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All You Need to Know Logic (Part 1 - Relations in Logic)
By philobean | June 25, 2007
Everyone needs logic because everyone uses it, consciously or as a matter of habit. Clearly, this argument is a little circular, arguing in effect that we need logic because we need logic. Yet this is an immutable and apparent truth. We need logic because our language is grounded on both rules of syntax (grammar) and rules of sense (logic). We need logic because our world is seen as intrinsically logical (we poke fun at every ironic, paradoxical and oxymoronical thing that happens around us). We need logic because we rely on it when we make decisions, when we judge circumstances and other persons, and when we assess the moral consequences of thoughts and actions.
But logic is simple enough. This is the first of a three part post on the basics of formal logic. This post covers logical relations, the second rules of replacement, and the last rules of inference. With these, you should be able to do just one important thing–determine the validity or invalidity of an argument. That’s it.
So, let’s begin.
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Axiom 1: Statements are either true or false depending on whether or not it conforms to a state of affairs in reality. [Note: While there are some statements that are neither true nor false, we ignore these types of statements in this discussion. To read more about the other type of statements, follow the following link to J.L. Austin’s Speech Act Theory.]
Axiom 2: Arguments (which are a series of logically related statements) are either valid or invalid. They cannot be either true or false. Arguments are valid if its conclusion(s) follow logically from its premises. Premises are the assumptions from which conclusions are drawn out logically. For example, in the argument,
If it rains, then the ground is wet.
It rains.
Therefore, the ground is wet.
The first two statements are premises while the last is the conclusion.
Axiom 3: Valid arguments whose premises are true are sound arguments.
Based on the preceding three axioms, you can now use the words true, valid and sound correctly. Please feel free to instruct others who seem to be using these words interchangeably of the precise use of each.
So, then, on to the four types of logical relations. They are simple, commonsensical and are the following: And, Or, Condition, and Equivalence.
Samples:
- (And) It rained yesterday AND it will rain today.
- (Or) It rained yesterday OR it will rain today.
- (Conditional) If it rained yesterday, then it will rain today.
- (Equivalence) It rained yesterday if and only if it will rain today.
These logical relations connect statements (which, remember, are either true or false). Thus, we can determine whether the compound statement (the whole combined statement) is either true or false depending on whether each statement comprising it is either true or false and the logical relation that connects these statements together. Here are the very simple rules.
AND Logical Relation
| Statement 1 | Statement 2 | Combined Statement |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
OR Logical Relation
| Statement 1 | Statement 2 | Combined Statement |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Conditional Logical Relation
| Statement 1 | Statement 2 | Combined Statement |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Equivalence Logical Relation
| Statement 1 | Statement 2 | Combined Statement |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Consider these relations for a while and why the combined relations have the assigned truth values. It may seem odd at first but all the results should be quite commonsensical (after all, we are all ‘logical’ beings, pun intended).
Topics: Academic |