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In [Aristotle’s] formal logic, thought is organized in a manner very different from that of the Platonic dialogue. In this formal logic, thought is indifferent toward its objects. Whether they are mental or physical, whether they pertain to society or to nature, they become subject to the same general laws of organization, calculation, and conclusion — but they do so as fungible signs or symbols, in abstraction from their particular “substance.” This general quality (quantitative quality) is the precondition of law and order — in logic as well as in society — the price of universal control.
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p. 136

 

Herbert Marcuse

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By virtue of the universal concept, thought attains mastery over the particular cases. However, the most formalized universe of logic still refers to the most general structure of the given, experienced world; the pure form is still that of the content which it formalizes. The idea of formal logic itself is a historical event in the development of the mental and physical instruments for universal control and calculability. In this undertaking man had to create theoretical harmony out of actual discord, to purge thought from contradictions, to hypostatize identifiable and fungible units in the complex process of society and nature. Under the rule of formal logic, the notion of the conflict between essence and appearance is expendable if not meaningless; the material content is neutralized; the principle of identity is separated from the principle of contradiction (contradictions are the fault of incorrect thinking); final causes are removed from the logical order. Well defined in their scope and function, concepts become instruments of prediction and control. Formal logic is thus the first step on the long road to scientific thought

 

Herbert Marcuse

 

Whenever logical processes of thought are employed— that is, whenever thought for a time runs along an accepted groove— there is an opportunity for the machine. Formal logic used to be a keen instrument in the hands of the teacher in his trying of students' souls. It is readily possible to construct a machine which will manipulate premises in accordance with formal logic, simply by the clever use of relay circuits. Put a set of premises into such a device and turn the crank, and it will readily pass out conclusion after conclusion, all in accordance with logical law, and with no more slips than would be expected of a keyboard adding machine.

 

Vannevar Bush

 

It has been a long road from Plato's Meno to the present, but it is perhaps encouraging that most of the progress along that road has been made since the turn of the twentieth century, and a large fraction of it since the midpoint of the century. Thought was still wholly intangible and ineffable until modern formal logic interpreted it as the manipulation of formal tokens. And it seemed still to inhabit mainly the heaven of Platonic ideals, or the equally obscure spaces of the human mind, until computers taught us how symbols could be processed by machines.

 

Allen Newell

 

The methods of logical procedure are very different in ancient and modem logic, but behind all difference is the construction of a universally valid order of thought, neutral with respect to material content. Long before technological man and technological nature emerged as the objects of rational control and calculation, the mind was made susceptible to abstract generalization. Terms which could be organized into a coherent logical system, free from contradiction or with manageable contradiction, were separated from those which could not. Distinction was made between the universal, calculable, “objective” and the particular, incalculable, subjective dimension of thought.

 

Herbert Marcuse

 

So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premisses which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary.

 

Bertrand Russell