How to Read Mathematical Formulas

How to Read Mathematical Formulas In order to understand the mechanics of mathematical formulas, it's important to know the conventions regarding the order in which logical operators apply. As a quick summary, the precedence order for applying logical connectives is as follows: (∀𝐿) (….) (∃𝐿)(….) Quantifiers are the words that bind tightly, that hold things together. A quantifier applies to whatever comes immediately after it. Typically, what comes after the word "then" is a group of other items, such as "and's and or's and not's." You would put those items in parentheses, brackets, square brackets or whatever. Red is a predicate that applies to all balls. If we said, "For all balls Red B," then "Red" would be the subject and "B" would be the predicate. ∀𝐵 𝑅𝑒𝑑 (𝐵) As shown by the following example, the for all relation applies only to sets that are defined as a subset of their containing set. If we wanted the for all relation to apply to that set, we would need to include parentheses around it. (∀𝐵) 𝑅𝑒𝑑 (𝐵) The quantifier "not" is used to negate a statement. When negating a statement, the negation symbol followed by parentheses should be placed immediately after the quantifier, and everything between the parentheses should be negated. For example, if you want to say that 3 is neither greater than 0 nor less than 0, then you would write "not (3>0) and not (3<0). ¬(3 > 0 ∧ 3 < 0) If 3 is greater than 0, and 3 is less than 0, then the conjunction of these two statements is false. But if the conjunction is false, then its negation must be true. So this sentence is true. But suppose we had written it differently: The case in which 3 is greater than 0 and 3 is less than 0. In this case, both clauses are true but the entire statement is false because they are mutually exclusive and jointly exhaustive—one cannot be true without the other one being false. ¬(3 > 0) ∧ (3 < 0) T If the parentheses did not include both of those, then what we have is a conjunction—a set of statements that are true together. In this case, the two statements are both false, so we have something that is false. ¬((3 > 0) ∧ (3 < 0)) F F So these are clearly not the same, because this one is true and that one is false. Here, the negation applies to everything between them, making it true. ¬(3 > 0) ∧ (3 < 0) The issue between these two sentences wasn't whether there were parentheses around the 3 greater than or less than 0. The issue was whether the parentheses governed everything that was next, or just the one thing that was next. These are not the same. ¬((3 > 0) ∧ (3 < 0)) ≢ ¬(3 > 0) ∧ (3 < 0) The next one is a conjunction: 3 is greater than 0, and 3 is less than 0. 3>0 ∧3 < 0 The fact that 3 is greater than 0 and that 3 is less than 0 can be expressed using the conjunctions "greater than" and "less than," which are known as quantifiers. (3 > 0) ∧ (3 < 0) Some mathematicians argue that conjunction and disjunction have the same strength. Some people say that conjunction is stronger than disjunction, but this argument is not particularly strong. In any case, it is best practice to always use parentheses when you want to disambiguate yourself from your statement. The point is that whenever one should apply a grouping rule such as conjunction or disjunction, one should always use parentheses around it. (…..) ∨ (….) And likewise, if you have an implication or a conditional, the whole thing will be the antecedent and this will be the consequent. (…) ⇒ (…) Now, in here, there may be all sorts of conjunctions and disjunctions and stuff. In here you may find quantifiers, negation signs and a whole lot more. So whenever you look into the sort of basic thing with all of these is when you've got a quantifier or a negation symbol or a conjunction or a disjunction or an implication. ⇔ If one assumed that A/C had a tighter binding than B/D, they could write something like A and B or C and D. 𝐴∧𝐵 ∨𝐶 ∧ 𝐷 And if you put space in there, it is fairly clear that it meant to be either A or B, or C or D. (𝐴 ∧ 𝐵) ∨ (𝐶 ∧ 𝐷) The golden rule is to put parentheses around the quantifiers. [( ) ( ) (∀𝐿) ∃𝑆1 𝑉𝑎𝑙𝑖𝑑(𝐿, 𝑆) ⇒ ∀𝑆2 𝑉𝑎𝑙𝑖𝑑(𝐿, 𝑆2) ] Now, the parenthetical phrase there teams up with the parenthetical phrase there. And we've them not as parentheses but as square brackets to make it absolutely clear. [( ) ( ) ] (∀𝐿) ∃𝑆1 𝑉𝑎𝑙𝑖𝑑(𝐿, 𝑆1)⇒ ∀𝑆2 𝑉𝑎𝑙𝑖𝑑(𝐿, 𝑆2) The argument is going to say that, for any given licence L, if there exists at least one state in which the licence L is valid, then it will hold that L is valid in every possible state. For every licence, L, if a valid state is any state in which L is valid, then there exists a state where L is valid. This second sentence simply applies what was previously said to this particular situation. So it literally says: "For every licence, if the valid state is a state in which that licence is valid, then the exists here simply applies to this." The exists binds what's next to it: "There exists a state where…" And what that means is that there must be a state where the given licence is valid. [( ) ( ) (∀𝐿) ∃𝑆1 𝑉𝑎𝑙𝑖𝑑(𝐿, 𝑆1)∧ ∀𝑆2 𝑉𝑎𝑙𝑖𝑑(𝐿, 𝑆2) ] What is the difference between the two? Well, in this sentence the for all applies to everything inside the parentheses. This sentence has a conjunction, so let's see what that would mean. In this sentence, the binding is the same: the for all applies to everything in parentheses. This exists applies to this thing; and this for all applies to this thing. In both cases, L is still bound—which means that as was determined in (1) above, L inside here is bound. For every licence L, there is a state in which the licence is valid, and there is a state in which the licence is valid in all states. For any licence L, there is a state in which L is valid and there is a state in which L is valid in every state. But this does not mean that the same applies to every single licence. It is important to know that not every license is valid. For example, if you go to California and try to drive with too much alcohol in your bloodstream, you will find yourself with an invalid license. It is the first thing that was a problem. For every licence, there is a state in which it is valid. Well, that's simply not the case. Already the first conjunct makes it invalid. The part that says it is valid in a state was the antecedent; therefore, if it is valid in a state, then it is valid in all states. So that said, for every licence—if it is valid in a state—this says that all licences are valid in some states. But this is not always true; not all licences are valid. Therefore, there is a difference between these two and the difference is meaningful in terms of validity of licenses and so forth. ( ) ( ) (∀𝐿) ∃𝑆1 𝑉𝑎𝑙𝑖𝑑(𝐿, 𝑆1) ⇒ ∀𝑆2 𝑉𝑎𝑙𝑖𝑑(𝐿, 𝑆2) We don't have these brackets here, so let's look at what's going on. For every licence there is a state that it is valid in. There is a line here. The for all and the exists do not apply to that line; they apply to what comes next. And there was no bracket, so it does not include here. So what this actually says is that for every licence there is a state in which the licence is valid. So what this really says is that all licences are valid somewhere. That isn't true, and it isn't what we want to say either; if that were the case then for all states 2 this would say that L is valid in all states 2. In fact, as we'll see later, there are many kinds of licences that aren't valid in any jurisdiction. So the antecedent of this conditional is false and the conditional itself is undefined. If you know what L is, you can assign meaning to it. And once you've assigned a meaning to it, if that L refers to my license and if that L is my license then we would have a true conditional. But as it stands, you've just got an unbound or free variable—it doesn't have any real meaning until you assign a value to it. [ (∀𝐿)(∀𝑆1)(∀𝑆2) 𝑉𝑎𝑙𝑖𝑑(𝐿, 𝑆1) ∧ 𝑉𝑎𝑙𝑖𝑑(𝐿, 𝑆2) ] Thus, for every licence and for all pairs of states, the license is valid in one state and the licence is valid in two states. That means all licences are valid in all states. Again, this is not the case in the United States because you can have invalid licenses so we've got something that's actually false. There's redundancy here because the second S adds nothing new. It simply says that for all licences and for all states, the licence is valid in that state and it's valid in the other state. So we could just scrap that and scrap that bit and we'd have the meaning without any of that stuff. In order to distinguish between cases that are commonly confused by beginners, let me write down a transcription of what it means. In short, for all x, if P(x) then Q(x). If P is true then Q is true. (∀𝑥)[𝑃(𝑥) ⇒ 𝑄(𝑥)] For every real number, if that number is nonnegative, then it has a square root.This occurs frequently in mathematics.Once you've established that "for all" there is a relationship between P and Q. (∀𝑋)[𝑃(𝑥) ∧ 𝑄(𝑥)] It says that every x satisfies P and Q. That's a fairly strong statement, because it only means that if something is an x, then it also satisfies P and Q. So it's not too frequent, because you could equally well say for all x P(x) and for all x Q(x), which is equivalent to saying everything satisfies P and everything satisfies Q. ∀𝑥𝑃(𝑥) ∧ ∀𝑥𝑄(𝑥) Notice that the statement above is valid, because of the binding from the quantifier for all to what's next to it. The quantifier for all cannot bind anything else in this case. This is because in mathematics, such a statement would never appear unless it were confined by parentheses indicating which variables are being quantified over; however, since an overlap would be confusing here and therefore impractical, this statement does not require any additional parentheses. (∃𝑥)[𝑃(𝑥) ∧ 𝑄(𝑥)] This says that for any x, if P of x and Q of x, then there is an x that satisfies both P and Q. This is quite a strong statement, as it says you can find a single x that has both property P and property Q. So you can find an x that has the property P and has the property Q. So it's strong simply because it makes a claim about all x satisfying both properties. (∃𝑥)[𝑃(𝑥) ⇒ 𝑄(𝑥)] There is at least one x such that if P of x, then Q of x. If you see yourself writing an existence with an implication, the chances are very high that you've sort of got confused. This says for every x if it satisfies P, then it satisfies Q. Now that's making a strong statement. For every x, there's an implication. This simply says, there's one x for which there's an implication. Well, in a sense, it's almost vacuous then. One thing to say for example is that if you can find an x that does not satisfy P, in other words, you can find an x anywhere for which P of x is false. If you can find an x anywhere for which P of x is false then you have a conditional that's necessarily true. If you're trying to make an existence statement, then you can do so simply by finding an x that does not satisfy P. Because if you can make that part false, the conditional becomes true. So if you're trying to make a strong statement, forget that one. On the other hand, if you see an implication following the word exists and an existence claim comes before it, then chances are you've gotten confused. In general, these two symbols are significant: If a formula has either of these two symbols, it is usually significant because it reduces to two separate claims or possibilities. However, this one is less significant because it really just reduces to the two separate things. This one is also pretty weak because it really just restates what's already stated in the previous sentence. Therefore, if you see an implication following exists with an existence claim before it in your own writing, flag these parts and ask yourself if what you meant is actually there.