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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...?

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  • $\begingroup$ What is "truth"? That is a question for the ages. $\endgroup$
    – Somos
    Commented Oct 19, 2020 at 15:49
  • $\begingroup$ IF the axioms are true and the theorems are proved with correct deductive inferences, then YES, they (the theorems) will be true. What "true" means form a mathematical point of view ? It means that in every "domain" (better: interpretation) where the axioms are true, also the theorems will be... provided that the axioms are consistent. $\endgroup$ Commented Oct 19, 2020 at 16:20
  • $\begingroup$ From a philosophical point of view, Truth is a thorny issue. $\endgroup$ Commented Oct 19, 2020 at 16:24

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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.

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    $\begingroup$ Thank you for your answer. If I may ask, why was the truth-functionality aspect desired in maths? Particularly, why not leave the controversial 3rd and 4th column in the material implication's truth table as undefined blanks (when the antecedent is false)? I do understand one idea, and it's that with truth-functionality, we are working only with general truth values with not much interest in relations and semantics. $\endgroup$ Commented Oct 19, 2020 at 17:33
  • $\begingroup$ @MohamadHusseinNaim There are logics that treat the material conditional exactly as you mention, i.e. a material conditional with a false antecedent is said to be undefined. I see this as having very little real applicability though, and we start to run into trouble pretty easily: for example, $A\land B \to A$ is undefined whenever $B$ is false - in a real world setting, however, this makes little sense. Without a truth-functional approach we just wouldn't get very far in maths - I think it's the best explanation as to why classical logic is the way it is. $\endgroup$ Commented Oct 19, 2020 at 18:41
  • $\begingroup$ @MohamadHusseinNaim Here's a little thought experiment if you wish: try to come up with a sketch of an alternative formal system and plug in some simple mathematical statements to see how the material implication behaves. This should hopefully give you some intuition as to why classical logic suits mathematics best :) but as I said, it's not the only useful logic out there: the modal operator in modal logic is non-truth-functional, for example, but this is used to express things like uncertainty or differing temporality, which we have no need of in traditional mathematics. $\endgroup$ Commented Oct 19, 2020 at 18:50
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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

enter image description here

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.)

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