Be Reasonable – Part 1
Much of this site’s content centers around the attempt to put together reasonable arguments in support of one position or another. We haven’t really spent much time exploring what ‘reasonable argument’ are, however, and one of the quickest ways to end a potentially constructive conversation is to let basic terms go unexamined. It might seem like an understanding of logic and reasoning should be common sense, but within the philosophical arena, there are fundamental differences about the very nature of logic and reasoning that aren’t just academic hand-wringing. Those differences often spill out into people’s everyday discourse – as do the errors that drive logic professors crazy. So what I’d like to do is start a sort of primer to basic structures of logic, and touch on some of the related issues.
What I’m discussing here is very basic, formal deductive logic. It’s formal not in the sense that it wears a three-piece suit, but in that it’s concerned with the form, or structure, that an argument takes – how its parts fit together, and what does and does not follow from given pieces of information. A logician will often work with symbols rather than actual arguments to keep this emphasis clear. It’s deductive because it works from given information to determine what other facts absolutely must be true – there are no shades of gray or degrees of probability. Of course, not every argument that one encounters will fit neatly into a particular formal structure or be amenable to a strict yes-or-no evaluation. Much of the reasoning we do in everyday life is of the inductive variety, which factors degrees of probability into the mix. But many of the underlying principles are the same, which makes the study of formal deductive logic worthwhile.
Arguments are built from statements. A simple statement is a basic piece of information that can’t meaningfully be broken down into smaller pieces of information – it’s like the atom of the logical world. “My name is Dave� is an example of a simple statement. Statements have ‘truth-values’ assigned to them. Most basic logic uses a two-value truth system, so this is a fancy way of saying a statement is either true or false. (It is possible to have a system of deductive logic that incorporates more than two truth values, but that’s more complexity than we need for this discussion.)
A complex statement is built from combining simple statements together using one or more of a small set of operations which establish some relationship between the simple statements. With one exception, each operation connects two statements together and makes some claim about the truth-values that are assigned to the two. The basic operations are:
NOT (~)
Here’s the exception. NOT only operates on one statement, and its job is to reverse the truth value of the modified statement. “I do NOT live in Pittsburgh� means that the statement “I do live in Pittsburgh� is not true.
AND (&)
An AND statement is true when both of the connected statements are true. “My name is Dave AND I live in Philadelphia� is a true statement because “My name is Dave� and “I live in Philadelphia� are true statements. “My name is Dave and I live in Pittsburgh� would be a false statement.
OR (v)
An OR statement is true when at least one of the connected statements is true. “My name is Dave OR I live in Pittsburgh� is true; so is “My name is Dave OR I live in Philadelphia.� OR does not state that one statement is true and the other false, even though that’s usually how we tend to think about it.
IF-THEN (the symbol is a sideways U)
An IF-THEN is true when, in every instance that the IF statement is true, the THEN statement is also true. “IF it is raining, THEN I will take my umbrella� is true if there is never an occasion where it rains (making the first statement true) and I do not take the umbrella (making the second statement false.) This is one of the areas where the distinction between deductive and inductive reasoning makes itself felt. In an inductive argument, if I forget my umbrella once, you can still attach some weight to the statement if it describes a general pattern. With deductive reasoning, one exception is all it takes to render the statement false.
There are two kinds of statements in a deductive argument. The premises are the statements that we take as given; the conclusion is the statement that we claim follows from the premises. A deductive argument is considered valid when every time the premises are all true, the conclusion is also true. A valid argument doesn’t necessarily ‘prove’ the conclusion, though, because if it turns out one of the premises isn’t true, then the whole thing falls apart. An argument that has a valid structure and premises that are all true is called a sound argument, and that’s the goal of any process of deduction.
For example, the following:
Premise: My name is Dave.
Premise: I live in Pittsburgh.
Conclusion: My name is Dave and I live in Pittsburgh.
is a valid argument – if you assume both premises are true, the conclusion has to be – but it’s unsound, because the second premise is false. Change ‘Pittsburgh’ to ‘Philadelphia’ and you have an argument that is both valid and sound.
These five basic operations lend themselves to a number of argument structures, and also contain the seeds of some common misconceptions about logic and reasoning. I’ll discuss both next time.
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.