So last time I mentioned about going over some of the basics in mathematical logic, and what vacuous truth meant. But in order to understand that, two simpler concepts must be understood, which are modus pollendo ponens and modus tollendo tollens. These two terms form the most basic and intuitive understanding of implicative statements, and I will deal with them in a more formal manner.
So what is modus pollendo ponens
? Actually, the name is pretty long, as you can see, so the most common form of the phrase that most people use (including mathematicians) would be modus ponens (and the same goes to the other phrase, shortened to modus tollens). But what modus ponens means is just this: for statements A and B, whenever both “A implies B” and A are true, it is necessary for statement B to be true as well. This is the essence of the literal translation of the Latin phrase, which was roughly to prove assert something to be true due to other true assertions. And so following in the same vein of thought, modus tollens acts in the exact opposite way – whenever “A implies B” is true but B is false, then A is false. In a nutshell the summary goes as follows: starting with “A implies B” is true, then the rule when A is true, then B is true is called modus ponens, and the rule when B is false, then A is false is modus tollens. Doesn’t this all sound blatantly obvious? Well, at least the first rule. It’s how we always thought of what if-then statements worked, especially in a world where causality is mostly taken as a unquestioned assumption. Modus tollens isn’t all that hard to figure out once the rule is considered at hand. Here’s an example to consider:
Suppose that you had a unlabeled bottle in front of you, and you are very thirsty. You want to drink it, as you’re dying from thirst, but do not want to die in a more painful way if the bottle happens to contain poison of some kind. Thus the thought “if it is poisonous, then I’ll die by drinking it” isn’t all that irrational. But then more time passes along and you decide “oh what the heck” and drink that bottle of mystery liquid. Hours and hours pass, but you don’t die, you actually feel better. Thus since you didn’t die, then the bottle was shown to be not poisonous.
I know many people will look to this example and then try to point out “oh, but it’s a slow working poison, he wouldn’t have know yet” or other trivial arguments against it. However, in the context of mathematical logic, and if the statement “he didn’t die” be taken as a rigorous mathematical fact, these are some things that be drawn out of the example:
- The drink is poisonous is either a true or false fact, and truth and falsity is exclusive of one another, (meaning that this statement is a proposition)
- Poisonous drink implies death (from now on, a true proposition will be stated as is – only when it is false will I say for a given proposition A, “A is false“
- No death occurred
Following from these statements then, we can consider the case when “the drink is poisonous” is true and see where that leads us. If that is true, then by modus ponens death should had occurred, but I had already noted that no death occurs. Since a statement cannot be both true and false at the same time, then the case where “the drink is poisonous” can never happen, so the statement must be false as well. This is the principle of modus tollens – using modus ponens in a state of contradiction to create a rule in a counterintuitive manner. Doesn’t this also sound familiar? Anyone with a basic high school math background should recognize this concept with another term: contrapositive. But in any case, now you know that when people use these implicative, or even accusatory, statements, we now have to tools to diffuse any bad logic employed by them. But how to stop people from being irrational is another story… >,<
Next time: modus tollendo ponens and modus ponendo tollen
). While I enjoy talking to people spontaneously, my short-term memory is crappy, and anything cool or meaningful insight I think of is lost if I don’t record it down soon enough. That and my thoughts are so jumbled sometimes that writing helps organize my thoughts as well.
