Why are conditionals with false antecedents considered true?

Question

I don't understand what conceptual sense this scenario makes, or what the motivation behind the decision to make conditionals with a false antecedent true was. Can anyone help me understand this?

Also, I have a related but ultimately separate question on something read in a book. I am reading "Introduction to Logic: and the Methodology of Deductive Sciences" by Alfred Tarski, which gave the following example after defining implications.

If 2*2 = 4, then New York is a large city; 
If 2*2 = 5, then New York is a large city; 
If 2*2 = 4, then New York is a small city; 
If 2*2 = 5, then New York is a small city. 

It then says the following, "In everyday language, these sentences would hardly be considered as meaningful, and even less as true. From the point of view of mathematical logic, on the other hand, they are all meaningful, the third sentence being false, while the remaining three are true."

What is it that makes the third sentence false? No further explanation for this is given in the book.

Answer 1

The standard "conceptual explanation" is that if it were to obtain that 2*2=5 then it would mean that we know nothing anymore, and so any consequence should follow. This is more of a fable invented by teachers to make students feel better. The real reason the material conditional is accepted is that it is the closest (to intuition of implication) operation which is compositional, i.e. its truth value is determined exclusively by the truth values of its terms. Compositional operations obey simple rules (of Boolean algebra), and therefore are much handier to deal with, intuition takes the second seat to operational utility. We similarly accept that empty set is a subset of every set, or that the product of negatives is a positive not for conceptual reasons but because it makes the algebra work better. For more on this trade off see Shouldn't statements be considered equivalent based on their meaning rather than truth tables? So "if 2*2 = 4, then New York is a small city" is false simply because the premise (2*2 = 4) is true while the conclusion (New York is a small city) is false, which is precisely when the material conditional is defined to be false by its compositional definition.

That the conditional of the natural language is not compositional is well known of course. For example, unless salt is actually placed in water "if salt is placed in water it does not dissolve" is true in classical logic, and so is "If 2*2 = 5, then New York is a small city", but not in natural reasoning. The conditional of natural if-then reasoning is called the indicative conditional, but that can hardly be formalized given the vagueness of natural language and reasoning. Some approximation of it is given by the conditional in relevance logic, which demands in particular that the premise be relevant to the conclusion for the conditional to hold. So in contrast to the material conditional, "If 2*2 = 5, then New York is a small city" is relevantly false. However, "If 2*2 = 5, then 2*2+2 = 7" is relevantly true, and therefore the relevant conditional is non-compositional, its truth value depends on the content of its terms, not just their truth values.

Another alternative known as the strict conditional was introduced by Lewis specifically to deal with counterfactuals like the salt in water example above. It combines the material conditional with the necessity operator in modal logic. Since "if salt is placed in water it does not dissolve" is not true necessarily, i.e. in every possible world, interpreted as the strict conditional it fails to be true. On the other hand, "if salt is placed in water it dissolves" obtains necessarily, trivially in the worlds where it is not placed in water, and due to the properties of salt and water where it is. So this strict conditional does hold, as intuitively expected. However, 2*2 = 5 is false in every possible world, so "If 2*2 = 5, then New York is a small city" holds materially in all of them, and therefore it holds also strictly.

Answer 2

On the simplest level, the answer is because every proposition in sentential logic ("propositional calculus") must be either true or false. And this appears to be the best possible resolution given the options of True or False.

To state it in more detail, sentential logic normally follows the following three laws (Aristotle was possibly the first to state them):

1. Law of identity - any term we use keeps the same meaning in the context of our argument (A= A).
2. Law of non-contradiction - we do not accept contradictory claims as both being compossible. (not (A and not A)).
3. Law of the Excluded middle - for every proposition (sentence), it must be either true or false (A or not A)

Given these three laws as our foundation, we can come up with operators that work based on the truth or falsity of multiple sentences. Generally, people use "and" , "or" , "xor" , "if, then", "biconditionality" etc.

For the conditional, the question we're asking about the truth of the sentence refers to the whole. For many accounts of sentential logic, we are asking, "does this conditional obtain in the real world"? Let's say the two terms are A and B.

We consider this FALSE when we have A but not B. i.e. the claim that "if it rains, then it pours." for it to be true, it would require us to be able to confirm that every time it rains, it does in fact pour. But if ever it does rain, but it does not pour (A = true, B = false), then we can say this claim is false about the world.

Similarly, we consider the statement to be TRUE about the world when both the antecedent and consequent do occur. e.g., If I am wearing a blue shirt, I am wearing a shirt. Moreover, it always obtains that when I wear a blue shirt, I am wearing a shirt.

Turning to cases with FALSE antecedents, remember that we must decide that they are either true or false. One option would be to give up -- i.e. to change one of the three laws. Doing so, however, takes us outside of classical logic.

The other option is to realize something about what we are saying.

First off, we are not saying A causes B. Instead, we are merely saying that whenever A occurs, B also occurs. In other words, "if/then" does not make a causal claim about the world. Instead, it makes a factual claim about the world where we can even say it backwards and still be right. E.g., if am extending my umbrella, then it is raining.

Given this, think about the false antecedent claims differently. If the question is "have I found a case where A happens and B does not?" If I have not, then I have not disproved whenever A, then B. Thus, I have no reason to believe this is false. To put it another way, a conditional with a false antecedent is not up for a test. I do not know that it does not obtain in the world. Thus, there's no reason for me to say it is false.

Given then that we have only the choices of TRUE and FALSE, I should consider it TRUE.

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All of that to say, however, that this is one of the most common worries people have about sentential logic. I'd say it's a point where one needs to recognize a limitation in the tool set rather than think the toolset is broken for what it does. The material conditional in sentential logic is not identical the English construction "if, then" in scope or function.

Answer 3

Ultimately the symbols for various "logical operators" like the implication symbol -> or the "and" symbol ^ are just defined by their truth tables, so you shouldn't place too much weight on the ordinary-language meaning of "implication" or any other name given to a logical operator. However, one way of thinking about it is that "P implies Q" is logically equivalent to "(P and Q) or (not-P)", which is at least somewhat intuitive--either P is true and therefore Q is true as well, or P is false so it doesn't matter if Q is true or false.

And thinking a little more about mathematician's motivation in calling this "implies", suppose we wanted an alternate logical operator, let's call it "alt-implies", such that "P alt-implies Q" would be false if both P and Q were false individually, but the same as the standard implication operator for the other entries on the truth table. Thinking about logically equivalent ways to describe this, it would be equivalent to the rule "when evaluating the proposition 'P alt-implies Q', simply ignore P and define the proposition to be true if Q is true, false if Q is false." This is why what I called the "alt-implies" operator is just called the "q" operator on the chart of all possible binary logical operators here. And you would still need some name for the operator that mathematicians currently label "implies", so since what I called "alt-implies" above is most easily understandable as just a "q" operator (and calling it that makes the it symmetrical with the similar 'p' operator whose truth value is just identical to P), it seems reasonable to use the word "implies" for the operator which has the value "true" when both P and Q are false but otherwise has the same truth-value as Q.

Answer 4

Suppose it is raining and it is not raining. Then it is raining. Hence, it is raining OR the moon is made of cheese (1). Since it is not raining, the moon is made of cheese (2).

If you accept:

• (1) disjunction introduction, i.e. P -> (P or Q)
• (2) disjunctive syllogism, i.e. ((P or Q) and not P) -> Q

then you need to accept that contradictory assumptions imply everything.

Also there is a neat mathematical reason, why contradiction "should" imply everything. You can put a (pre)-order on the set of propositions by saying P is less then or equal to Q if and only, if P => Q. Then contradiction is the minimum in this ordered set, because it is less then every other proposition.

Answer 5

We often encounter situations where we say "if something unlikely were true, something else unlikely would be true". I chat up a girl and say "if I was a multi-millionaire, I would buy you a car". She will completely understand that it is very unlikely that I am a multi-millionaire, and she isn't going to get a car. If I said "if I was a multi-millionaire, I wouldn't buy you a car" she would equally understand this, and probably be a bit annoyed. Now I say "If 2 + 2 were 5, I would buy you a car". Logically this is almost the same, but people tend to be confused by it. The likelihood of getting a car has gone down from practically zero to zero, which makes very little difference, but people don't get it. As shown by the question asked.

Answer 6

Material Implication, Logical Equivalencies, Converse, Inverse, Contrapositive

Given the material conditional P -> Q, P is termed "antecedent" (condition) and Q is termed "consequent" (consequence), where P -> Q reads "P implies Q" and means that P is a sufficient condition for Q which in its turn means that Q is a necessary condition for P.

If the antecedent (P) is true, then the material implication (->) holds only if the consequent (Q) is also true. That is, a true antecedent/premise/condition (P) can only imply a true /consequent/conclusion/consequence (Q).

If the antecedent (P) is true and the consequent (Q) is false, then the implication does not hold (true). That is, a true antecedent cannot imply a false consequent: truth cannot imply falsity. This is the only option for which the material implication does not hold, i.e., the operator/connective (->) outputs false if and only if the antecedent (P) is true and the consequent (Q) is false.

If the antecedent (P) is false, then P materially implies Q, regardless of whether Q is true or false. From falsity anything follows. That is, a false antecedent (P) implies the consequent (Q) in the case when Q is false as well as when Q is true. For more information, please look up the "principle of explosion" which states in Latin: "Ex falso sequitur quodlibet" = "From falsity follows anything".

Example 1. (with a false consequent): "If 2+2 = 5, then I am god" both P and Q are false, yet the implication holds (true).

Example 2. (with a true consequent): "If Julius Caesar invades North America, I speak some Latin" also holds true. (I do speak a little Latin).

Example 3.: "If you write a great post, then I('ll) give you $10." This conditional constitutes a promise. Let us see for which truth value combinations of P and Q, the promise (implication) holds (true).

Let us analyze Example (3):

• Let: P := You write a great post, and
• Let: Q := I give you $10.
• Case 1. P is true and Q is true.
• Case 2. P is true, and Q is false.
• Case 3. P is false, and Q is true.
• Case 4. P is false, and Q is false.

Suppose case 1 is the case: "You write a great post, and I give you $10." Does the implication hold or have I broken my promise? I have fulfilled my promise in response to your great post. The implication holds (true).

Suppose case 2 is the case: "You write a great post, but I do not give you $10". Does the implication hold or have I broken my promise? In fact, I have broken my promise, because I have not fulfilled the consequent of the conditional, given a true antecedent. Therefore, for this option, the implication does not hold (true), i.e., the implication outputs a truth value of false.

Suppose case 3 is the case: "You do not write a great post, and I give you $10." Have I broken my promise? My promise was predicated on your writing a great post, and it says nothing about what should happen if the antecedent were false. My promise (implication) merely states what should happen if the antecedent were true. If you do not write a good post, but I nonetheless give you $10, then, strictly speaking, I have not violated my promise. Therefore, the implication holds (true), with a false antecedent (P) and a true consequent (Q).

Suppose case 4 is the case: "You do not write a great post, and I do not give you $10". Here, even though both antecedent (P) and consequent (Q) are both false, the implication nonetheless holds (true). A falsehood can imply a falsehood, because from falsity anything follows.

P -> Q means "P materially implies Q" which is stated as the following material conditional (if-then) statement: "If P, then Q". The material conditional P -> Q implies that P is a sufficient condition for Q.

Given the material conditional (if-then) statement (P -> Q), the operator/connective (->) is called material implication, which sets up a material conditional "If P (is the case), then Q (follows)", which can equivalently stated as "Q if P", which in its turn is equivalent to stating "P only if Q", which implies that P is a sufficient condition for Q, which is represented as follows: P => Q.

Moreover, the sufficiency of P for Q is logically equivalent to the necessity of Q for P: [P => Q] <= logically equivalent to => [Q <= P].

Consider the following four options:

• P => Q: P is sufficient for Q.
• Q <= P: Q is necessary for P.
• P <= Q: P is necessary for Q.
• Q => P: Q is sufficient for P.

Henceforth, let the symbol (≡) denote logical equivalence, and (≡|≡) denote logical non-equivalence.

• Options (1) and (2) are logically equivalent: (1) ≡ (2)
• Options (3) and (4) are logically equivalent. (3) ≡ (4)

Consider the original conditional (P -> Q) with the (forward) material implication (->), connecting P to Q, so that P is set up as a sufficient condition for Q.

Let us refer to P -> Q as the "original" material conditional:

A1. Original material conditional: (P -> Q), with a "forward implication", i.e., from P to Q.

A2. Converse of Original: (Q -> P), with a "reverse implication", i.e., from Q to P. reverse of "forward".

A3. Inverse of Original: (~P -> ~Q), with a "forward implication" and with both antecedent (P) and consequent (Q) negated.

A4. Contrapositive of Original: (~Q -> ~P), with a "reverse implication" and with both antecedent (P) and consequent (Q) negated; therefore, contrapositive = the inverse of the original in reverse; that is, contrapositive (of original) = converse of inverse (of original). Note that:

• The contrapositive (A4) is logically equivalent to the original (A1): A1 ≡ A4.
• The converse (A2) is logically equivalent to the inverse (A3): A2 ≡ A3.

The inverse (A3) can be derived by the contraposition of the converse (A2): by reversing the direction of the implication in (A2) and negating the P and Q. Note, that just as the contraposition of the original yields the contrapositive (of the original), where the contrapositive is logically equivalent to the original, so too does the contraposition of the converse yield the inverse (of the original), which is logically equivalent to the converse. Therefore, the contraposition of a conditional (C) validly yields another conditional (C*) that is logically equivalent to the conditional (C): C ≡ C*.