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.::oDrew -> A Consice Guide to Symbolic Logic (5/18/2009 12:27:44)
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A Concise Guide to Symbolic Logic
Everyone uses logic –often without even realizing it. However, a much smaller percentage of the population takes it upon themselves to learn and understand the precise intricacies and mechanics used in so-called “formal” logic. In this guide, I hope to provide you with at least a basic understanding of various symbolic logic systems, so that perhaps from there you may go on to greater depths of logical comprehension.
I. Important Terms &Definitions
Logic – Simply put, “logic” is the tool we use to construct, analyze, evaluate, affirm, and/or counter arguments. A debate without logic is like a football game without rules or referees – there might be two teams with clear goals, but without any rules, there’s no objective way of deciding who the winner is.
Argument – The term “argument” in logic does not have quite the same meaning as we commonly understand it. In logic, “argument” does not refer to a fight or disagreement, but rather, a set of premises leading to a conclusion. A “well-formed” argument has true premises that lead to a true conclusion, is clear in its meaning, and does not make jumps in reasoning.
Premise – A premise is the basic component of an argument. It does not prove anything on its own, but in a valid argument, each of the premises lead up to the conclusion. “If it’s raining, then the ground is wet” is a premise. This doesn’t prove anything, but it’s clearly true, and will lead us to the…
Conclusion – What the argument claims. If I want to prove to you that blood is thicker than water, my conclusion will be, “blood is thicker than water.” Many people “signal” their conclusion using words such as “therefore,” “so,” “ergo,” and so on. If we take the premise above, “if it’s raining, then the ground is wet,” and add another premise, “it’s raining,” the conclusion logically follows: “therefore, the ground is wet.”
Validity – Validity, that is, whether or not an argument is valid, is perhaps one of the most crucial aspects of logic. If an argument is valid, it follows all of the “rules” of logic, and vice versa. As a matter of fact, the goal in virtually any debate is to prove that your opponent’s argument is invalid; that his (or her) argument breaks the rules of logic. We’ll see much, much more about validity throughout this guide, but for now, be sure to remember this: An argument is valid if it would be contradictory to have the premises all true and the conclusion false. That is, if your argument is valid, and all of the premises are true, then your conclusion will be true. No ifs, ands, or buts about it.
Soundness – This is confusing for some people, but validity does not imply truth. “If I am tall, I am blue; I am tall, therefore I am blue,” is a valid argument (because it follows the rules of logic), but clearly does not make a true claim. An argument is sound if it’s valid and every premise is true.
Fallacy – Fallacies are common errors in logic. As long as you are able to determine the validity (or invalidity) of an argument, there’s no need to memorize fallacies, but being able to recognize some of the more frequently occurring ones might help you work more efficiently. As a quick example, the following argument commits the fallacy of “denying the antecedent:” If it’s raining, then the ground is wet. But it’s not raining. Therefore, the ground is not wet. As (I hope) you can see, although this argument might seem to be valid at first glance, it would be absurd to think that the ground cannot be wet if it isn’t raining.
I-a. Symbols
Here’s a quick list of some of the symbols used in symbolic logic. They will be explained in more detail later.
.: | Conclusion marker – used to indicate if and when a conclusion has been reached. Essentially, a shorthand “therefore.”
⊃ | If-then – Read “A ⊃ B” as, “If A, then B.”
≡ | If and ONLY If – equivalent to “A ⊃ B and B ⊃ A.”
• | Conjunction – “A • B” = “Both A and B.”
∨ |Disjunction – “A ∨ B” = “Either A or B.”
˜ | Negation – “~A” = “Not A”; “~(A • B)” = “Not both A and B.”
II. Propositional Logic
Propositional logic is the most widely used form of formal logic. Arguments in propositional logic rely largely on “if-then” statements; the example argument about rain uses propositional logic. Let’s take a closer look at this argument. First, we’ll place each premise and the conclusion on their own lines.
If it’s raining, then the ground is wet.
It’s raining.
.: (Therefore,) The ground is wet.
Now, as you might imagine, rewriting each premise of an argument on its own line can quickly become confusing and time-consuming when we’re dealing with lengthy, complex arguments. So to simplify our arguments, we replace each statement with a letter.
If R, then W.
R.
.: W.
To further simplify the argument, we substitute our logic symbols where appropriate:
R ⊃ W
R
.: W
And there you have it, a valid propositional argument in symbolic logic.
Let’s consider why this argument works. The first premise is an “if-then” claim; it says that the truth or existence of one thing causes and/or implies another. In this case, “R” is the antecedent – it is the cause, the origination. “W” is the consequent – it is the consequence, or effect, of “R.” The second premise affirms the existence of the antecedent. Without this affirmation, all we have is speculation. But now we know that R causes W, and we have R, so we must have W – and we have our conclusion. This argument form is known as modus ponens.
Now let’s try switching up this argument a bit. Consider the following:
R ⊃ W
~W
What conclusion, if any, follows these premises? Let’s think about it. If R exists, we must have W. The ground will be wet if it is raining. But apparently, the ground is dry. So where does this leave us?
R ⊃ W
~W
.: ~R
If you saw where this was going, great work. If you’re still not quite sure, here’s an explanation. Think of this argument like a math problem. The first premise makes the claim, “If 1 = b, then 1 + b = 2.” Sounds logical, and is. But the second premise says, “1 + b ≠ 2.” Well, if 1 + b ≠ 2, then “b” cannot equal 1, otherwise 1 + b would equal 2. So clearly, it’s not the case that 1 = b. Make sense? This argument form is called a modus tollens.
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