Lecture Note
University
Stanford UniversityCourse
Introduction to Mathematical ThinkingPages
10
Academic year
2022
Murphyal
Views
31
Implication Implication in Mathematical Thinking is the idea of "if...then". This can be seen in many different ways. For example, if you add two numbers together, then the sum will always bebigger than or equal to either of the original numbers. If a person is taller than someoneelse, then they are also older than them (unless of course they are twins!). Another waythis can be seen is if there is a solution to an equation, then it always exists (i.e. there is atleast one solution). You may experience several days of confusion while the idea seeps into your mind. It'ssimilar to learning how to ride a bike—all the actual learning occurs before you can finallyget the hang of it. So too, most of the benefit from understanding the way our language is used inmathematics comes from trying to figure it out. This extra process of trying to understandcan help you develop your mathematical thinking ability and will ultimately help you learnmore about mathematics. Once you are able to use language fluently, it will help you think mathematically. And oncestudents learn to think mathematically, they will be able to use language well too. However,the part of language that helps students with mathematical thinking is largely over by then. We will need to consider other tasks to develop your mathematical thinking ability forever.As you work through this unit, keep in mind that the payoff for struggling with issues issignificant in terms of being able to think like a mathematician. Here we go. In mathematics, it is common to see expressions of the form Phi implies Psi. Well,implication is the means with which we prove results in mathematics, starting withobservations or axioms. So let's explore the logic behind the term "implies." How does the truth or falsity ofstatement Phi imply that of Psi depend upon the truth or falsity of Phi and Psi? We can say that Phi implies Psi if the truth of Psi follows from the truth of Phi, it seemsobvious. But is that what we want? Here is an example. We now know that phi of the statement loop 2 is a rational. And we know that psi of thestatement is 0 is less than 1. So we should ask ourselves if the statement phi implies psi istrue. Well, phi is true. Later we will prove that. The fact that 0 is less than 1 and the fact that psi is true both hold true. However, this doesnot imply that phi also holds true. It is not true that there is a relationship between phi and psi. This takes some effort toprove, as you'll see. We know this one. It is true that this is true and that's true. However,the truth of this does not follow from the truth of that. Now, we can see that there is a complexity to implication that we did not see before.Implication involves causality.
Causality is an issue of great complexity, which philosophers have been discussing forgenerations. Now, however, we're facing a problem that didn't arise before: it didn't matterwhether there was any kind of relationship between the two conjuncts or the two disjunctswhen we were dealing with conjunction and disjunction. For example, let us consider the sentence "Julius Caesar is dead" and conjoin it with themathematical sentence 1 + 1 = 3. Let us do the same with disjunction. The claim that the conjunction and disjunction of two statements is true if both are true andfalse if both are false does not require a relationship between these two statements.Clearly, they are independent. One is a statement about a long-dead individual, and theother one is a mathematical statement. Incidentally, why do we have these items in front of us? A quick quiz: are both statementstrue or are they false? What do you think? Pay attention, the definition of a conjunction isthat it is true if both conjuncts are true. It is true that this one's true and this one's false, so the conjunction is false. It is also truethat this one's true and this one's false. In order for a disjunction to be true, it needs atleast one of its disjuncts to be true; however, in this case there was no meaningfulrelationship between the two conjuncts or between the two disjuncts, making it easy todetermine what the truth value was. It was a purely logical consideration, but it's not sufficient to consider implication; we haveto consider causality. So let's express that explicitly. The term implication has two parts: a truth part, and a causation part. We are going toignore the causation part, which we can leave to philosophers if you like, and focus on thetruth part instead. Throwing away the assumption that there is a causal relationship, we can now focus onwhat is true. It may seem dangerous to throw out this important assumption, but when wefocus on the truth part we are left with enough to save our leaves in mathematics. Somuch so that we are going to give this a name. We will call it the conditional or,sometimes, the material conditional. This is the part we will be focusing on.
We will divide implication into two parts: conditional and causation. The first part,conditional implication, can be defined entirely in terms of truth values. Ultimately, thesecond part of this discussion belongs to the philosophers. The symbol we use to denote conditional probability, at least the symbol we’re going touse now, is a double arrow. Many textbooks use a single arrow to represent many otherthings as well, but we might be using that later. In order to avoid confusion, I will use this notation for the conditional. So I will writeconditional expressions like this: phi implies psi. When we have a conditional, we call phithe antecedent, and we call psi the consequent. We're going to define the truth of phi conditional psi in terms of the truth of phi and thetruth of psi. You might worry that by throwing away a notion, we're left with a notion that's really of nouse whatsoever. However, even though we're throwing away something of greatsignificance, the truth part leaves us something useful. Since genuine implications are the only circumstances in which we are ultimatelyinterested, whenever we have a genuine implication, the truth behavior of the conditional isthe correct one. It captures what we might experience when we have a genuine implication. It may seemmysterious at this stage, but once we look at some examples, it should become clear. The issue of causation, the square root of 2 and the 0 less than 1 example, and the truthor falsity of those two statements is not what matters—the relevant question is whetherthere is a relationship between them.
Now, that's a complex issue. But because we're going to define the conditional purely interms of the truth value of its two constituents, the antecedent and the consequent, it turnsout that we'll always be able to define the conditional regardless of how complicated it maybe. The definition of the conditional will agree with the way implication behaves if we have agenuine implication, and if we don't have one, then the conditional will still be definedanyway. It may seem somewhat perplexing when we describe the concept in this way. However, aswe take a look at some examples, you will begin to see how it works. A little quiz for you now. The truth of the conditional from phi to psi is defined in terms ofwhether 1, the truth of phi and psi, or 2, whether phi causes psi, or both. The answer isnumber 1; we define conditional truth as the truth and falsity of antecedent andconsequent. Because we can express the truth of the conditional in terms of truth and falsity, theconditional has a truth table. Let's see if we can figure out what it is. We have already examined this part. If we define the conditional as the truth part ofimplication, then an implication has a property that a true conclusion follows from a truepremise in all cases where the premise is true. As a result, we must fill in these values withtruths throughout the top level.
Let's examine the first row of the truth table. Suppose phi is the statement N is bigger than7 and psi is the statement N squared is bigger than 40. If phi is true, in other words, if N isbigger than 7, then N squared certainly is bigger than 40. Indeed, phi implies psi here.This is because if N is bigger than 7, then N squared is biggerthan 40.Thus it is true that if N is bigger than 7 then N squared is bigger than 40.Everywhere we have the truth. And this is consistent with the truth table. Now let us examine a different example. Let phi be the statement Julius Caesar is dead.Let psi be the statement pi is bigger than 3. Phi is true. Psi is true. According to the truthtable, it follows that phi conditional psi is true. Saying differently, Julius Caesar is dead,conditional on pi being greater than 3. Now, if you read the statement, "Julius Caesar is dead" as implying that pi is greater than3, then you are in a nonsensical situation. But remember: This is not implication; this is justthe truth part of implication. In terms of the truth part, there is no problem. In the first example, there is a meaningfulrelationship between phi and psi. When we know that N is bigger than 7, then we canconclude that N squared is definetely bigger than 40. If there is a connection between two things, then the behavior of the conditional is certainlyconsistent with what's really going on. In the second case, there's no connection betweenthe two. The conditional is true but it has nothing to do with one thing following fromanother. Even though this has no meaning in terms of implication, its truth value is defined. In bothsituations, we have a well-defined truth value. In the first case, it's a meaningful truth value. That does follow from that. In the secondcase, it's purely a defined truth value—but that won't cause us any problems becausewe're never going to encounter this kind of thing in math. We encounter this kind of thing all the time. But we're not going to encounter this kind ofthing. So all we've done is extend a notion to be defined under all circumstances and in away that's consistent with the behavior we want when something meaningful is going on.
This is common in mathematics to extend the domain of definition of something so that it'salways defined. Provided we do the definition correctly, it really doesn't cause anyproblems. In fact, it solves a lot of problems and eliminates a lot of difficulties if we extendthe definition so that it covers all cases. It is an important part of modern advanced mathematics. Incidentally, if you think this isjust playing games, let me mention that the computer system that controls your next flightdepends upon the fact that expressions such as these are always well-defined. Software control systems do not require knowledge of whether Julius Caesar is dead,because they do not depend on such facts. Computer systems, by and large, do notdepend on an understanding of cause and effect because they do not need to understandcausation in order to function. It is important for computer systems to have well-defined elements. For example, theexpression phi conditional psi can be found in many software systems, and it literallydepends on those elements' being clearly defined. It does not depend on whether acomputer knows whether Julius Caesar is dead or alive. Now let's look at the second row of the truth table. What goes in here? T or F? We haveonly two options so we can "guess" but we are going to figure it out and get the correctanswer. Let us consider what would happen if we put a T here. Let us place it lightly, as it seemslike it will be a T. If this were true, when we think about it in terms of genuine implication,because we are trying to capture the truth-value behavior with genuine implication,remember. So if phi implies psi, then if we interpret that statement as real implication, the truth of psiwill follow from the truth of phi. That's how we began to remember.
Real implication means that if the antecedent is true, then the consequent must be true aswell. If we have a T here and we're using real implication, then the truth of this would followfrom the truth of that. But we don't have a T here. We have an F. So we can't use realimplication here because it contradicts what real implication means. So this has to be an F. If we were to put a T there, we would create a notion that does notagree with the real implication. For the conditional to agree with real implications, it must be an F. If a truth, then we wouldhave a true antecedent and a false consequence from a true implication. We have concluded that this statement is false, which means that F must be true. Do not lose the point. If there were an implication involving the phi-psi relationship that were true, and if it weretrue that phi implies psi, then psi would have to be true. But since we cannot have phi truewhile psi is false, if there were an implication involving the phi-psi relationship that weretrue, then we could not have it be true that phi Implies psi. What this means is that in the case where phi is true, but psi is false, we have to have afalse here. It can be messy or tricky. Sorting out implications and then extracting conditional formimplications might not seem obvious. Let's see if we can fill in the last two entries in the truth table for the conditional. It will bethe final move. Probably you have no thoughts now. What to put here? And the reason you have no intuition is that you're used to dealing with implication, butyou've never dealt with an implication where the antecedent was false.
You have never dealt with a true assumption because you are only interested in drawingconclusions from true assumptions. So you don't have any intuition about a false one ofthese guys. But you do have intuitions about this guy. And, the reason this will help us is that negation swaps around Fs and Ts. So, based onthe Fs here, when we look at this guy we'll have truths. Based on this guy we'll have Ts. Soyou're used to dealing with this. Given this principle, the solution to this problem will be equivalent to solving the inverseproblem—dealing with not that when this is false. This is equivalent to swapping truth andfalsity within the logical statement, so you must look at negation rather than implication. The fact that phi is true does not imply that psi is true. Just think about that for a minute,the fact that phi is true does not imply that psi is true. This is how you know that this thing holds. You know that phi implies psi only if you cancheck that phi is true and psi is false. It is the only way you can conclude this thing is notfalse. In all other cases not covered by the above, the statement will be false. If phi is false, psiwill be false. One more step to the conclusion. If the F and T are swapped, negation will be false and this guy will be true. This means ifwe have T and F, then negation is false and this guy is true. In all other cases, this guy istrue, so those are true as well. And that’s it. That was really complicated. Keep rethinking it and it will become more clear. One more quiz to solve. Which of the following are true? Phi conditional psi is true whenever, (1) Phi and psi areboth true. (2) Phi is false and psi is true. (3) Phi and psi are both false. And (4), phi is trueand psi is false.
The task is to check all that are true. So, which of these four are the case? Which of thesefour conditions tell you when phi, psi is true? The answer is 1, 2, and 3. The phi conditional psi is true when one of the following four conditions is true: 1, 2, or 3. Itis false only in the case of 4. Thus, correct answer is therefore 1 , 2, and 3 without 4. So far we've defined a notion, the conditional, that captures only part of what impliesmeans. To avoid difficulties, we have based our definition solely on the notion of truth and falsity.Our definition agrees with our intuition concerning implication in all meaningful cases. Asfor a true antecedent, the definition is based on an analysis of the truth values of genuineimplication. If we consider false antecedents, we see that their truth value depends on thatof does not imply. By defining conditional statements in the way we do, we do not end up with a notion ofimplication that contradicts the notion of genuine implication. Rather, we obtain a notion ofconditional statements that extends genuine implication to cover those cases where theclaim of implication is irrelevant because the antecedent is false or meaningless whenthere is no real connection between the antecedent and the consequence. In the meaningful case where phi and psi are related, and in addition, phi is true, the truthvalue of the conditional will be the same as the truth value of a material implication. Remember that the fact that the conditionals always have a well-defined truth value makesthem important elements in mathematical reasoning, since in mathematics we cannotallow ourselves to deal with undefined values. The last quiz. And do not forget about the importance of discussion. The antecedent is true, and the consequence is true, so the conditional is true. In fact,there's a deeper result going on here. Providing you take a positive number instead of pi,any positive number, then if that number's square is bigger than 2, it must be bigger than1.2.
As the square root of 2 is 1.4, etc., so for positive numbers it does not have to be pi, butcan be any positive number whose square is greater than 2, it must be bigger than 1.2. The conditional "if p then q" is true no matter what value p is assigned, as long as q is true.In this case, it's enough that the antecedent is false for the conditional to be true. While theconverse isn't always true—meaning that if q is false, p might not be true—it'snevertheless correct in this example. The antecedent of the conditional "if pi equals 3 thenq" is false because pi does not equal 3. But since the antecedent is false, the consequentmust be true; thus the conditional statement is true. The quantity (pi squared) is not less than 0, so the statement is true. The secondstatement is false, so the first statement's consequent must be false as well. Finally, thethird statement is also false. A false conclusion cannot be drawn from a true assumption. If a statement is true, then itsconsequence must also be true. The statement "triangles have four sides" is not true. The statement "squares have fivesides" is also not true. However, anything with a false antecedent is true by virtue of havingno contrary, so anything with a false antecedent is true. It is true. We’ve got false, conditional, false, and that's always true. And what with thisone? Euclid's birthday is unknown. Rectangles are quadrilateral figures with four sides. Either this is true, or it's false. Eitherway, we know that if the statement is true in either case, then the thing is true. And sincethe consequence of both statements turning out to be true is also true, then we can saythat our original statement is true.
Concept of Implication in Mathematics
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