Pretabular logics are those that lack finite characteristic matrices, although all of their normal proper extensions do have some finite characteristic matrix. Although for Anderson and Belnap’s relevance logic R, there exists an uncountable set of pretabular extensions :1249–1270, 2008), for the classical relevance logic \\rightarrow B\}\) there has been known so far a pretabular extension: \. In Section 1 of this paper, we introduce some history of pretabularity and some relevance logics and their algebras. In Section 2, we introduce (...) a new pretabular logic, which we shall name \, and which is a neighbor of \, in that it is an extension of KR. Also in this section, an algebraic semantics, ‘\-algebras’, will be introduced and the characterization of \ to the set of finite \-algebras will be shown. In Section 3, the pretabularity of \ will be proved. (shrink)

It is quite common among historians of Arabic logic to attribute MS Aḥmad iii, no. 3401, entitled Al-manṭiq al-kabīr, to Fakhr al-Dīn Rāzī. This view is expressed explicitly...

The rule of contraposition has been investigated thoroughly by Arabic logicians. In this paper, we study the work done by Fārābā and Avicenna, the fathers of Arabic logic. Fārābā studied contraposition of universal affirmatives, discussed its four forms, and discovered a relation between one form and the conversion of negative universals. Although Fārābā and logicians before him have used contraposition only for conditionals, as well as for indefinite and universal affirmative categorical propositions, Avicenna generalized the rule to all the four (...) Aristotelian quantified categorical propositions. However, many of his ideas on contraposition were opposed by his 12th and 13th century followers, which are to be investigated later. (shrink)

KR is Anderson and Belnap’s relevance logic R with the addition of the axiom of EFQ: \ \rightarrow q\). Since KR is relevantistic as to implication but classical as to negation, it has been dubbed, among many others, a ‘classical relevance logic.’ For KR, there have been known so far just two pretabular normal extensions. For these pretabular logics, no simple axiomatizations have yet been presented. In this paper, we offer some and show that they do the job. We also (...) introduce some Routley–Meyer semantic conditions for these axioms. All these may facilitate realizing our desire to discover other classical pretabular extensions over KR, if such extensions exist. (shrink)

We exhibit infinitely many semisimple varieties of semilinear De Morgan monoids (and likewise relevant algebras) that are not tabular, but which have only tabular proper subvarieties. Thus, the extension of relevance logic by the axiom $$(p\rightarrow q)\vee (q\rightarrow p)$$ ( p → q ) ∨ ( q → p ) has infinitely many pretabular axiomatic extensions, regardless of the presence or absence of Ackermann constants.

The logical rule of contraposition as applied to a particular affirmative proposition (I-contraposition), despite its rejection in the medieval Latin logic, had a different history in the medieval Arabic logic, varying from common acknowledgement to total dismissal (it was accepted by Avicenna and by all of his followers in the eleventh and twelfth centuries and rejected by all of Arabic logicians in the late thirteenth century onwards). This paper is a narrative of the fate of I-contraposition in the early Arabic (...) logic. I will study Avicenna’s, al-Suhrawardī’s, al-Rāzī’s, al-Kashshī’s, and al-Khūnajī’s views on this rule. Although Avicenna explicitly acknowledged I-contraposition in a brief note, and in an apparently Meinongian language, the other four figures found I-contraposition problematic and listed some counterexamples to it. Nonetheless, they all attempted to justify Avicenna’s acceptance of the rule by restricting its application conditions. These efforts led to interesting syntactic and semantic analyses but gathered little attention from later medieval logicians. Finally, in the middle Arabic logic, the revised versions of I-contraposition were also declared invalid. (shrink)

Recently, in ‘An inquiry on Fakhr al-Dīn Rāzī’s authorship of Al-manṭiq al-kabι¯r (MS Aḥmad iii, no. 3401)’, we refuted the quite common view among historians of Arabic logic as to attribute Al-manṭiq al-kabι¯r (Major [book on] logic) to Fakhr al-dīn Rāzī; however, in that paper, we could not identify the real author of the book. In this paper, we try to show that Al-manṭiq al-kabι¯r was written by Burhān al-dīn Nasafī (1203–1288), who authored another book, Sharḥ Asās al-kiyāsa whose logical (...) part is very similar to Al-manṭiq al-kabι¯r in the following significant features: the titles of chapters and sections, the style of writing, some technical as well as some non-technical terms, many sentences word by word. None of these features we find in any other Arabic logical work. So, Al-manṭiq al-kabι¯r and Sharḥ Asās al-kiyāsa are so close to each other that we might most possibly identify the authors of the books as the same scholar, i.e. Burhān al-dīn Nasafī. Fortunately, this has been verified recently by two manuscripts of a medieval Arabic logical book. (shrink)

In his second period of logical research, which includes several books and treatises, Athir al-Din al-Abhari is the only Avicennan logician who vindicated as invalid one of the two most important Avicennan novelties: iqtirani conditional syllogism. His exposition and reasons for this invalidity in these books and treatises are various and rooted in numerous kinds of conditionals and their truth-conditions and their developments in the books and treatises. His most important reason to deny conditional syllogism is notice to the manifold (...) assumptions used to prove the syllogisms. In this paper, we deal with the main reason, studying the mentioned differences and developments. Citing successor logicians’ points of views on Abhari’s claim and assessment of all of them is out of the purposes of this paper and in need to other researches. (shrink)

Contingent conditionals, at the comparative logical works, has yielded different, and often inconsistent, analyses. At contemporaries’ works, there are two approaches to the contingent conditionals: modal and truth-table methods. At the former, there has been used modal connectives of necessity and possibility to analyze the contingent conditionals; but at the latter, some truth-tables has been proposed. At the paper, besides presenting and criticizing the extant theories, we gain new analyses and formulations of the disputed subject matters.

Pretabularity is the attribute of logics that are not characterised by finite matrices, but all of whose proper extensions are. Two of the first-known pretabular logics were Dummett’s famous super-...

Modal Logic’s theorems and rules are valid in possible worlds but Relevant Logic’s theorems and rules are valid, respectively, in logical worlds and situations. Robert Meyer in 1974 removed this asymmetry between the theorems and the rules of Relevant Logic by establishing a logical system, whose theorems and rules are valid in all situations. Introducing a new kind of truth and falsity operators, the authors in this article sketch a logical system defined on the basis of Relevant Logic. Such a (...) system can not only preserve symmetry, but also remove some inconsistency between Modern Classical Logic and Relevant Logic, because the latter, like the former, puts possible worlds as a criterion for validity. (shrink)

The intra-school controversies in Hanafī school are remarkable. These controversies pose a risk for legal safety and stability and creates difficulties for muqallīd Hanafī judges and muftīs. In the historical process, different types of literature such as mukhtaṣar and fatwa (legal opinion) books, and applications such as aṣṣaḥ-ı aqvāl and maʻrūdhāt, emerged to solve this problem. One of the last example of these applications is the codification movement. The subject of this study is the relationship between the Majalla al-Aḥkām al-ʿAdliyyah, (...) which is the first product of codification based on fiqh (Islamic jurisprudence), and the intra-school preference. In order to describe this relationship, it is briefly emphasized that one of the aims of codification of Majalla is to satisfy the need for intra-school preference. The function and content of Majallaʼs necessary reasons, (asbāb al-mūjiba) was analyzed in terms of intra-school preference. The interventions on the preferences of the Majalla Commission are also emphasized. The Majalla is a text of the intra-school preference. The analysis of its content in relation to intra-school preference is very important. Therefore, in order to reveal what kind of method is used in intra-school preferences and to determine different aspects of intra-school preference in the Majalla, the chapter about ‘sale’ (Kitāb al-Buyūʿ) of the Majalla are examined.Summary: The intra-school controversies in Hanafī School, which is quite important in terms of freedom of ijtihād (independent legal judgment) and legal wealth, possible to threaten legal safety and stability. These controversies also make difficult for judges to reach a judgment or fatwa in controversial affairs. For this reason, determination of the rājiḥ (preferred opinion) in controversial issues in the school, is a need that must be eliminated.The intra-school preference can be described as "to determine the superior, preponderant of different opinion or narrations about a specific issue in the school". In the historical process different types of literature such as mukhtaṣar and fatwa books, and different applications such as asaḥḥ-ı aqvāl and maʻrūzāt, emerged to meet this need. One of the applications aimed to meet this need is the codifications in the last period of the Ottoman State.The relationship between the Majalla al-Ahkam al-Adliyyah, which is the first example of codification based on fiqh (Islamic jurisprudence), and the intra-school preference; and in this context, while preparing the Majalla how a method was followed in the intra-school preference; at what rate be obeyed the rājiḥ (preferred) opinion; articles contrary to the rājiḥ (preferred) opinion in Majalla and their reasons; whether legal arrangements have been made in all controversial issues; at what rate be met the need of determination of rājiḥ (preferred) opinion are the issues that need to be examined.One of the main reasons for the need for a law text such as Majalla al-Ahkam al-Adliyyah is the multitude of the intra-school controversies in Hanafī School. Therefore, the goal of Majalla is to prepare a code that contains rājiḥ opinion.When Majallaʼs necessary reasons, i.e. asbāb al-mūjiba, are examined in terms of the intra-school preference, it is seen that there are two basic functions of necessary reasons. The first of them is, emphasizing that the code is prepared substantially in accordance with the râjih opinion in the school. In fact, that this issue in the asbāb al-mūjiba of the first ten books of Majalla is expressly stated. The other function of the asbāb al-mūjiba is to justify that why rājiḥ opinion are not be confirmed in some article. In the necessary reasons it is stated that twelve articles are prepared in the direction of unrâcih (unpreferable) opinion. Considering that Majalla is formed of 1851 article, it will be better understood that how the rājiḥ opinions are effective in Majalla's content. Essentially, the conditions of the period also make it necessary. The preference of the non-preferred opinion of Züfer (d. 158/775) in 692nd article of Kitāb al-Ḥavala, caused the reaction of some circles mainly Şeyhülislam Hasan Fehmi Efendi (d.1298/1881) and for this reason Ahmet Cevdet Pasa (d.1895) was dismissed from the ministry of Divan al-Ahkâm al-Adliyyah and the chairman of the Majalla Commission.When the asbāb al-mūjibas of contrary articles to rājiḥ opinions are examined, it is seen that maṣlaḥa and customs are determinative in these preferences. It is also seen that the preferences of some jurists are also mentioned as a reason of preference. The most widespread preference’s reason in Hanafī sources is in congruity with the texts. On the contrary, it is noteworthy that, besides the interdiction of extravagant (safih), it is not operated as a preference’s reason in Majalla.Some preferences of the Majalla Commission were intervened by other institutions. The chapters prepared by the commission were examined both by Majlis-i Wukalā and Mashihat, and these chapters were subject to some revisions at this time. Here, two articles prepared on the basis of the non-preferred opinion were intervened by Majlis-i Wukalā. The first of these is 216th article that written in the direction of opinion of Mashāyikh Balkh, which argues that it is permissible to sell separately ḥaqq al-murūr (the right of passage) and ḥaqq al-shirb (water right), the second is the 611th article which regulates the liability to damage of ajīr al-mushtarak (joint salaried employee).When the content of Kitāb al-Buyū, the first and the most voluminous book of Majalla, is examined in terms of the intra-school preference, it is seen that the rājiḥ opinions in the school are generally taken as basis. It is stated that in the asbāb al-mūjiba of the Kitāb al-Buyūʿ the four issues are prepared basing on the non-preferred opinion and these preferences are justified with custom and maṣlaḥa. However, the contrary articles to the rājiḥ opinion in the Kitāb al-Buyūʿ are not limited to these. According to our research, in addition to these, three articles were prepared with the preference of non-preferred opinion. These are the 118th article regarding that the bayʻ bi-l-wafā is permissible in chattels, 165th article on the definition of al-ghabn al-faḥiş (lesion), and the 309th article concerning the provision of the khiyār al-sharṭ (right of cancellation). These three articles show that the regulations contrary to the rājiḥ opinion in Majalla are not just those explained in the asbāb al-mūjiba and this situation is contrary to the method that is followed when preparing the Majalla.In some controversial issues in the Hanafi School, authorized jurists to preference (asḥāb al-tarjīh) disagreed over which opinion is rājiḥ. It wasn’t explained that how a method would be followed in Majalla when the legal arrangements were made in these matters. When these issues are examined, it can be said that the preferences of the majority, the providing of convenience and the preference by some jurists are effective.Preferences in Majalla have not been occurred only among the sharʻī (related to Islamic Law) provisions. In addition, the definitions and the inscription in the definitions are also the subject of preference. In this context, the preferences related to the definition of the sales contract, the suspended contract (al-ʻaqd al-maqqūf) and the al-ghabn al-faḥiş in Kitāb al-Buyūʿ can be mentioned as an example.Some controversial issues in the Hanafi School are not regulated in Majalla. When considering the multiplicity of controversial issues and therefore the difficulty of making legal arrangements to cover all of them, this situation is natural. Moreover, some of these issues may not have been regulated in Majalla because they are not frequent and necessary. However, it is not possible to explain all the controversial issues that have not been regulated for these reasons. Because some of the controversial issues that have not been legislated are basic, relatively important issues and they are included in the mukhtaṣars. This indicates that when preparing the Majalla, instead of making preference on some controversial issues, the method of non-regulation of these matters is adopted. It can be said that this method is applied as a way of creating law text which is clear of controversy. Thus, the determining of the rājiḥ opinion in these matters has been left to the judges. This situation is not suitable to meet the need of determining the rājiḥ opinion, which is the goal of Majalla. But it is the result of the difficulty of making a law that encompasses all legal issues. (shrink)