Mathematical arguments are based on axioms of classical logic

Question

I'm recently digging deeper into logic and its philosophy. I'm familiar with classical logic and predicate calculus that we can learn from discrete maths textbooks.

What I would like to know is why such type of logic with its truth-functional structure, and especially the material implication, is used in math the way they are used. How can I be sure that I'm indeed reaching a really 'true' conclusion when using rules of inference based on that system (a valid argument is defined to be a tautological material implication between the premises and the conclusion). By true, I do not mean in the sense of the propositional truth in this system, but as in why such system we decided to stick with is superior in its interpretations and truth-seeking potential.

I don't have much rigor in logic, so sorry if I'm asking superficial questions. Was it done backwards; as in first verifying if material implication does a good job in inference then proceeding to use it...?

Answer 1

Logic (or rather a particular logic) is only an abstract model of human thinking, so to claim absolute correctness for any given logical system would be wrong. This is one of the reasons that mathematicians/logicians actually talk of different logics, e.g. the classical first-order logic that you mention, but there are other types: modal logic is a very prominent example. Some logics have outgrown their standard scope of application as the foundation of mathematics and have found diverse applications, even as a way to model and make inferences about human decision-making or moral constructions: check out deontic logic.

Note, too, that not all mathematicians agree with classical first-order logic as the basis of all mathematics. Fierce debates raged in the first half of the 20th century between some of the most renowned mathematicians of the time: perhaps the most lasting and significant of those happened between David Hilbert and L.E.J. Brouwer, where the former defended the classical (the broader philosophical system that underpins this view is called the formalist theory) logic you're familiar with, while the latter espoused a system that is now called intuitionistic logic. Quite a few people are carrying out research in intuitionistic logic to this day, whether they believe it to be "superior" to classical logic or not. We are used to thinking that classical logic is the one and only golden set of rules of inferring truth (mathematical or otherwise), so it may be baffling at first to see some of the classic laws of thought rejected by other logical systems. Consider the law of double negation, i.e. $$P \models \lnot \lnot P$$ in logical notation. The intuitionists actually reject this law, which is why intuitionistic logic is often referred to as weaker than the classical one.

The dispute as to which logic is "superior" is fairly meaningless and is unlikely to ever be resolved. It is, largely, a matter of opinion and applicability. Classical logic, although not necessarily devoid of paradoxes, has done a great job in its modelling of mathematical inference. Whether it is the ultimate instrument for deriving the truth, however, is a question that will forever remain unanswered.

Answer 2

Beginners in logic often confuse "implies" with "causes." Consider for example the implication: "If it is raining, then it is cloudy."

$~~~~~~Raining \implies Cloudy$

This does not mean that rain causes cloudiness. It means only that, at the moment, it is not the case that it is both raining and not cloudy.

$~~~~~~Raining \implies Cloudy~~~\equiv~~~\neg (Raining \land \neg Cloudy)$

From this definition, it should be apparent why the implication $Raining \implies Cloudy$ can be true even if it is not raining.

When material/logical implication is introduced in introductory textbooks or courses, it is often simply defined by a truth table or logical equivalence.

The Truth Table

The Logical Equivalence

$~~~~~~A\implies B~~~\equiv~~~\neg (A\land\neg B)$

How can we be sure these will give us the "correct answers?" It turns out that both can be derived from other more fundamental rules of logic. See my blog posting in this topic: If Pigs Could Fly. (Some knowledge of the basic methods of proof is required.)