Can someone provide me with the simplest possible argument for why the material conditional has the truth table it does?
Googling doesn't help at all, providing only flimsy and clumsy arguments. I want the argument to be simple, and yet as convincing as possible.
The question is slightly out of focus: it is conventional to say that the material conditional is defined by the truth table that it has. One does not need to argue why it has this truth table, though it is relevant to ask why such a truth function is useful and important.
In the propositional calculus, all connectives are truth functions, and it is a principle of compositional semantics to try to account for the meaning of a compound sentence as a function of the meanings its parts, as far as possible. One of the truth functions we would like to have is one that minimally captures certain relationships in deductive logic, such as modus ponens and modus tollens. If we posit a two place truth function A * B, we want a relation that satisfies A * B; A therefore B (modus ponens) and A * B; ¬B therefore ¬A (modus tollens). These constraints are not enough on their own, because they are consistent both with the material conditional A→B and the material biconditional A↔B. A further constraint is that affirming the consequent is invalid, i.e. A * B; B does not entail A. The material conditional is the only truth function that satisfies all three. Consequently, it serves a useful role within the propositional calculus for expressing implication relationships.
An important feature of the material conditional is that it is purely a truth function and cannot express the full richness of what is meant by if/then sentences in natural languages. This is not a problem, since we can define all kinds of other conditionals if we wish, and the literature is replete with examples. For example, there are strict conditionals, which are just material conditionals under the scope of a modal operator. There are analogs of material implication in non-classical logics that have different rules of inference, e.g. in intuitionistic logic, where the → conditional is naturally interpreted as "I can prove that a proof of A can be transformed into a proof of B". There are conditionals in non-classical logics with different definitions, e.g. those of Kleene and Łukasiewicz. There are semantically defined conditionals, such as those given by Frank Jackson, Robert Stalnaker and Hartry Field. Also, the conditional probability P(B|A) is a kind of conditional that is defined within probability theory and has its own logic.
All of these are tools in the logician's toolbox and it is part of the logician's art to know which tool to use for which job. It is important not to suppose that the material conditional is the only way to represent a conditional relationship. It is one tool in the box, and usually the first conditional you are introduced to when learning logic, but not the only one.
The various forms of entailment (e.g. "from P we can deduce Q") is a preordering on propositions. Implication is the propositional form of this preordering.
Since the truth of the preordering relation is already two-valued, in the special case of assigning truth values to propositions in a two-valued logic, the truth value assigned to Q→R is precisely the truth of Q ≤ R, and T≤F is the only time ≤ can be false.
More generally, the defining property of implication is
P∧Q ≤ R if and only if P ≤ Q→R
from which we infer Q→R is the largest proposition amongst all X with the property that X∧Q ≤ R.
For those with category theory experience, you might recognize this as saying that if we form the category of all propositions with the arrows being entailment, the category is Cartesian closed: P∧Q is the product P×Q and Q→R is the exponential RQ
The rationale is simple---the material conditional has the truth table it does in order to provide a truth-functional logical connective that would let us represent the modus ponens and modus tollens inferences from natural language.
More formally:
(1) If there is a truth-functional logical connective -> to represent modus ponens and modus tollens, then that connective has the truth table the material conditional does.
(2) There is a truth-functional logical connective -> to represent modus ponens and modus tollens.
(3) Therefore, the logical connective -> has the truth table the material conditional does.
The crucial premise here is (1). To see why it's true, think back to the definition of "truth-functionality". We say a connective is truth-functional if and only if the truth value of the molecular sentence composed of the connective and the atomic sentences it connects is a function of the truth values of the atomic sentences. Since the material conditional is a binary connective, it takes two atomic sentences as inputs, each of which can have two possible truth values (T or F), so we get four possible combinations of inputs. The crucially important thing about the truth-functionality of the connectives is that the whole molecular sentence must a truth value if the atomic sentences have truth values, which they do. This means that we have to fill out each row of the truth table below.
A B A->B
T T T
T F F
F T T
F F T
Rows 1 and 2 look obviously correct, since they match our intuitive, natural language reasoning use of "if . . . then". We intuitively know Modus ponens and modus pollens are valid inference rules, and that's what is represented in lines 1 and 2 here.
Rows 3 and 4 require more comments.
Since we want a truth functional logic, we have to fill in something for the truth value of "A->B" on these lines, and we have only two choices (T or F).
So, let's see what happens if we fill in F for line 3. (i.e. let's assume A is false, B is true, and A->B is false.) Then the following argument would be valid (i.e. the argument would have the property that the truth of its premises would guarantee the truth of its conclusion), but intuitively the argument is not valid.
(I) ~(A->B)
(II) ~A
(III) B
(I*) It is not the case that (if it is raining, the street is getting wet).
(II*) It is not the case that it is raining.
(III*) Therefore, the street is getting wet.
Suppose (I*) is true because somebody's put a tarp over the street to keep the rain off. How would the fact that there's a tarp down and the fact that it isn't raining entail that the street is getting wet?! This argument would be valid if A->B is true on line 3 of the truth table, but obviously this argument isn't valid; therefore we must hold A->B to be true on line 3 of the truth table.
We'll do the same procedure to show what the truth value of A->B should be for line 4 of the truth table. Let's start by assuming that A->B is F, A is F and B is F. Then we can also construct an argument that would also be "valid" like so:
(IV) ~(A->B)
(V) ~B
(VI) ~A.
(IV*) It is not the case that (if it is raining, the street is getting wet).
(V*) It is not the case that the street is getting wet.
(VI*) Therefore, it is not the case that it is raining.
Again, assume that (IV*) is true because somebody's spread a tarp over the street. Now how would that fact, plus the observation that the street is not in fact getting wet in (V*), entitle us to conclude it wasn't raining (VI*)?! Obviously it couldn't. But if A->B were false on line 4 of the truth table, we could validly infer that. Therefore, A->B must be true on line 4 of the truth table.
The subargument for (2) is easy. It is necessary that there be such a connective, because what we're trying to do in logic is represent certain basic forms of inference that we intuitively know are valid (modus ponens and modus pollens) and we're trying to create a mathematical structure that will let us build up a systematic, scientific logical theory from these intuitive foundations. Thus, we have to have a truth-functional connective to represent the conditional since we use such conditions in natural language reasoning all the time and the only way to make such a connective truth functional, as we've just seen, is for it to have the same truth table as the material conditional.
Hence, (3) follows by modus ponens.
The truth value of an assertion A -> B is determined only in circumstances in which A. But circumstances in which ~A allow only that the assertion is neither proved nor disproved, regardless of the status of B. Since A -> B is a positive assertion, we do not take it that the assertion is not proved (that is, ~T) but that it is not disproved (~~T). An assertion that is ~~T must be reduced to T in a truth table.
The interesting problem is that an A -> B assertion treated truth-functionally introduces this bias: in the absence of the concrete circumstance in which A, the fact of the assertion where before there was none causes us to err on the side of T even though there is no more justification for this than for its contrary.
I agree with Shane that:
Suppose (I*) is true because somebody's put a tarp over the street to keep the rain off. How would the fact that there's a tarp down and the fact that it isn't raining entail that the street is getting wet?!
But it also seems that if it is not raining (~A), but the street is getting wet (B), then the assertion 'it is not the case that (if it is raining, the street is getting wet)', or '~(A -> B)', far from being intuitively invalid, is simply not proved. In fact, intuitively, it begins to appear more sensible than the assertion that A -> B.
The focus in classical logic and mathematics is essentially conservative. We want to preserve the truth of the premises, and avoid inconsistencies and ambiguity. We do not necessarily intend to trace cause and effect properly. (The implicit lack of temporality should be an adequate indication of this. How can one really care about cause and effect, if you have already dismissed time?)
Given those goals, it is safe to fill up the indeterminate cases left over from syllogistic logic with 'false-implies everything' because one should not be deducing anything from a false premise to begin with. Whatever the reason for mistrusting it, overstating the mistrust is better than understating it. By taking up the convention that any false premise implies all absurd things, you are overstating the reasons for your mistrusting it as broadly as is possible, which is an optimally conservative position.
It turns out that we can maximize the excessive safety without resulting in any extraneous contradictions, by simply replacing all the ambiguous outcomes with 'false'.
So the convention is conservative, and definite. It will not lead you into asserting too much if your premises are weak, but it removes the ambiguity of a third truth value, even if that leads to inappropriate attribution of causes.
The truth table for the material conditional P -> Q expresses a mathematical ordering relation among the truth values for P and Q; that is, it is
When true, it signifies that Q is not less true than P, or Q is at least as true as P. This applies to the truth values only. Any other relationships among the two statements is ignored.
In terms of deductive reasoning, this assures that if P -> Q is true, and we assume that P is true, drawing the conclusion that Q is true does not introduce an error that was not present in the initial assumptions. If P -> Q is true but P is false, Q may be either true (more true than P), or equally false, and we cannot conclude anything about its truth value.
Let's suppose that I have a cat that loves purple. She only lies on the mat if the mat is purple. If the mat is any other colour, she will sleep elsewhere. Evidently, she also sometimes sleeps on the floor, or on the couch, even if the mat is purple. But if she is on the mat, then the mat is purple.
So, lets look at the composite proposition P1, of the P2 → P3 kind:
(P2: The cat is on the mat; P3: The mat is purple.)
Let's compare this with the following possible real events:
If this is true, it doesn't contradict P1. So we have that P2 and P3 can both be true, and the implication holds.
This also does not contradict P1. We have said that if the cat is on the mat, the mat is purple; we didn't discuss what colour the mat is if the cat is not on it. She still only lies on the mat if it is purple, but at the moment she is doing something else, sleeping on the couch, hunting invisible critters, or inspecting the garbage basket. So we can have that P2 is false and P3 true, and the implication holds.
Still this does not contradict P1. She only lies on the mat if it is purple; the mat is not purple, so she is not on it. So we can have that both P2 and P3 are false, and the implication still holds.
So here we get to the case where P1 can no longer be true. If the cat is on the mat, then the mat is purple. But we have now empirically verified that the cat is on the mat, and the mat is blue, or pink with yellow polka dot. So, when P2 is true, and P3 false, the implication itself is false.
This gets confused because when in common parlance we hear "if... then" we tend to associate this with causation, as if P → Q meant "P causes Q". But in logic-ese "if.. then" doesn't work like that. The mat isn't purple because the cat is on it, and the cat isn't on it because it is purple (she is on it because she is tired, or sleepy, or bored, or some other cat reason; the mat being purple is a condition, but not the cause, of the present cat-on-the-matness). What P → Q means in the context of logic classes is exactly this: "P is never the case when Q is not the case".
Hope this helps.
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:
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.
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.
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.
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 that are are 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.
