Be Reasonable – Part 2
Picking up on our discussion of deductive logic, the five basic operations we discussed last time have certain properties that make one statement identical to another – there are multiple ways to express the same basic idea. This is important because one way of phrasing a statement might suggest or make clear a way of developing the argument that a perfectly equivalent phrasing might not. Some of these are very basic, such as the property of commutation – ‘p v q’ is the same thing as ‘q v p.’ Others are obvious, like the rule of double negation: ‘~~p’ is the same thing as ‘p.’ But others are slightly more complicated.
The De Morgan rules, for example, illustrate the relationship between & and v. If you have the statement ‘~p v ~q,’ you can turn that into the statement ‘~(p & q).’ The parentheses mean that the ~ is applied to the entire statement enclosed within, so ‘~(p & q)’ means that “The statement ‘p & q’ is false.� Why is that? Because an & statement like ‘p & q’ is only true if the two statements connected by the & are both true. But our original statement, ‘~p v ~q,’ says that at the very least, either p or q is false. Likewise, ‘~p & ~q’ can be turned into ‘~(p v q),’ because in order for ‘p v q’ to be true, either p or q has to be true, and we know that they’re both false.
Beyond the rules of substitution are rules of inference. These are rules that say if we have some set of statements, we are allowed to infer some further statement. Unlike the rules of transformation, which are applied to single statements, rules of inference require a set. Some of the basics are:
Modus ponens
This is the basic rule that lets us make use of conditional, IF-THEN statements. If we are given:
1. IF p THEN q
2. p
We can naturally infer
3. Therefore, q
Because statement 1 tells us that any time ‘p’ is true, ‘q’ is also true, and statement 2 tells us that ‘p’ is, in fact, true. What we can’t do is go backwards, and say
1. IF p THEN q
2. q
3. Therefore, p
Even given the IF-THEN, it’s entirely possible for q to be true and p to be false. If I say, “If I am at the baseball game, I will be wearing my Phillies cap,� and then you see me at the mall wearing said cap, you wouldn’t say, “Oh, you must be at the baseball game.� This common mistake is referred to as affirming the consequent.
Modus tollens
This is another rule that works with conditionals, and is a counterpart to modus ponens. Its structure is:
1. IF p THEN q
2. ~q
3. Therefore, ~p
Statement 1 tells us that any time ‘p’ is true, ‘q’ is true. But statement 2 tells us that ‘q’ is false. So p must be false as well – because I know that if p were true, q would be as well. The only way that ‘q’ can be false is if ‘p’ isn’t around to make it true. There’s a rule of substitution that takes advantage of this called transposition. You can turn a conditional statement into its counterpart, or contrapositive; ‘IF p THEN q’ is equivalent to ‘IF ~q THEN ~p.’
There’s also a common fallacy associated with this rule of inference as well, called denying the antecedent:
1. IF p THEN q
2. ~p
3. Therefore, ~q
Again, look at my Phillies cap statement above – ‘I’m at the baseball game’ is false, but ‘I’m wearing the hat’ is true. And there’s absolutely nothing wrong with that. This fallacy gets a lot of play in modern conversation. Someone will say, “If the Eagles stop the run, they’ll win the game. The Eagles didn’t stop the run, so therefore they lost the game.� The Eagles showed the error of that conclusion throughout the regular season. You also see it in more serious circumstances than sports, where someone will argue that a certain number of conditions indicate that we should take some course of action. Someone then argues that one of the conditions is false, and therefore the suggested course of action is flat out wrong. This is not good reasoning, because there may well be some other reason to take the course of action. Rather, it’s better to say that if we show the antecedent to be false, there’s no information one way or the other about the consequent – more research is needed.
Hypothetical syllogism
This rule helps us relate conditional statements to one another. Its structure is:
1. IF p THEN q.
2. IF q THEN r.
3. Therefore, IF p THEN r.
From statement 1, we know that every time p is true, q is also true. Statement 2 tells us that every time q is true, r is true. Therefore we can skip the middle man and say that every time p is true, r is true. We’re basically making a train of conditional statements, where the consequent of one statement is the antecedent of the next, and in the end showing the connection between the first and last items in the train.
Constructive dilemma
Less often used, but still helpful, is this rule which requires three initial statements to make an inference:
1. p v q
2. IF p THEN r
3. IF q THEN s
4. Therefore, r v s
This rule is usually used as a way of rephrasing a disjunction. It may seem obvious, but sometimes the second disjunction is the one you really want to consider, and so the rephrasing is important. For example,
1. I will pass this course or fail it.
2. If I pass, I will graduate on time.
3. If I fail, I won’t be able to continue my studies.
4. Therefore, I will graduate on time or I won’t be able to continue my studies.
Statement 4 does much more to illustrate the gravity of the potential outcomes than statement 1, so that’s probably the one I want to keep in mind. Any time someone is making an argument as to whether we should pass or reject some proposal, it’s likely that there’s a constructive dilemma involved somewhere, to remove the focus from the technical matter of voting yea or nay and onto the practical consequences of the decision.
This is not a fully comprehensive list of the rules of deductive logic or how they are used; there are some very basic rules I didn’t spell out because they’re fairly obvious and we’re not in a setting where we’re going to be making technical proofs. In addition, all of these rules apply to what’s called ‘sentential logic,’ which is obviously concerned with the relationships between sentences. There’s another dimension to deductive logic called ‘quantitative logic,’ which is more concerned with what items belong in certain categories and what properties those categories and items have. The famous syllogism associated with Plato is an example of quantitative logic:
1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.
The first statement introduces a class – men – and then ascribes a property to that class – mortality. The second statement says that a particular thing – Socrates – is a member of the class mentioned in the first statement – men. Therefore, the third statement concludes that the particular thing in question – Socrates – must have the property mentioned in the first statement – mortality. The symbols and rules involved in deductive quantitative logic are probably beyond the scope of the discussion here, although we can certainly discuss it if there turns out to be a clamor for such a conversation.
We’ll wrap this whole conversation up next time with some further discussion of inductive logic and the fallacies sometimes associated with it, and exactly how we should treat these rules of logic.
About Dave Thomer
Born and raised in Philadelphia. Studied philosophy at Fordham University in the Bronx, where I made many great friends and met my wonderful wife. We moved back to Philadelphia, where I studied philosophy some more and became a teacher. Online and off, I'm looking for ways that we can be better citizens and build stronger communities.