Bir Mantık Sistemini Aforoz Etmek: İkinci Derece Mantığın Statüsü Hakkında Yeni Tartışmalar

Abstract

Biçimsel mantık sistemleri bir ifadenin değillenmiş biçiminin nasıl kurulacağını biçimsel yolla gösterir, her ne kadar çeşitli mantık sistemlerinin değilleme anlayışları farklı olsa da. Peki, bir mantık sisteminin kendisi nasıl değillenir? Diğer bir ifadeyle, bir argümanlar topluluğunu geçerli kılan bir kalkülüs, ne zaman, yani hangi koşullarda (sözcüğün belirli bir anlamında) “yanlış” görülüp terk edilir? Mantık tarihi incelendiğinde bunun açık bir ölçütünü bulmak güç bir iştir. Ancak bu çalışmada, böyle bir terkin henüz tamamen gerçekleşmemiş olsa da olası bir örneği incelenmektedir. 20. yüzyılın başında biçimsel mantığın doğal bir sistemi olarak görülen ikinci derece mantık (İDM), görece uzun zamandır mantıktan aforoz edilmektedir. Bu aforoz uluslararası eğitim kurumlarında ve pedagojik yapıtlarda büyük oranda tamamlanmıştır. Buna karşın çeşitli matematik ve felsefe çevreleri ikinci derece mantığın terk edilmemesi gerektiğini savunmaya devam etmektedir. Bu çalışmada işaret edilen tartışmalar teknik boyutlarıyla birlikte incelenmekte ve konuya dair son yıllardaki önemli gelişmeler de okuyucuya sunulmaktadır.

Keywords

• Niceleme
• mantıksalcılık
• adcılık
• çoğul niceleme
• metamantık

Excommunicating a Logical System: New Debates on the Status of Second-Order Logic

Abstract

Formal logical systems demonstrate, by formal means, how to construct the negated form of a statement—even though different logical systems may conceive of negation differently. But how might a logical system itself be negated? In other words, under what conditions (in a particular sense of the word) is a calculus that validates a given body of arguments deemed 'wrong' and subsequently abandoned? When one looks to the history of logic, it becomes apparent that there is no readily available or well-defined criterion by which such abandonment is decided. Nevertheless, this study examines a possible instance of such an abandonment, even if it has not yet been fully realized. Second-order logic (SOL), which was once considered a natural extension of formal logic in the early 20th century, has for some time been excommunicated from the domain of logic. This excommunication has been largely accomplished within international academic institutions and pedagogical texts. Nevertheless, various mathematical and philosophical circles continue to argue that second-order logic should not be forsaken. The present study investigates the technical dimensions of these debates and presents the reader with recent significant developments on the subject.

Keywords

• Quantification
• logicism
• nominalism
• plural quantification
• metalogic

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