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I will not retell the main postulates and definitions for a long time, this can be found on Wikipedia and on the Internet in general, but I will briefly try to formulate what, in my opinion, is the difference between these two things.
Logical positivism is an epistemological methodology, if you will, an ideology of cognition, based on rationalism and logical-linguistic analysis, designed to bring to light and eliminate metaphysical constructions. And, thus, to build a rational picture of the known world, approximately corresponding to the picture obtained by natural sciences. This was such a large-scale program to rewrite the entire philosophy, which turned out to be impossible in its “strong” version, but in general gave rise to the entire analytical philosophy.
Logical atomism is one of the first ontological theories based on the methodology of logical positivism. It is based on the assumption that all “reality” can be decomposed into logical atoms. In the first version of atomism, B. Russell assumed two types of atoms-individuals and properties, then tried to represent individuals as an assembly of atoms of properties, well, there were different options. In general, he proceeded from the complete descriptibility of reality in logical language and actually moved from describing the events of reality to analyzing the events of language. Its atoms strongly resemble elementary events in probability theory, and in general, you can't drink your mathematical training.
That is, to summarize: L. positivism is a methodological program, and L. atomism is an ontological theory (one of) created on the basis of this methodological program.
These are very different phenomena.
The very expression “logical atomism” belongs to Bertrand Russell, but it is also often called metaphysics and philosophy of language of early Wittgenstein, set out in the” Logical-philosophical treatise”, because of their superficial similarity. The central idea of logical atomism is, in general, the idea that the meaning of simple statements is determined by correspondence with simple facts – “logical atoms” – in the world. Each pair of contradictory statements corresponds to a single fact. In a logically perfect language, atomic statements describe combinations of logical atoms, and complex statements are combinations of atomic statements. However, statements of logically imperfect natural languages mask this by their form, so they need to be analyzed. On the surface, the philosophical systems of Russell and Wittgenstein are very similar, but they have fundamental differences – about the nature of logical atoms and propositions, the well-known disagreement about “speaking and showing” , etc. – which take a long time to delve into.
Sometimes logical atomism is mentioned together with / confused with logicism – an unsuccessful attempt (in particular, by Frege and Russell) to prove that mathematics is a branch of logic and to develop a theory that puts mathematics on the basis of logic.
Logical positivism is not a specific philosophical system, but rather a movement created by two groups of scientists-at the University of Vienna (Schlick, Carnap, Godel, etc. – the so-called Vienna Circle) and Berlin (Reichenbach, Hempel, etc.), among others, they were significantly influenced by Wittgenstein's “Treatise”, especially the idea of “clarifying” and rejecting meaningless statements of the role of philosophy and a strict description of any phenomenon in the world using the language of formal logic. In general, the goal of logical positivism was to create a” scientific ” philosophy based on formal logic. The main postulates of the movement include verificationism (the view that any statement about the world has meaning only if it is empirically verifiable or logically necessary), the distinction between analytical (true only in the plane of its linguistic meaning, tautologies) and synthetic (true depending on whether they correspond to experience) statements, the denial of the existence of Kantian a priori synthetic statements, and empirical reductionism (the view that any statement about the world can be reduced to a statement about one's direct experience). Statements that did not conform to verificationism were disparagingly called metaphysics, which is ironic, because some authoritative researchers of Wittgenstein (for example, Anthony Kenny) call the Treatise one of the most metaphysical works in philosophy. When the Nazis came to power, many logical positivists fled to the United States, and critically influenced the development of Anglo-American analytical philosophy, especially the philosophy of science. But this is not to say that their project was crowned with success, and later the central ideas of logical positivism were criticized by Popper( who himself for some time had an attitude to them), Quine, and others.