Although there are a lot of intimidating mathematical looking signs in this post, I’ve explained the basics of logic in a simply manner.
After spending some time establishing that there is some degree of absolute truth, we then must ask ourselves what are the laws of logic (or if you prefer, the rules of reason) and to what degree does reason and religion go together? These two questions are important as it is vital for us to understand what defines what must logically be true to reality and whether those principles also hold true to religion.
Before we dive into the rules of logic, let’s go over some of the basic shorthand commonly used in logic:
“p” is the common letter used for the first propositional phrase (similar to using “x” in algebra, though any letter can be used). In our examples, we will use the phrase “It is Friday.”
“q” is the common letter used for the second propositional phrase (again, like using the letter “y” in algebra. I’m not sure why they could not use those letters as it would make life simpler). In our examples, we will use the phrase “We will party.” Subsequent letters in the alphabet are used if any more phrases are necessary.
“T” indicates that the statement is true. “F” indicates a statement is false.
“¬ or ~” (kind of like a pixelated gun) indicates the negation of, the opposite of, or “not.” So, ¬ p or ~p would be “It is not the Friday.” Or perhaps you might prefer the negation of the Friday, whatever floats your boat.
The floating period “•” is written as “p•q” or “p ∧ q” stands for a statement made that both p and q are put together in combination (the fancy word used is “conjunction”) with each other. Typically we would this in a word problem as “and.” So written out: “It is Friday and we will party.”
Conversely, there is also the opposing or disjunction phrase written as “p ∨ q” (“or”) that would be written as “It is Friday or we will party.”
In making a conditional statement, we write “p→q” or “if p then q.” If it is Friday, then we will party. This can also be written as p ⊃ q. We can also find out that if ~q, then ~p (if we don’t party, then it is not Friday).
Two common fallacies that can occur here to affirm the consequence (if p, then q. q is true, therefore p is) in the 1st example and denying the antecedent in the 2nd (“if p, then q. ~q, therefore ~p):
(Fallacy 1) We party, therefore it must be Friday.
(Fallacy 2) It is not Friday, therefore we do not party.
These statements are false because we can party everyday, not just because it’s Friday.
If we wanted to make a statement that is dependent on both clauses being true (a biconditional statement) or “p if and only if q,” we would write it either as “p ↔ q” “p ºq” or “p iff q” (iff = if and only if). This would then be written as “Today is Friday if and only if we party.” In other words, both statements either need to be true or not true in order for the statement to be true.
We can then create the following table, often called a truth table:
| p | q | ~p | p•q | p ∨ q | p→q | p ↔ q | |
| T | T | F | T | T | T | T | |
| T | F | F | F | T | F | F | |
| F | T | T | F | T | T | F | |
| F | F | T | F | F | T | T |
To read this table, simply start with the first two columns and use whatever statement you are making to come up with your answer. So, if we make the statement “if it is Friday, then we will party (p → q)” and if p is true (it is the Friday), then q is true (we will party). Note that the original statement must be true in order for the “if…then” statement to work; this logical statement won’t work if we have party poopers who may or may not party if it is Friday.
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In beginning to understand the premise of logic, we can then start to know the three classical laws of thought: the law of identity, the law of non-contradiction, and the law of the excluded middle.
The law of identity states that every thing is the same with itself and different from another: p is p and not ~p. Simply put, Friday will always be considered Friday and never Sunday. One may label Friday differently (such as when someone says “Friday” in another language), however it is still the same day.
I have already mentioned about the law of non-contradiction before, but I’ll also state it here: a statement cannot both be true and not true in the same respect at the same time (p cannot be ~p). In other words, it cannot be both Friday and not Friday at the same time. Last time I showed that a Buddhist cannot simultaneously desire to reach enlightenment as enlightenment requires someone to be void of both desire and pain.
Finally is the law of the excluded middle. With this law, there is no possibility for a 3rdstatement since p + ~p = 100%. Today is either Friday or not Friday, but there is not a 3rdpossible answer to the question, “What day of the week is it?” Frankly, people just want to know if they are going to be partying or not.
With these laws in mind, it will help us to further understand what truth is.
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Reason and Religion is a post from: Looking Toward The Future
