Socratic Proofs

Journal of Philosophical Logic - Our aim is to express in exact terms the old idea of solving problems by pure questioning. We consider the problem of derivability: “Is A derivable from...     

The Method of Socratic Proofs for Modal Propositional Logics: K5, S4.2, S4.3, S4F,S4R,S4M and G

The aim of this paper is to present the method of Socratic proofs for seven modal propositional logics: K5, S4.2, S4.3, S4M, S4F, S4R and G. This work is an extension of [10] where the method was presented for the most common modal propositional logics: K, D, T, KB, K4, S4 and S5. Presented by Jacek Malinowski     

A Loop-Free Decision Procedure for Modal Propositional Logics K4, S4 and S5

The aim of this paper is to present a loop-free decision procedure for modal propositional logics K4, S4 and S5. We prove that the procedure terminates and that it is sound and complete. The procedure is based on the method of Socratic proofs for modal logics, which is grounded in the logic of questions IEL.     

Quine and Slater on Paraconsistency and Deviance

In a famous and controversial paper, B. H. Slater has argued against the possibility of paraconsistent logics. Our reply is centred on the distinction between two aspects of the meaning of a logical constant ^*c^* in a given logic: its operational meaning, given by the operational rules for ^*c^* in a cut-free sequent calculus for the logic at issue, and its global meaning, specified by the sequents containing ^*c^* which can be proved in the same calculus. Subsequently, we use the same strategy to counter Quine's meaning variance argument against deviant logics. In a nutshell, we claim that genuine rivalry between (similar) logics ^*L^* and ^*L^* is possible whenever each constant in ^*L^* has the same operational meaning as its counterpart in ^*L^* although differences in global meaning arise in at least one case.     

An Operational Logic of Proofs with Positive and Negative Information

The logic of proofs was introduced by Artemov in order to analize the formalization of the concept of proof rather than the concept of provability. In this context, some operations on proofs play a very important role. In this paper, we investigate some very natural operations, paying attention not only to positive information, but also to negative information (i.e. information saying that something cannot be a proof). We give a formalization for a fragment of such a logic of proofs, and we prove that our fragment is complete and decidable.     

Finding Missing Proofs with Automated Reasoning

This article features long-sought proofs with intriguing properties (such as the absence of double negation and the avoidance of lemmas that appeared to be indispensable), and it features the automated methods for finding them. The theorems of concern are taken from various areas of logic that include two-valued sentential (or propositional) calculus and infinite-valued sentential calculus. Many of the proofs (in effect) answer questions that had remained open for decades, questions focusing on axiomatic proofs. The approaches we take are of added interest in that all rely heavily on the use of a single program that offers logical reasoning, William McCune's automated reasoning program OTTER. The nature of the successes and approaches suggests that this program offers researchers a valuable automated assistant. This article has three main components. First, in view of the interdisciplinary nature of the audience, we discuss the means for using the program in question (OTTER), which flags, parameters, and lists have which effects, and how the proofs it finds are easily read. Second, because of the variety of proofs that we have found and their significance, we discuss them in a manner that permits comparison with the literature. Among those proofs, we offer a proof shorter than that given by Meredith and Prior in their treatment of ukasiewicz's shortest single axiom for the implicational fragment of two-valued sentential calculus, and we offer a proof for the ukasiewicz 23-letter single axiom for the full calculus. Third, with the intent of producing a fruitful dialogue, we pose questions concerning the properties of proofs and, even more pressing, invite questions similar to those this article answers.     

Critical thinking: A Socratic model

A concept of critical thinking is developed based on the Socratic method and called accordingly a Socratic model. First the features of critical thinking stressed in this model are stated and illustrated. The Socratic method is presented and interpreted, then taken to yield a model of critical thinking. The process of internalization by which the Socratic model helps us to become critical thinkers is described. Argument analysis is considered as a widely used instructional strategy adaptable for teaching critical thinking on the Socratic model. This Socratic model is advanced as one helpful way of organizing our ideas about critical thinking, helpful in unifying disparate factors and anchoring them in the humanist tradition.     

Quantified Propositional Logic and the Number of Lines of Tree-Like Proofs

There is an exponential speed-up in the number of lines of the quantified propositional sequent calculus over Substitution Frege Systems, if one considers proofs as trees. Whether this is true also for the number of symbols, is still an open problem. This revised version was published online in June 2006 with corrections to the Cover Date.     

Descriptive Study of the Socratic Method: Evidence for Verbal Shaping

In this study we analyzed 65 fragments of session recordings in which a cognitive behavioral therapist employed the Socratic method with her patients. Specialized coding instruments were used to categorize the verbal behavior of the psychologist and the patients. First the fragments were classified as more or less successful depending on the overall degree of concordance between the patient’s verbal behavior and the therapeutic objectives. Then the fragments were submitted to sequential analysis so as to discover regularities linking the patient’s verbal behavior and the therapist’s responses to it. Important differences between the more and the less successful fragments involved the therapist's approval or disapproval of verbalizations that approximated therapeutic goals. These approvals and disapprovals were associated with increases and decreases, respectively, in the patient’s behavior. These results are consistent with the existence, in this particular case, of a process of shaping through which the therapist modifies the patient’s verbal behavior in the overall direction of his or her chosen therapeutic objectives.     

On Irony Interpretation: Socratic Method in Plato's Euthyphro

Socratic Method in the Euthyphro can be fruitfully analysed as a method of irony interpretation. Socrates' method – the irony of irony interpretation – is to pretend that Euthyphro is an ironist in order to transform him into a self-ironist. To be a self-ironist is to ironize one's knowledge of virtue in order to bring an intuitive and unarticulated awareness of virtue to mind. The exercise of the capacity for self-irony is then a mode of striving for the good.     

哲人的面具——从《游叙弗伦》看苏格拉底的反讽

《游叙弗伦》讨论的是虔敬,然而,要真正理解虔敬的意义以及虔敬在两位对话人身上的表现,苏格拉底的反讽是关键。何以认定对话中暗含苏格拉底的反讽?除了其中众多不可小视的细节,还可以从二人定义虔敬的整个过程中发现某些意味深长的标记。由于反讽的运用,虔敬问题变得更复杂了;反过来,借着对虔敬的讨论,苏格拉底的反讽也展示了它多层面的意义。     

The Class of Extensions of Nelson's Paraconsistent Logic

The article is devoted to the systematic study of the lattice εN4_⊥ consisting of logics extending N4_⊥. The logic N4_⊥ is obtained from paraconsistent Nelson logic N4 by adding the new constant ⊥ and axioms ⊥ → p, p → ∼ ⊥. We study interrelations between εN4_⊥ and the lattice of superintuitionistic logics. Distinguish in εN4_⊥ basic subclasses of explosive logics, normal logics, logics of general form and study how they are relate. The author acknowledges support by the Alexander von Humboldt-Stiftung and by Counsil for Grants under RF President, project NSh - 2112.2003.1.     

惯用语理解的多种心理机制：不同的证据

惯用语的理解机制一直是心理语言学研究的热点问题。来自语言学和神经生理学的证据都表明,对惯用语的加工存在多样化趋势,而且加工策略和手段会随个体卷入社会生活的程度而变化。惯用语的加工受加工者自身的隐喻知识以及惯用语本身性质的影响。另外,惯用语加工激活的脑区表明句法和语义分析在惯用语理解中均发挥重要作用,惯用语并没有词汇化,但是不能使用统一的加工模型来整合惯用语的理解机制。惯用语自身性质的多样化导致惯用语理解的多种心理机制。     

A Note on Priest's Finite Inconsistent Arithmetics

We give a complete characterization of Priest's Finite Inconsistent Arithmetics observing that his original putative characterization included arithmetics which cannot in fact be realized.     

Recall and recognition hypermnesia for Socratic stimuli

In two experiments, we investigate hypermnesia, net memory improvements with repeated testing of the same material after a single study trial. In the first experiment, we found hypermnesia across three trials for the recall of word solutions to Socratic stimuli (dictionary-like definitions of concepts) replicating Erdelyi, Buschke, and Finkelstein and, for the first time using these materials, for their recognition. In the second experiment, we had two “yes/no” recognition groups, a Socratic stimuli group presented with concrete and abstract verbal materials and a word-only control group. Using signal detection measures, we found hypermnesia for concrete Socratic stimuli—and stable performance for abstract stimuli across three recognition tests. The control group showed memory decrements across tests. We interpret these findings with the alternative retrieval pathways (ARP) hypothesis, contrasting it with alternative theories of hypermnesia, such as depth of processing, generation and retrieve–recognise. We conclude that recognition hypermnesia for concrete Socratic stimuli is a reliable phenomenon, which we found in two experiments involving both forced-choice and yes/no recognition procedures.     

Dialetheism and the Impossibility of the World

This paper first offers a standard modal extension of dialetheic logics that respect the normal semantics for negation and conjunction, in an attempt to adequately model absolutism, the thesis that there are true contradictions at metaphysically possible worlds. It is shown, however, that the modal extension has unsavoury consequences for both absolutism and dialetheism. While the logic commits the absolutist to dialetheism, it commits the dialetheist to the impossibility of the actual world. A new modal logic AV is then proposed which avoids these unsavoury consequences by invalidating the interdefinability rules for the modal operators with the use of two valuation relations. However, while using AV carries no significant cost for the absolutist, the same isn't true for the dialetheist. Although using AV allows her to avoid the consequence that the actual world is an impossible world, it does so only on the condition that the dialetheist admits that she cannot give a dialetheic solution to all self-referential semantic paradoxes. Thus, unless there are any further available modal logics that don't commit her to the impossibility of the actual world, the dialetheist is faced with a dilemma. Either admit that the actual world is an impossible world, or admit that her research programme cannot give a comprehensive solution to the self-referential paradoxes.     

Ontological Proofs of Existence and Non-Existence

Caramuels’ proof of non-existence of God is compared with Gödel’s proof of existence.     

Chunk and Permeate,a Paraconsistent Inference Strategy. Part I: The Infinitesimal Calculus

In this paper we introduce a paraconsistent reasoning strategy, Chunk and Permeate. In this, information is broken up into chunks, and a limited amount of information is allowed to flow between chunks. We start by giving an abstract characterisation of the strategy. It is then applied to model the reasoning employed in the original infinitesimal calculus. The paper next establishes some results concerning the legitimacy of reasoning of this kind – specifically concerning the preservation of the consistency of each chunk – and concludes with some other possible applications and technical questions.     

Justification logics and hybrid logics

Hybrid logics internalize their own semantics. Members of the newer family of justification logics internalize their own proof methodology. It is an appealing goal to combine these two ideas into a single system, and in this paper we make a start. We present a hybrid/justification version of the modal logic T. We give a semantics, a proof theory, and prove a completeness theorem. In addition, we prove a Realization Theorem, something that plays a central role for justification logics generally. Since justification logics are newer and less well known than hybrid logics, we sketch their background, and give pointers to their range of applicability. We conclude with suggestions for future research. Indeed, the main goal of this paper is to encourage others to continue the investigation begun here.