A crash course in logic

[aperson]

I was in IRC and someone posted an argument proving god to be true. I was bored and decided to deconstruct it and point out where the argument fails using formal logic.

Now for the crash course:
In predicate logic (PL) Sentences are similar to things such as Ta. Now Ta can mean whatever I define it to mean, so let's define it to mean Adam is tall. you can also have predicates that have more than one input, let's say we have Tab. This could translate to Adam is taller than Barry (or Barry is Taller than Adam, it all depends on how you define it).

Additionally, in predicate logic you can have variables, variables are generally reserved to the last few letters of the alphabet; I will use w, x, y and z. Variables aren't actually elements of the language, like Adam or Barry, but they can stand for any object in the domain. Variables, connected with quantifiers are a fundamental part of PL.

Quantifiers are of the form Ax or Ex (actually the A is upside down and the E is backwards, but those aren't in the standard character set so I'm choosing to write them like this). Ax translates to 'For all x' and Ex translates to 'there exists at least one x.' If my universal domain was colors and Bx meant x is black, then the statement (Ex)Bx would be true (There exists at least one color which is black). (Ax)Bx would be false, because not all colors are black. If our universal domain was people, and Px means x is a person, then (Ax)Px is true... and as an extension of this Pa is true, and Pb, Pc, Pd, Pe, etc... because they are all people as specified by our universal domain.

statements in logic are made with the use of logical operands. They are &, \/, >, and <->. The actual look of the operands differs by preference, but the meaning between them is the same. A & B means that both A and B have to be true, A \/ B means that either A or B must be true. A > B means that if A is true, B must be true. A <-> B means that A is true if and only if B is true.

So the statement (Ax)Bx & (Ex)Bx would be false: Not all colors are black. Conversely (Ax)Bx \/ (Ex)Bx would be true, because there is at least one color which is black.

As an interesting case, (Ax)Bx > (Ex)Bx is also true, since the first statement is false the statement is automatically true. (Basically, if the 'if' in your if-then is false then it doesn't matter what the then is, the statement is true in logic). (Ax)Bx <-> (Ex)Bx is false, (Ax)Bx is F while (Ex)Bx is T.

Now, what we are going to be doing is constructing a derivation. In a derivation you attempt to start with an assumption, and then reach a conclusion by doing certain allowed operations on your assumptions. In derivations, each new assumption is marked by another scope line, so if I were to assume that some sentence A is true, I would write
| A

If I additionally assumed B was true, I would write
| A
| | B

If I start with A and come to conclusion B I can do this:
| A
...
| B
A > B
Do you see why? if A is my only assumption and I am allowed to reach B by our logical rules, then I know that if A is true then B is true. The simplest case is this:
| A
| A --- repetition
A > A

Or you could do:
| A
| A v B (v Introduction)
A > (A v B)
Which would read: If A is true, then A or B is true. The v Introduction is one of the logical rules; if you know that A is true, then no matter what B is, even if it is always false, A v B is true. This shows a fundamental aspect of all the rules in these derivations: they are truth-preserving


Now you have all the basics required to interpret the logical argument below. Consider it a game, see if you can make sense of the logical construction and understand why and where the contradiction arises.



-----------------------------------


1.God is that than whom no being is greater.
2.Assume God exists in the mind and physically.
3.That which exists physically is greater (contains more objective reality) than that which exists only in the mind
4.Assume God exists only in the mind.
5. This God would be less than than a God that exists in the mind and
6. This is contradictory, and due to this, statement 2 is affirmed.

Gx = x is a God, Kxy = x is greater than y, Mx = x exists in the mind, Px = x exists physically.

01: (Ax)[Gx > (Ay)~Kyx] --- Def. of god (claim 1)
02: | (Ax)[Gx > (Mx & Px)] --- Ass. (claim 2)
03: | (Aw)(Az){[Mw & ~Pw & Pz & Mz] > Kzw} --- Def. of objectivity (claim 3)
04: | [Ma & ~Pa & Mb & Pb] > Kba --- Universal elimination (3)
05: | | (Ax')[Gx' > Mx' & ~Px'] --- Ass. (claim 4)
06: | | Ga > Ma & ~Pa --- Universal elimination (5)
07: | | Gb > Mb & ~Pb --- Universal elimination (5)
08: | | | Ga --- Ass.
09: | | | | Gb --- Ass.
10: | | | | Ma & ~Pa & Mb & ~Pb --- > elimination (6,7), & Introduction
11: | | | | Kba --- > elimination (4) [FALLACIOUS]
We are required to build Ma & ~Pa & Mb & Pb to complete the > elimination to get Kba, we don't have Pb, we have ~Pb and we cannot make this elimination. Therefore no contradiction occurs and the proof is fallacious.

As an exercise, here is actually the completion of his proof (ie how he wraps it up after his mistake). It is significantly more difficult from here... But the hilarious part, another contradiction arises in the reductio ad absurdium construction.

12: | | | Gb > Kba --- > Introduction 09-11
13: | | Ga > Gb > Kba --- > Introduction 08-12
14: | | (Ga & Gb) > Kba --- Importation
15: | (Ax')[Gx' > Mx' & ~Px'] > [(Ga & Gb) > Kba] --- > Introduction 05 - 14
16: (Ax)[Gx > (Mx & Px)] > (Ax')[Gx' > Mx' & ~Px'] > [(Ga & Gb) > Kba] --- > Introduction 02 - 15
17: {(Ax)[Gx > (Mx & Px)] & (Ax')[Gx' > Mx' & ~Px']} > [(Ga & Gb) > Kba] --- Importation
18: {[Ga > (Ma & Pa)] & [Ga > Ma & ~ Pa]} > [(Ga & Gb) > Kba] --- Universal elimination x2, 17
19: | {[Ga > (Ma & Pa)] & [Ga > Ma & ~ Pa]} --- Ass.
20: | | Ga --- Ass.
21: | | Ga > (Ma & Pa) --- &E 19
22: | | Ga > (Ma & ~Pa) --- &E 19
23: | | Ma & Pa --- >E 20, 21
24: | | Ma & ~Pa --- >E 20, 22
25: | | Pa --- &E 23
26: | | ~Pa --- &E 24
27: | | Pa & ~ Pa &I 25, 26
Well well, this is a contradiction... we can do some rather ridiculous stuff when these come up:
28: | | Q & ~Q
29: | Q & ~Q
30: (Ax)[Gx > (Mx & Px)] > (Q & ~Q)
Since you have an if then with a contradiction on the right, the left can never be true:
31: ~(Ax)[Gx > (Mx & Px)]

Therefore, the assumptions made in the original claim are actually inconsistent. How ridiculous is that?

Finally, let this be a lesson that even though an argument might appear immediately consistent in the English language, pointing it out as such formally can actually be a much longer process.

[lol_usernme]

Originally Posted by aperson

1.God is that than whom no being is greater.
2.Assume God exists in the mind and physically.
3.That which exists physically is greater (contains more objective reality) than that which exists only in the mind
4.Assume God exists only in the mind.
5. This God would be less than than a God that exists in the mind and
6. This is contradictory, and due to this, statement 2 is affirmed.

Wow, this sounds a lot like the Ontological argument, which is centuries old and which is completely ******y and gay.

It's never clear in the argument what "greater" means, and why existing in the mind would make God less great. If God existed only in the mind, then there would be nothing to assign greatness to, since he would not exist. The whole thing is just a semantic mess, and it doesn't take formal logic to point out vagueries in the words being used.

[aperson]

Originally Posted by lol_usernme

Wow, this sounds a lot like the Ontological argument, which is centuries old and which is completely ******y and gay.

It's never clear in the argument what "greater" means, and why existing in the mind would make God less great. If God existed only in the mind, then there would be nothing to assign greatness to, since he would not exist. The whole thing is just a semantic mess, and it doesn't take formal logic to point out vagueries in the words being used.

The amusing thing about the formal proof, though, is that I give him as much credit as possible for what 'greater' could mean and the argument still falls on its face in a variety of places.

[flamingspinach]

Originally Posted by lol_usernme

it doesn't take formal logic to point out vagueries in the words being used.

Right you are. But the word is "vagaries", and it doesn't mean quite what you think it does.

-fs

[Squeek]

Flaming, there are times when spelling and grammar are not important to correct.

This is one of them. It's assumed that he means multiple instances of a vague concept.

In other news: mmm, discrete math. My new favorite math.

[Kilgamayan]

This isn't discrete math, it's symbolic logic.

It's even better than discrete because of how piss easy it is.

[Squeek]

This was the first thing I did in Discrete Math. It's a part of it.

Unless my professor was just messing with me.

[flamingspinach]

Originally Posted by Squeek

Flaming, there are times when spelling and grammar are not important to correct.

This is one of them. It's assumed that he means multiple instances of a vague concept.

In other news: mmm, discrete math. My new favorite math.

.

It strikes me somehow as extremely hilarious that you find a thread about LOGIC in a forum called CRITICAL THINKING to be a place in which one is obliged to make ASSUMPTIONS about what other people mean! Really, now. Pull yourself together.

-fs

[Squeek]

With regards to spelling, yes.

I'd use "vagueries" to describe multiple instances of "vaguery" as well.

He could have re-written it, but you understood what he meant anyway because nobody has a clue what "vagaries" are without looking it up.

[flamingspinach]

Just because you don't know a word doesn't mean nobody else does. "Vagary" is a relatively common word - it most often appears in phrases such as, most often, "the vagaries of chance"; "the vagaries of the American legal system"; "the vagaries of the Royal Personage's whims"; the list goes on. Read more books.

-fs

[Squeek]

Thank you for correcting one unprovable statement by using another.

Now, back to the topic at-hand.

[FoJaR]

more like: | | Ga --- Ass.

[flamingspinach]

Originally Posted by Squeek

Thank you for correcting one unprovable statement by using another.

Now, back to the topic at-hand.

I corrected an error in a post pertaining to the topic at hand. My "unprovable statement" about how common a word is was in response to a post - yours, in fact - that was not pertaining to the topic at hand. You're the one who went off topic in the first place, so I'd thank you not to make supercilious comments.

-fs

[slipstrike0159]

And yet you persist in the off-topic arguement, which is the more stupid one?

In any case, logic doesnt need to be so complicated (even if you dont think it is). You can use a very basic logic to prove any side on an arguement but that doesnt prove the entire side correct.

[nforcer06164]

Originally Posted by slipstrike0159

In any case, logic doesnt need to be so complicated (even if you dont think it is). You can use a very basic logic to prove any side on an arguement but that doesnt prove the entire side correct.

I did something similar to this when we had this very same discussion about a year ago.