5

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.

  • The third sentence is false because it has a true antecedent and a false consequent. – Eliran May 10 '16 at 0:53
  • @virmaior from my inexperienced perspective my question is different than the one you linked. The linked question seems to be asking more generally about what makes an implication true or false, while my question is asking for the justification of one specific scenario that logicians claim produces a true implication. – IgnorantCuriosity May 10 '16 at 1:47
  • Okay... there should be several questions on that already (though I didn't find one in the brief interruption between revising a manuscript for publication)... The lengthy question I suggested this duplicates does explain what's going on here, but it might be too hard to follow. Though, if you're reading Tarski ... – virmaior May 10 '16 at 2:18
  • @virmaior the Tarski book I'm reading is a very approachable introduction to the subject written for someone with no background knowledge. – IgnorantCuriosity May 10 '16 at 2:48
4

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.

  • Thank you for the reply. Could you possibly clarify your last sentence for me? I'm not sure how you are using the word "obtains". – IgnorantCuriosity May 10 '16 at 1:25
  • @IgnorantCuriosity I rephrased, "obtains" is logician's word for saying that something is fulfilled or satisfied. What I meant is that 2*2 = 5 is false in every world, therefore necessarily, therefore the material conditional is fulfilled necessarily, therefore the strict conditional is fulfilled (holds, is true). – Conifold May 10 '16 at 1:38
3

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.


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.

2

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.

  • Thank you for the response. As for your opening point, I separated my question about false antecedents and the 2*2 example with this line "Also, I have a related but ultimately separate question on something read in a book." – IgnorantCuriosity May 11 '16 at 3:34
  • Ah, now I see, sorry I missed it--I removed that opening paragraph from my answer. – Hypnosifl May 11 '16 at 3:43
0

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.

0

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.

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