Intuitionistic modal logics incompatible with the law of the excluded middle

Studia Logica ◽  
1981 ◽  
Vol 40 (2) ◽  
pp. 103-111 ◽  
Author(s):  
Dimiter Vakarelov
Keyword(s):  
Modal Logics ◽  
The Law ◽  
Excluded Middle ◽  
Intuitionistic Modal Logics

Reconsidering Kant’s Rejection of Indirect Arguments in Transcendental Philosophy

2021 ◽  
pp. 1-19
Author(s):  
Marcel Buß
Keyword(s):  
Immanuel Kant ◽  
Explanatory Power ◽  
Transcendental Philosophy ◽  
Point Of View ◽  
Transcendental Arguments ◽  
Mathematical Proofs ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Excluded Middle ◽  
Insight Into

Abstract Immanuel Kant states that indirect arguments are not suitable for the purposes of transcendental philosophy. If he is correct, this affects contemporary versions of transcendental arguments which are often used as an indirect refutation of scepticism. I discuss two reasons for Kant’s rejection of indirect arguments. Firstly, Kant argues that we are prone to misapply the law of excluded middle in philosophical contexts. Secondly, Kant points out that indirect arguments lack some explanatory power. They can show that something is true but they do not provide insight into why something is true. Using mathematical proofs as examples, I show that this is because indirect arguments are non-constructive. From a Kantian point of view, transcendental arguments need to be constructive in some way. In the last part of the paper, I briefly examine a comment made by P. F. Strawson. In my view, this comment also points toward a connection between transcendental and constructive reasoning.

7. Deducing

2020 ◽  
pp. 74-88
Author(s):  
Timothy Williamson
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Logical Inference ◽  
Disjunctive Syllogism ◽  
Philosophical Concept ◽  
The Law ◽  
The Difference ◽  
Excluded Middle

Detective work is an important tool in philosophy. ‘Deducing’ explains the difference between valid and sound arguments. An argument is valid if its premises are true but is only sound if the conclusion is true. The Greek philosophers identified disjunctive syllogism—the idea that if something is not one thing, it must be another. This relates to another philosophical concept, the ‘law of the excluded middle’. An abduction is a form of logical inference which attempts to find the most likely explanation. Modal logic, an extension of classical logic, is a popular branch of logic for philosophical arguments.

Constructivism in mathematics

2018 ◽  
Author(s):  
David Charles McCarty
Keyword(s):  
Twentieth Century ◽  
Real Analysis ◽  
Constructive Logic ◽  
Mathematical Objects ◽  
Mathematical Ontology ◽  
The Law ◽  
And Mathematics ◽  
Excluded Middle

Constructivism is not a matter of principles: there are no specifically constructive mathematical axioms which all constructivists accept. Even so, it is traditional to view constructivists as insisting, in one way or another, that proofs of crucial existential theorems in mathematics respect constructive existence: that a crucial existential claim which is constructively admissible must afford means for constructing an instance of it which is also admissible. Allegiance to this idea often demands changes in conventional views about mathematical objects, operations and logic, and, hence, demands reworkings of ordinary mathematics along nonclassical lines. Constructive existence may be so interpreted as to require the abrogation of the law of the excluded middle and the adoption of nonstandard laws of constructive logic and mathematics in its place. There has been great variation in the forms of constructivism, each form distinguished in its interpretation of constructive existence, in its approaches to mathematical ontology and constructive logic, and in the methods chosen to prove theorems, particularly theorems of real analysis. In the twentieth century, Russian constructivism, new constructivism, Brouwerian intuitionism, finitism and predicativism have been the most influential forms of constructivism.

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