[Bertrand Russell wrote that, ‘As logic improves, less and less can be proved.’ In the following passage from his book Human Knowledge: Its Scope and Limits, 1948, it is also clear that strict logic is much more complicated and cumbersome than the instinctive non-demonstrative inferences that all of us employ every day. Fortunately, that looser reasoning usually suffices.]
The purely logical analysis of “dogs bark” soon reaches complexities which make it incredible that ordinary folk can seem to understand anything so remote, mysterious, and universal. The first stage, for the logician, is to substitute: “Whatever x may be, either x is not a dog or x barks.” But since dogs only bark sometimes, you have to substitute for “x barks” the statement “there is a time t at which x barks.” Then you must substitute one or other of the two alternative definitions of “t” given in Part IV. In the end you will arrive at a statement of enormous length, not only about dogs, but about everything in the universe, and so complicated that it cannot be understood except by a person with a considerable training in mathematical logic. But suppose you have to explain your statement “dogs bark” to such a person, but as he is a foreigner with only a mathematician’s knowledge of English, he does not know the word “dog” or the word “bark.” What will you do? You will certainly not go through the above logical rigmarole. You will point to your dog and say “dog”; you will then excite him till he barks and say “bark.” The foreigner will then understand you, although, as a logician, he has no business to do so. This makes it clear that the psychology of general propositions is something very different from their logic. The psychology is what does take place when we believe them; the logic is perhaps what ought to take place if we were logical saints.
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