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.

Related content:

  1. Inductive arguments establish objective facts, so how can they be considered subjective?
  2. I would take a standard textbook on math, where all the propositions are correct. Write down 99 correct mathematical statements. And then add “Zeus exists”, and compile a text. Then I would argue, that if we have a box, from which we sample randomly 99 balls and they are have the property of being black, we can think with good reason that the next one will be black. And therefore, since 99 of the math propositions in the texts are have the property of being correct, there is good reason to think that “Zeus exists” is also correct. It seems wrong somewhere. But where?
  3. Just recently saw the following argument in a logic book: all lions are herbivores all zebras are lions ————– therefore all zebras are herbivores this seems to be logically valid syllogism, but it is disturbing.
  4. 1) The claim “There is extraterrestrial life in the universe, because my father said so” is an example of an appeal to authority. But it can be viewed as an enthymeme, where the hidden assumption is that “my father is always right”. In such a case, there is no logical problem with the argument. Do you agree? 2) Do you think that to say that : “Person A is biased , therefore\what he says is wrong” is fallacious? It can be interpreted as “person A is biased, therefore his information cannot be trusted. Therefore what he says is wrong”. 3) Are errors of logic errors of psychology as well? Or perhaps, only errors of psychology? Appeal to authority besides being a logical fallacy, has a whole psychology and sociology besides it.
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