On one hand, arguments are supposed to be objective – something which is true is always true, for everyone. On the other hand, if person says “P exists because X,Y,Z”, while he personally has seen the evidence (x,y,z) for P, and another person says “P exists because X,Y,Z” and he has only read about X,Y,Z from second sources – their knowledge is actually very different. Where is that difference (crucial one) reflected in logic?

I think your question is rooted in the fact that we generally consider two distinguishably different subjects under the umbrella of logic. The first is the art of argument as it takes place in natural language. The second is an deliberately constructed system, first proposed by Aristotle, and later greatly extended and expanded by figures such as Boole and Frege, which is intended to capture some of the essential qualities of natural argument, but to do so in a way that has mathematical rigor, precision and surety.

In ordinary language some of the arguments we use are based on matters of fact, the perceived truth of which cannot help but be subjective. Other arguments are largely definitional and thus deductive –they express relations of ideas that follow directly and inevitably from agreed-upon definitions.

In formal logic, the only arguments allowed are of the second type, and they exist within a framework carefully constructed so as to eliminate all remaining ambiguities of context, meaning and truth value. Thus, they can truthfully be described as objective and sure to the same extent and with the same assurance as mathematical truths such as “2+2=4” –at least (and this is an important caveat!) within the agreed-upon framework.

Your example of [(x AND y AND z ) IMPLY p ], however, shows the effects of mixing the two types of argument in ordinary language. Let us assume that there is a valid deductive relationship between x,y and z and p. In that case, given x,y, and z, it would be objectively the case that p would also have to be true. But all arguments are essentially hypothetical statements. The conclusion is only as good as the premises. If we have reason to doubt x,y and z, then we have equal reason to doubt p (assume we have no independent reason to believe p).

In general, deductive arguments are “objective”, inductive arguments are “subjective.” But one should be aware of an additional opportunity for confusion stemming from the concept of “mathematical induction,” which, despite its name, is an objective and sure process for establishing mathematic truths. This concept encompasses all arguments of the form “f(0), f(n) IMPLIES f(n+1), therefore all f”. It is so-named because of superficial similarities to ordinary language induction, but it gains a rigor on its way into mathematics that moves it into the realm of surety.

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