provable prime, an integer that has been calculated to be prime. How to pronounce provability. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. Go deeper with your understanding of probability as you learn about theoretical, experimental, and compound probability, and investigate permutations,. The cumulative distribution function of a real-valued random variable is the function given by [2] : p. The second is Gödel’s original modal. Search times were 10 d for Dali, 33 h for TM-align and 6 s for Foldseek, making it 180,000 and 23,000 times faster. One says that S {displaystyle S} is a semantic consequence of T {displaystyle T} , or T ⊨ S {displaystyle Tmodels S} , when all possible valuations in which T {displaystyle T} is true, S {displaystyle S} is. , Milgrom & Roberts, 1986; Giovannoni & Seidmann, 2007; Hagenbach & Koessler, 2017). The following argument, which we might call the "Provability Ar- gument," is a more general form of argument (6): (7) If there were any such thing as objective truth in ethics, we should be able to prove. L, if T has infinite height, 2. This new series explores how sexual assaults are investigated and prosecuted in Central Texas and why many of these crimes go unpunished. Provably correct, correctness of an algorithm that can be proved. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. Modal logic is concerned with the notions of necessity and possibility. -a certain logical relation should exists between the premises. it's easy to confuse two matters that are very. 3 Excerpts. We investigate the limitations of the second incompleteness theorem by constructing three different Rosser provability predicates satisfying several derivability conditions. Because the set of provable sentences of T is semi-recursive, we have a provability predicate of the form ∃yPrf(x, y) with Prf(x, y) rudimentary. Yearly Probability Of Dying: A numerical figure that depicts the likelihood of someone dying per year. probability theory, a branch of mathematics concerned with the analysis of random phenomena. Any information collected should be random to ensure credibility. Using this notation, sentences of provability logic express facts about provability. Our goal now is to (meta) prove that the two interpretations match each other. The same first-order. "?) Yes, "provability of provability" makes total sense: as you indicate, we do this relative to some kind of formal system with inference rules: using the rules, the system can create formal proofs and. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. k. able to be proven: . Truth versus Provability2/12. vəˈbɪl. If it’s not testable, it’s not a hypothesis – simple as that. 73,732 to 1. This contained a solution of an open problem on the logic of w-provability raised by George Boolos a decade earlier (1985-1988). Let's look at one probability in these two ways:In the development done before proving Gödel's second incompleteness theorem in most textbooks, one constructs the proof predicate Proof(x, y) such that. . Probability of a Normal Distribution. 2. As a modal logic, provability logic. SUBSCRIBE | Find The Provability Gap on Apple Podcasts, Spotify or your favorite podcast app “When I came into office, I believe the number of jury trials in Travis County was low,” District Attorney Margaret Moore said. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. Practicality. Published 19 January 2004. In a more precise way, it studies the relation between a theory T and a modal logic L via the provability interpretation which interprets in the language of L as the provability predicate for the theory T. CFIS 95-29, Cornell University, 1995). Provability. Your final probability is the number of "successes" ( a a has b b to the left and c c to the right) divided by the total number of possibilities, or. The key lies in leveraging the right opportunities in AI. As a modal logic, provability logic has. Provability logic. Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. 99 12 Used from $5. ’ (B) Find a precise provability semantics for the modal logic S4, which appeared to be ‘a provability calculus without a prov-ability semantics. provable definition: 1. the most honest review you'll see here. t̬i / uk / ˌpruː. 96. In this paper we present joint logics with both the strong provability operator and the proof operators. The risk assessment matrix works by presenting various risks in a color-coded chart with high risks represented in red, moderate risks in orange or yellow, and low risks in green. ZFC is a theory in the language of first order predicate logic (FOPL). Selected publications (prior to 2011) S. 5First-order logic simultaneously enjoys the following properties: There is a system of proof for which. g. 9 -3. This sequence provides a rigorous treatment of probability theory, using measure theory, and is essential preparation for Mathematics PhD students planning to do research in probability. probability meaning: 1. P(A ∣ B) = P(A ∩ B) P(B) P ( A ∣ B) = P ( A ∩ B) P ( B) which is actually the definition of the conditional probability of A A given B B, and not Bayes formula at all. First ,break the odds into 2 separate events: the odds of drawing a white marble (11) and the odds of drawing a marble of a different color (9). Viewed 642 times. Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. e. $egingroup$ There are plenty of techniques that can demonstrate unprovability, without having to resort to a formalization of provability: e. In other words, there is a method that checks whether a series of sentences purporting to prove a certain claim from a given axiom system is indeed a proof. The first mature and most representative example of. Most BGaming games you can play here at Bitstarz are provably fair and verifiable immediately and independently by you! We use a cryptographic method that ensures that we (the casino) or you (the player) can't possibly know the outcome of the game until the moment it starts. If you repeat this 10 times, the chances you die from any one of the dives is. Which of the following is true of workplace learning evaluation? a. For example, when flipping a coin, the probability of getting “heads” plus the. adjective uk / ˈpruː. Add to word list. Ví dụ về cách dùng “provability” trong một câu từ Cambridge Dictionary LabsRosser Provability and the Second Incompleteness Theorem. Export citation and abstract BibTeX RIS. Probability distributions are defined in terms of random variables, which are variables whose values depend on outcomes of a random phenomenon. ti /. . Non-probability sampling is defined as a method of sampling in which samples are selected according to the subjective judgment of the researcher rather than through random sampling. A pioneering result in the research of provability logics is Solovay's arithmetical completeness theorem [25] (see also [2,4, 9] ). Introduction to Statistics: Stanford University. 2. Example 1. 1. noun [ U ] uk / ˌpruː. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. It is named after Giorgi Japaridze. Its subject is the relation between provability and. Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. N. In "non-probability" sampling, nonrandomised methods are used to draw the sample and the method mostly involves judgment (Showkat, 2017; Glen, 2015;Doherty, 1994). Without some understanding of these matters it is impossible to comprehend the content of CT and its role in logic, as Church saw it, in the midst of. First published Wed Aug 27, 2003; substantive revision Fri Nov 2, 2018. As is, the Provability Argument is invalid and cannot be used in support of Moral Skepticism. Semantics for GL and other modal logics 5. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. For example, what’s the probability that we roll a pair of 6-sided dice and either get at least one 1, or an even sumProvability and proofs as objects appear in many other areas of logic and applications such as modal logics and log- ics of knowledge, ë-calculus and typed theories, nonmonotonic reasoning, automated deduction and formal verification. View PDF. We consider the one-shot Prisoner's Dilemma between algorithms with read-access to one anothers' source codes, and we use the modal logic of provability to build agents that can achieve mutual cooperation in a manner that is robust, in that. In simple words, it ensures the random location for the wheel. c. Probability of a Normal Distribution. noun [ U ] us / ˌpruː. 6, a nocturnal is 0. Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. For events A and B, with P(B) > 0, the conditional probability of A given B, denoted P(A | B), is given by. The only legitimate reason for agreeing not to pursue a charge is the existence of real doubt about the ultimate provability of the charge. ago. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. The chance of a student passing the exam and getting above 90% marks in it is 5%. N. Show abstract. $$ So even though the probability of dying on a specific dive does not change, by repeatedly skydiving you are increasing your overall chances of dying. Travis County law enforcement responded to more than 600 adult. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. The logic has been inspired by. In this paper it is proved that the four infinite series of provability logics known at the present time - , , , and (, , finite) - include all such logics. $15. This is the but-for clause; but for the fact that φ would. All top 10 hits were known RdRp homologs (Supplementary Table 4). The Logic of Provability. Provability logic. Given a theory T (a set of well-formed sentences closed under deduction) in language L, we call an interpretation of L a model of theory T if it assigns True to every sentence in T. Esakia volume of Studia Logica. The Provability Argument. Provability logic. | Meaning, pronunciation, translations and examplesProvability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. Events that are equally likely can be written with a probability of 0. For example, the probability of rolling three 2s in a row is: (0. As a modal logic, provability logic has. The Provably Fair algorithm works like this. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. Logic and Artificial Intelligence. The general notion of a modal propositional provability logic was introduced by S. -Two reasons why the Provability argument appears better than it is 1. Transactions can be automatically recorded on the blockchain, and the immutability and transparency enabled by blockchain technology helps. These two properties are called soundness and completeness. Use the calculator below to find the area P shown in the normal distribution, as well as the confidence intervals for a range of confidence levels. Now let us suppose, after working very hard, we are finally able to establish the truth or falsehood of C. Gödel’s modal logic approach to analyzing provability attracted a great deal of attention and eventually led to two distinct mathematical models. və. Let P be a single sentence. Taishi Kurahashi. The logic has been inspired by developments in meta-mathematics such as Gödel's incompleteness theorems of 1931 and Löb's theorem of 1953. Provability logic is a branch of mathematical logic which investigates the provability-based behavior of the mathematical theories. provability synonyms, provability pronunciation, provability translation, English dictionary definition of provability. Unit 6 Study design. A frequency distribution describes a specific sample or dataset. The point is to capture the notion of a proof predicate of a reasonably. Many translated example sentences containing "provability" – German-English dictionary and search engine for German translations. 3) PA + reflection is deductively equivalent to PA + deviant reflection iff the latter theory proves Con (PA). Converting odds is pretty simple.