Critique of Logic

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Critique of Logic

Post by Remus Bleys on Sat Mar 22, 2014 7:15 pm

(p ≡ q) is logically equivalent to ((p ⊃ q) • (q ⊃ p)). What this means is that the statement "p if and only if q" is has the same truth value (truth value being is this statement true or false) as the compound statement "p if q" and "q if p." If we are given the truth value of p then we know the truth value of q and if we are given the truth value of q then we know the truth value of p - this is true for both of those statements.

The statement (p ⊃ q) is logically equivalent to the statement (~q ⊃ ~p). This is because If p then q, but we are told q is false or that ~q (not q) is true, then we know for certain that ~p is true. We know this because if (p ⊃ q) it would mean that q would be true if p was true, but since ~q is true, that means that q is false, and thus p is false as if p was true q would have to be true. So, if ~q then ~p is true then If p then q is likewise true.

Because the statement (p ⊃ q) is logically equivalent to the statement (~q ⊃ ~p), the statement ((p ⊃ q) • (q ⊃ p)) can be rewritten as ((~q ⊃ ~p) • (~p ⊃ ~q)), this is rewritten as ((~p ⊃ ~q) • (~q ⊃ ~p)). Replace ~p with x and ~q with y, the statement ((x ⊃ y) • (y ⊃ x)) is deduced. Considering that (p ≡ q) is logically equivalent to ((p ⊃ q) • (q ⊃ p)), ((x ⊃ y) • (y ⊃ x)) is logically equivalent to (x ≡ y), which we shall rewrite as (~p ≡ ~q). Thus, (~p ≡ ~q) is logically equivalent to (p ≡ q).

In logic, there is a bit of a controversy over the truth tables of ⊃, or of the If-Then statement. There are four different interpretations on what this statement can mean:
Spoiler:

1. p ⊃ q
T T T
T F F
F T T
F T F
This means that if p is true and q is true the statement is true.
If p is true and q is false the statement is false.
If p is false and q is true then the statement is true.
If p is false and q is false then the statement is true.
Thus, p ⊃ q is true if and only if p and q or not p and q or not p and not q. This can be logically rewritten as (p ⊃ q) ≡ ((p • q) v (~p • q) v (~p • ~q))
Spoiler:

2. p ⊃ q
T T T
T F F
F F T
F F F
This means that if p is true and q is true the statement is true.
If p is true and q is false the statement is false.
If p is false and q is true the statement is false.
If p is false and q is false the statement is false.
Thus, For p ⊃ q to be valid both p and q must be true. This can be logically rewritten as (p ⊃ q) ≡ (p • q)
Spoiler:

3. p ⊃ q
T T T
T F F
F T T
F F F
This means that if p is true and q is true the statement is true.
If p is true and q is false the statement is false
If p is false and q is true the statement is true
If p is false and q is false the statement is false
Hence, for p ⊃ q to be true, then either p is false and q is true or p is true and q is false. This can be logically rewritten as (p ⊃ q) ≡ ((~p • q) v (p • q))
Spoiler:

4. p ⊃ q
T T T
T F F
F F T
F T F
This means that if p is true and q is true the statement is true.
If p is true and q is false the statement is false
If p is false and q is true then the statement is false.
If p is false and q is false then the statement is true.
For this to be true, either both p and q are both true or p and q are both false. This can be logically rewritten as (p ⊃ q) ≡ ((~p • ~q) v (p • q)).

So, take each of these 4 methods and apply them to the fact that (p ≡ q) is logically equivalent to ((p ⊃ q) • (q ⊃ p)).
Spoiler:
1.
(p ⊃ q) • (q ⊃ p)
(T T T) T (T T T)
(T F F) F (F T T)
(F T T) F (T F F)
(F T F) T (F T F)
Spoiler:
2.
(p ⊃ q) • (q ⊃ p)
(T T T) T (T T T)
(T F F) F (F F T)
(F F T) F (T F F)
(F F F) F (F F F)
Spoiler:
3.
(p ⊃ q) • (q ⊃ p)
(T T T) T (T T T)
(T F F) F (F T T)
(F T T) F (T F F)
(F F F) F (F F F)
Spoiler:
4.
(p ⊃ q) • (q ⊃ p)
(T T T) T (T T T)
(T F F) F (F F T)
(F F T) F (T F F)
(F T F) T (F T F)
The truth table of a biconditional must be by definition this table provided below. In order to be logically equivalent they must match their truth tables.
Spoiler:
p ≡ q
T T T
T F F
F F T
F T F
From this, we can deduce that the only two methods to truly follow this are the rules of 1 and of 4. (Coincidentally the 2 and 3 system each equal the other as well). Take this sample problem:
Spoiler:

Let (p ≡ q) be the equation for It is Ozone (p) if and only if (≡) it is composed exclusively of O3 Molecules.
This is for the purpose of breaking down this graph and explaining what each mean
The biconditional statement truth is inside of the parentheses

The first method tells us that the statement:

If Ozone then O3 (a true statement) is True according to this system
If not Ozone then not O3 (a true statement) is True according to this system
If not Ozone then O3 (a false statement) is false according to this system
If Ozone then not O3 (a false statement) is true according to this system

The second method tells us that the statement:

If Ozone then O3 (a true statement) is True according to this system
If not Ozone then not O3 (a true statement) is False according to this system
If not Ozone then O3 (a false statement) is False according to this system
If Ozone then not O3 (a false statement) is False according to this system

The third method tells us that the statement:

If Ozone then O3 (a true statement) is True according to this system
If not Ozone then not O3 (a true statement) is false according to this system
If not Ozone then O3 (a false statement) is True according to this system
If Ozone then not O3 (a false statement) is false according to this system

The fourth method tells us that the statement:

If Ozone then O3 (a true statement) is True according to this system
If not Ozone then not O3 (a true statement) is True according to this system
If not Ozone then O3 (a false statement) is False according to this system
If Ozone then not O3 (a false statement) is False according to this system
Thus, F ⊃ T is T results in a contradiction, results in the absurdity of two incompatible statements both being true.
In addition, the very structure of logic, that of if t then t is true, means any two statements can be added together. If the author of this has used "remus bleys" as a username then ww2 happened becomes a true statement. Surely, both of these statements are true. Surely, it is logically correct. However, this is an instance of the fallacy Post hoc ergo propter hoc. For, if I did not take the username, ww2 would still have happened. Such a statement is absurd, Marxism obviously has no space for the metaphysics that would try to prove that my username had caused ww2.

The only consistent one which is always factually true is in the case of the Fourth system. Because all other systems rely on biconditionals, but their bi conditionals are false as, because take for instance (p ⊃ q) ≡ (p • q), which is by definition true, but logically false. Thus this is both true and false, and from that anything can follow. Thus logic becames pointless, unless it perhaps uses the 4th system. The 4th system however has its flaws in that i can have q without having p. "In order to get an A I have studied." I could have guessed and have gotten an A. Just because I have an A doesn't mean that I will have studied, but if I studied it does mean i will get an A.
This is the start of a critique of logic. We must not turn to logic to replace dialectics in our critique of that philosophy as well. Logic has shown itself to be absurd, and while it would be inane to totally and completely abandon logic, it must be seen as yet another philosophy that must be transcended, anything salvagable kept and the rest ditched, in our real materialist understanding of things. Nor is this universal in my thought.
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Re: Critique of Logic

Post by Karinpon on Sun Mar 23, 2014 4:58 pm

I can tell this post must be good because I became hopelessly lost in the first paragraph. For my part, I do not subscribe to logic.
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Re: Critique of Logic

Post by Remus Bleys on Mon Mar 24, 2014 9:52 pm

It's pretty shit to be quite honest.  I know a lot of its wrong but I don't really know why and there is a couple things I need to elaborate on. However the basics of the critique is sound, I sent it to some logic professor before I did anything with it and he okayed it.

edit: also I'm trying to make this simpler so anything you don't get please point out
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Re: Critique of Logic

Post by Karinpon on Thu Mar 27, 2014 2:04 pm

For example I have no idea what ⊃ nor ~ mean. I am basically uneducated, though, so I'd wait for another person's response.
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Re: Critique of Logic

Post by Remus Bleys on Fri Mar 28, 2014 6:41 pm

⊃ and ~ are just logical shorthands. ⊃ (horseshoe) just means If...then... and ~ (negation) just means not, and • is a way of saying "and."
so p ⊃ q means If p therefore q, and ~p means not p. Or, to use my previous examples, "If it is Ozone, then it is composed exclusively of O3 Molecules" would be rewritten as "Ozone ⊃ it is composed exclusively of O3 Molecules" or "If p then q." Finally both of these can be rewritten as "p ⊃ q".

(p ≡ q) simply means "p if and only if q", or "Ozone if and only if it is composed exclusively of O3 molecules." This is the exact same as (q ≡ p), as "it is composed exclusively of O3 molecules if and only if Ozone" is simply a way of rewriting the original statement, they mean the same thing. This is what is meant by logical equivalence, it means the two are equivalent in all of their truth values, it means that they are interchangeable.

If one breaks down the statement "Ozone if and only if it is composed exclusively of O3 molecules" ("p ≡ q") we see that what can be formed from this is the statement: "If Ozone then it is composed exclusively of O3 molecules" (which is shortened to "p ⊃ q") and the statement "if it is composed exclusively of O3 molecules then it is Ozone" (which will be rewritten to "q ⊃ p"). This means that the "p ⊃ q" and "q ⊃ p," which will be shortened to ((p ⊃ q) • (q ⊃ p)), is logically equivalent to (p ≡ q). This is because if we are given that the object is Ozone, the biconditional statement (p ≡ q) informs us that it is composed exclusively of O3 molecules, and vice versa. In the same way, if we are given that it is Ozone, the horseshoe statement (p ⊃ q) informs us that it is made exclusively of O3 molecules; and vice versa with (q ⊃ p).

In a similar vein, the horseshoe statement "p ⊃ q" is logically equivalent to the phrase "~q ⊃ ~p," and this is because if we are given the fact that If p then q it necessarily means that if we are given p then q must necessarily be true (interestingly enough, if we are given q we are not necessarily given p*). Thus, since q must be true if p is true, then if we are told that q is false then we know that p can't be true.

Given the fact that that "p ⊃ q" is logically equivalent to the phrase "~q ⊃ ~p," and that ((p ⊃ q) • (q ⊃ p)) is logically equivalent to (p ≡ q), we can rewrite ((p ⊃ q) • (q ⊃ p)) as ((~q ⊃ ~p) • (~p ⊃ ~q)). This only makes sense, as If we are told that it is not ozone (~p) then we know that it can't be made exclusively out of O3 molecules, and this is shown by (~q ⊃ ~p), but it is also in (p ≡ q), as it can only be O3 if and only if Ozone, thus if it isn't Ozone, it isn't made exclusively out of O3.

I hope this reasoning for why things are logically equivalent, and what the symbol means, helps clarify all the "raw" logic that is in my OP, demonstrating why only systems 1 and 4 work for the Biconditional statement.


*This shows yet another problem in logic. Say that we are given that q is true but p is false. Thus, ~T ⊃ ~F means F ⊃ T, which is denied by the 2nd and 4th systems. To deny that F ⊃ T is a true statement is to state that q can't be true when p is false. If we set p to "you don't spend money on games" and q to "you will have more money," we can get the statement "If you don't spend money on games, you will have more money" or "p ⊃ q." However, it is fairly possible for p to be false but q is true, for instance you could continue to buy games, but you receive a raise at work - or whatever. The point is that just because q is true, we still don't know the value of p.
From this we deduce that only system 1 can even be taken seriously, and I will have to get back to my critique of system 1.
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Re: Critique of Logic

Post by madlogicbrah on Sat Apr 12, 2014 4:20 am

If logic is absurd, then how can you formulate a critique of it? Any critique of logic, requires logic and would therefore undermine itself.

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