Why do people use the material implicaton over alternatives?

Question

It isn't because it works. There are numerous alternative, fully-functioning implication systems.

If people have a problem with conditionals with false antecedents, and with true consequents, always being true, why not just change it according to what is found to be more intuitive?

Regarding equivalence with some disjunctions, every explanation I hear seems to make sense prima facie, but I think ultimately they make sense merely because it makes the whole system simple and easy to work with. For instance, (p->q)->(~pVq) is not as intuitively necessarily true as maybe (pVq)->(~p->q). Changing how implications behave simply reflects that some of these disjunctions would cease to be equivalent.

So again, why do we stick with the material implication? Is it because in it, they're equivalent to some disjunctions and are therefore translatable by something like DeMorgan's Laws and provide for tidy simplifications?

If you think this question is a duplicate, your intent with marking it as such is either malicious or you don't understand it.

Answer 1

why do we stick with the material implication?

First of all, we have to consider that the propositional connectives are a (very simple) mathematical model of natural language, suited for modelling formal arguments.

In the context of classical logic, their definition is through truth-table; having defined them, we may check how they are "proxing" natural language mechanism.

Someone do their job in a good way (negation and conjunction), someone with some arbitrariness (disjunction, inclusive : vel instead of aut); someone with a big approximation : the conditional.

We may find a useful the discussion in Stephen Cole Kleene, Mathematical Logic (1967), pag.9 and pag.58-on.

As Kleene says, a lot of controversies aroused around the truth-functional definition of the conditional connective.

The very relevant point to notice is that the mathematical model of "if A, then B" represented by truth-tables does not require any sort of "causal link" between them.

Having said that, we must take into account the key role played by the truth-functional connective "if ... then" and the inference rule of modus ponens that allows us to infer from the premises A and "if A, then B", the conclusion B.

We must read it as Gottlob Frege did in his Begriffsschrift (1879):

assuming as true both the premises, the assumption that "if A, then B" is True, rule-out the row T-F in the truth-table for implies, while the assumption that also A is True rule out two other rows (F-F and F-T, respectively). Then, the conclusion that B is True is licensed.

So, assuming the truth-functional definition of "if A, then B", we have that the truth of A is a sufficient condition for that of B.

This "mechanism" is what is used again and again in mathematical proofs: either having assumed some axiom A as true or having available some already proven theorem A, we may prove a new theorem B through a deductive argument showing "if A, then B".

And that is the reason why we:

use the material implicaton over alternatives.

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On "alternative" (i.e. non truth-functional) conditionals, see e.g. Indicative Conditionals, Counterfactuals, Strict conditional.

Answer 2

Short answer: (1) amongst our 16 possible binary operators, we have an operator at hand that never gives as output "Truth" when the input is ( Truth, Falsity) (2) we use it because this operator is the best tool we have to model the notion of logical implication (3) this is the reason we also call it implication with the lower status of " material implication" .

Be carefull not to admit this false assumption " being given that we have at hand an " if ...then" operator, how shall we define it". We have no " if ...then" logical operator at hand. What we first have at hand is a truth function that never gives " Truth" when the output is ( Truth, Falsity). And, after that, we decide to call it " if ...then" or ( material) implication because it helps us to define logical implication ( which is not a truth functional operator).

The question should not be " why do we define the expression " if ..then" by this truth function?" but rather, "how comes that this truth function has received the name " if...then" or material implication"?

NOTE : on the distinction between material and logical implication, see Seymour Lipschutz, Schaum's Outline Of Set Theory ( at archive.org).

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Suppose your goal is to define logical implication, this strong " IF.. THEN" meaning " it is impossible X to be true without Y to be true".

This strong " IF ..THEN " is the relation that holds between premises and conclusion in a valid reasoning.

Which logical operator will you use to define this " necessarily, if X then Y"?

You have 16 possible binary operators at hand.

You surely will choose this truth function from {T, F}² to {T,F} :

(T,T) --> T

(T,F) --> F

(F,T) --> T

(F,F) --> T

Why? Because this operator never leads you from true to false: when the truth value of the first proposition is "truth" and the truth value of the second is " falsity", this operator always yields " falsity" as result.

In virtue of this property, this operator will suit to represent what you mean by this expression " if.. then" in the logical implication sense.

Of course , it will not deserve the name of " strong if ...then" ( logical implication) but, nevertheless, since you will use it to define logical implication, you will call it implication , and more precisely, material implication; and you will attribute to it the symbol " --> " ( little arrow).

Now, with your " little if ...then" at hand, you will define your " strong IF ...THEN" and say :

X ==> Y ( read : X logically implies Y)

if and only if

the material implication (X --> Y) is necessarily true , that is, is true in all possible cases.

Indeed, why does A logically imply ( A OR B)?

Because the mlaterial conditional ( A --> (A OR B) ) is necessarily true ( true in all possible cases, whatever the truth value of A and of B may be) , as a truth table will show easily.

Answer 3

First of all, note that nothing prevents us from using multiple different kinds of implication at once; indeed, nobody (I think!) would quibble with the claim that most of the time when we use conditionals in natural language we are not invoking the material conditional. Even in mathematical contexts we can consider non-material implications.

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That said, material implication clearly enjoys a distinguished role - at least, in mathematics (which I"ll focus on here). This is because of how well it plays with quantification. Specifically, suppose I make a claim of the form

(#): For every x, if P(x) then Q(x).

The argument then is the following. Since (#) is a universal statement, it can only be false if for some x the statement "if P(x) then Q(x)" is false. However, thinking about how we use such statements in mathematical practice, it's clear that if x satisfies ~P(x) then x is not a counterexample to (#): for example, "Every prime greater than 2 is odd" is not falsified by taking x=4 since 4 doesn't satisfy the hypothesis "is prime >2."

Combining these observations we have that whenever ~P(x) is true the implication "P(x) implies Q(x)" is not false.

Now we "shift down" to the simpler language of propositional logic. The point is that propositional logic isn't actually a language we reason in; rather, it's a (framework for providing a) simplified model of some piece of reasoning we're interested in. From the above considerations and a general "minimization" principle (in particular we equate "not false" with "true" rather than introduce additional truth values), we wind up with the following idea:

While it doesn't play well with natural language, the material conditional is a pretty good model for implication in "natural mathematical language."

An important ameliorating component of this is the following:

While false antecedents yielding true implications is unintuitive, the no-counterexample-interpretation of (#) is quite intuitive.

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Some final comments:

First, of course none of the above is unobjectionable. In particular, one could easily dispute the "secondariness" I've ascribed to propositional logic or the "minimzation" goal (or its specific application to "not false = true"). But I think the argument above does capture why the vast majority of the mathematical community accept material implication as the "standard" interpretation of implication in mathematical contexts.

Second, as stated above nothing's preventing you from using different implications in a mathematical context - the only constraint is a general sociological convention that if you're going to use the term "implies" in a mathematical context, it will be interpreted as referring to material implication unless otherwise explicitly stated.

Finally, remember my claim earlier: "[propositional logic is] a (framework for providing a) simplified model of some piece of reasoning we're interested in." In different contexts we're interested in different pieces of reasoning, so even granting that the material implication is the "right" choice in mathematics we can still reject it in other contexts where it's clearly inappropriate. (More generally, we can - and I do - adopt various forms of logical pluralism.)

Answer 4

there is no alternative for truth functional classic logic. It's the only thing you can build out of the truth tables for propositional logic that even remotely resembles an indicative conditional, and it works in most contexts.

I recommend you look at the truth tables yourself and try to find another way to define it.

Answer 5

One attractive feature of the material conditional is that, in conjunction with the universal quantifier, it offers a natural way to translate classical logical statements of the form "all A are B" into propositions in first-order logic, like "for all x, A(x) -> B(x)". Consider for example the classic example "all men are mortal", translated as "for all x, Man(x) -> Mortal(x)". For this to be an equivalent translation, we can deduce the truth table of the material implication symbol even if we don't remember it offhand:

1. Suppose that within our domain of discourse, there is an x such that x is a man, and x is mortal. Obviously this is compatible with the statement "all men are mortal", so Man(x) -> Mortal(x) is TRUE when Man(x) is TRUE and Mortal(x) is TRUE.

2. Now suppose there is an x such that x is a man, but x is not mortal. This would falsify the statement "all men are mortal", so Man(x) -> Mortal (x) is FALSE when Man(x) is TRUE and Mortal(x) is FALSE.

3. And suppose there is an x such that x is mortal, but x is not a man (a cat, for example). This would not falsify the statement "all men are mortal", so Man(x) -> Mortal(x) is TRUE when Man(x) is FALSE and Mortal(x) is TRUE.

4. Finally, suppose there is an x such that x is not a man, and x is not mortal (a Greek God, for example). This would also not falsify the statement "all men are mortal", so Man(x) -> Mortal(x) is TRUE when Man(x) is FALSE and Mortal(x) is FALSE.

I don't know who was the first to actually define the material implication symbol (or something equivalent) with a clearly-specified truth table matching the modern one, so I don't know if their motive was to provide translations of propositions in classical logic in this way--if anyone knows about this history, please add a comment or your own answer. But at least this shows why something equivalent to material implication was probably bound to get a lot of use in first-order logic, despite the fact that it tends to confuse new learners since it's different from both logical implication and from if-then statements in ordinary language (indicative conditionals) which often have other meanings associated with them, for example they may be used in a sense closer to that of modal logic (like 'if I had a nickel for every time someone complained about how confusing material implication is, then I'd be a rich man').

It might have been simpler if the symbol was described differently in ordinary language (i.e. it wasn't called a 'conditional', and people didn't describe A -> B as 'A implies B' or 'if A then B'), but when the universal quantifier isn't there so you just have a proposition about a single entity like Man(Socrates) -> Mortal(Socrates), it seems difficult to come up with a non-clumsy description in English that suggests the right truth table. If the clumsiness of the expression isn't a big deal, one way I can think of is based on noting that A -> B has the same truth table as ~A or (A and B), so Man(Socrates) -> Mortal(Socrates) could be described in English as "either Socrates is not a man, or he's a mortal man". Another option could be based on the point above about using the universal quantifier to express "all A are B", so that a proposition that doesn't use the universal quantifier, and is just about a particular entity like Socrates, could be stated as something like "Socrates is not a counterexample to the claim that whenever something is a man, it is also mortal". Either one of these English-language formulations would indicate that if "Socrates" is the name of some entity that is not a man, then the statement is true regardless of whether that entity is mortal or not.

Answer 6

Mathematicians as well as logicians are quite happy with the material conditional since no real problems arise as long as the propositions linked by the conditional constructions have no intermediate probability values (i.e. as long as they are either absolutely certain or epistemically impossible).

Another reason is that proving a conditional intuitively involves assuming the antecedent and deriving the consequent, a type of reasoning central to mathematics; however it can be shown that conditionals stronger than material implication do not satisfy the deduction theorem and do not capture this way of reasoning. Material implication, however, has the deduction property.

Answer 7

This is at best only a quasi-answer, but I think it is in the ballpark of what you are contemplating...

You stated, "(p->q)->(~pVq) is not as intuitively necessarily true as maybe (pVq)->(~p->q)". I AGREE, especially when you consider the rendering of wedge as "unless", right? I think it would totally work in a system that eliminates double negation (introduction and elimination) as a rule of inference and demands (in translation from plain language) the immediate reduction of all odd numbers of tildes (negations) to one and all even numbers of tildes to none. BUT, that is to think way outside the box. LOL!

Answer 8

The main objection to material implication seems to be that it requires an implication to be true whenever its antecedent is false, regardless of the truth value of the consequent. This form of argument is almost never used in daily discourse. When we know that the antecedent is false, we don't usually give much thought as to the truth value of the implication in question. In classical logic, when dealing with propositions that are unambiguously either true or false in the present, such implications turn out to be unavoidably true. Here is my formal proof using a form natural deduction.

Why are we so reluctant to accept this principle of so-called vacuous truth in daily discourse? As some would have it, this would open the door to absolutely anything goes. Every proposition, no matter how absurd they say, would have to be true. This, of course, is simply not the case.

When the antecedent is false, the implication in question cannot be used to infer anything about the consequent. So, the notion that absolutely anything would go is out, even in formal logic--especially in formal logic. I know of no logical downside to the acceptance of vacuous truth in daily discourse. (You could be the life of the party by proving that all unicorns have eights legs! OK, so there may be a social downside.)