Formal logic question from beginner

[humanaction]

Hello, I am a complete novice at formal logic and so far removed from mathematical study that I have no idea how the following should be interpreted, or read, and what its implications are:

(a∨~a)∨~a⇒a∨(~a∨~a)⇒(a∨~a)⇒((a∧⊤)∨~a)⇒((a∧(b∨~b)∨~a)⇒(((a∧b)∨(a∧~b))∨~a)⇒
((a∧b)∨((a∧~b)∨~a))⇒(b∨(~b∨~a))⇒((b∨~b)∨~a)⇒⊤∨~a⇒(a∨~a)∨~a

Wikipedia tells me that the following symbols are; and “read as”:

() precedence grouping; “everywhere”
~ negation; “not” (though the context in which I’m getting this uses ~x to mean “the string x”)
∧ logical conjunction; “and”
∨ logical disjunction; “or”
⇒ material implication; “implies”
⊤ tautology; “top”

I realize I’m way out of my league, and that’s why I came here. If someone could generously explain this to me, and if it’s wrong, where it’s wrong, it would be much appreciated. I feel like this is pretty basic logic, but if there’s something you’d recommend I look into or research so I can understand this on my own, that would work too. Thank you.

P.S. Is there a mathematics forum better than physicsforums.com where I might better post this?

 

[Stephen Tashi]

(a∨~a)∨~a⇒a∨(~a∨~a)⇒(a∨~a)⇒((a∧⊤)∨~a)⇒((a∧(b∨~b)∨~a)⇒(((a∧b)∨(a∧~b))∨~a)⇒
((a∧b)∨((a∧~b)∨~a))⇒(b∨(~b∨~a))⇒((b∨~b)∨~a)⇒⊤∨~a⇒(a∨~a)∨~a

It isn't clear from your post whether you have made any progress in interpreting the above statement. It would be best if you asked a specific question about the first place that you have some confusion.

We can beginning at the very beginning with
$(a \lor \neg a) \lor (\neg a)$

$a$ represents a specific statement, such as "7 > 3" or "The USA has 50 states" or "9 > 100". The statement is either true or false. Statements do not contain variables. So "x > 3" is not a statement unless you have established one specific value for x. (When a variable is present in an expression, you have a "propositional function" instead of a statement.) You can make more picturesque verbal examples of statements. For example "Tom is lazy". But if you do that you must pretend that this statement is definitely true or false. "Tom" should not be a variable. There should be no question of him being "somewhat lazy and somewhat not lazy" - nothing wishy-washy is allowed.

The expression $(a \lor \neg a) \lor (\neg a)$ says " ($a$ or not $a$) or not $a$.

To make an example of that we could say:
"(7 is greater 3 or 7 is not greater than 3) or 7 is not greater than 3".

You can also make an example where $a$ is a false statement, such as
"(the USA has exactly 40 states or the USA does not have exactly 40 states) or the USA does not have exactly 40 states"

Is this much clear?

How did you bump into the problem of interpreting formal logic anyway?

P.S. Is there a mathematics forum better than physicsforums.com where I might better post this?

No, of course not! - not that I know about, anyway. Most other math forums will want you to ask very specific questions. If you can be specific, there are some other good math boards.

 

[Bacle2]

Hello, I am a complete novice at formal logic and so far removed from mathematical study that I have no idea how the following should be interpreted, or read, and what its implications are:

(a∨~a)∨~a⇒a∨(~a∨~a)⇒(a∨~a)⇒((a∧⊤)∨~a)⇒((a∧(b∨~b)∨~a)⇒(((a∧b)∨(a∧~b))∨~a)⇒
((a∧b)∨((a∧~b)∨~a))⇒(b∨(~b∨~a))⇒((b∨~b)∨~a)⇒⊤∨~a⇒(a∨~a)∨~a

Wikipedia tells me that the following symbols are; and “read as”:

() precedence grouping; “everywhere”
~ negation; “not” (though the context in which I’m getting this uses ~x to mean “the string x”)
∧ logical conjunction; “and”
∨ logical disjunction; “or”
⇒ material implication; “implies”
⊤ tautology; “top”

I realize I’m way out of my league, and that’s why I came here. If someone could generously explain this to me, and if it’s wrong, where it’s wrong, it would be much appreciated. I feel like this is pretty basic logic, but if there’s something you’d recommend I look into or research so I can understand this on my own, that would work too. Thank you.

P.S. Is there a mathematics forum better than physicsforums.com where I might better post this?

The interpretation often depends on the context. Some general issues to consider,tho,

are: is there a model for my sentences ? Basically, a model is a semantic image of your

syntactic set. Your sentences are basically formulas that follow precise syntactic

rules,without a natural/canonical/intrinsic meaning associated with them. An

interpretation is the standard method that I know of to assign meaning to sentences.

 

[humanaction]

Is this much clear?

How did you bump into the problem of interpreting formal logic anyway?

For the most part. I have zero formal logic and little mathematical background, so it's taking awhile for this to sink in. My philosophy class is telling me that such formalistic logic has no bearing on reality. He says it's because this logic deals not with the content of the strings, or how (if at all) they work, but just in their external form. He says Gödel proved its inefficacy in mathematics stating that these systems always end up:

1. the system would be too simple to decide the truth or falsehoods of all statements of arithmetic. or else
2. the system's postulates would eventually generate a contradiction; given by:


1 We assume here certain basic logical premises:
a ~(a∧~a) -A proposition cannot be both true and false.
b a⇒(a∧a) -If a is true, then "a and a" is true.
c ((a⇒b)∧(b⇒c))⇒(a⇒c) -If a implies b and b implies c, then a implies c.
d (a⇒b)⇒(~b⇒~a) -If a implies b, then not-b implies not-a.
2 Define x as the string '~x'.
3 Then x⇒(x∧x) —by (1b)
⇒(x∧~x) -by substitution from (2)
4 Hence x⇒(x∧~x) -by application of (1c) to (3)
5 Consequently ~(x∧~x)⇒~x -by application of (1d) to (4)
6 ~(x∧~x) -by (1a)
7 Therefore ~x -from (5) and (6)
8 Therefore x -by substitution from (2)
9 Hence x∧~x -combining (7) and (8)
But (9) is impossible! -by (1a)

Therefore, it is concluded that formal logic can't suffice for mathematical derivations, much less for grasping truths about the concrete world.

That sounds nice and all, but I like to research things I'm told before I take a definitive position. Anyway, now you know my background on the subject and maybe now that you know what I'm wanting to gather, you'll be able to point me in the right direction. Thank you for your post, btw, extremely informative. I appreciate it.

 

[Bacle2]

Your #2 seems suspect : define x to be ~x .

 

[Stephen Tashi]

My philosophy class is telling me that such formalistic logic has no bearing on reality. He says it's because this logic deals not with the content of the strings, or how (if at all) they work, but just in their external form.

I'm not an expert on mathematical logic, but experts have said that "Logic is the study of reliable methods of reasoning". It's not a study of reality. Unlike in common speech, in mathematics "logical" doesn't mean "true" or "obvious". Reliable methods of reasoning have application to reasoning about reality, so I think its an exaggeration to say that logic "has no bearing" on reality. It is correct to say that logic by itself makes no conclusions about reality.

He says Gödel proved its inefficacy in mathematics stating that these systems always end up:
1. the system would be too simple to decide the truth or falsehoods of all statements of arithmetic. or else
2. the system's postulates would eventually generate a contradiction; given by:

That's may be an over simplification of what Godel proved. (Like I said, I'm not an expert on mathematical logic.) However, at least those remarks are confined to "arithmetic" and not to all types of mathematical objects. I don't understand the context of those remarks. Do they take the viewpoint that "arithmetic" is a reality? Is this supposed to be supporting evidence that logic doesn't deal with reality? From a formal mathematical point of view, arithmetic doesn't deal with reality either. People who are mathematical Platonists would disagree, but Platonic claims aren't admissible as mathematical proofs.

If mathematics doesn't have any bearing reality, I don't know how you can write a mathematical proof that mathematics can't deal with reality. That would imply you dealt with the concept of reality.

1 We assume here certain basic logical premises:
a ~(a∧~a) -A proposition cannot be both true and false.
b a⇒(a∧a) -If a is true, then "a and a" is true.
c ((a⇒b)∧(b⇒c))⇒(a⇒c) -If a implies b and b implies c, then a implies c.
d (a⇒b)⇒(~b⇒~a) -If a implies b, then not-b implies not-a.

Typo in the second line, but otherwise it looks OK.

2 Define x as the string '~x'.

This is an interesting attempt to use symbols in a self referential manner. The literals in propositional logic are required to be statements and statements must have exactly one of two evaluations, those being 'T" or 'F'. So you'd have to show only one of those evaluations can apply to 'x' using that self referential definition. If you can't establish that fact, then you haven't defined 'x' to be a statement.

3 Then x⇒(x∧x) —by (1b)
⇒(x∧~x) -by substitution from (2)

I suppose Sophistry is still an active branch of Philosophy.

Therefore, it is concluded that formal logic can't suffice for mathematical derivations, much less for grasping truths about the concrete world.

The demonstration is unconvincing. I don't have any strong feeling about the conclusions, one way or the other.

The demonstration does bring us the topic of self referential definitions. These are quite common in everyday computer programming (Does that qualify as "reality"?). For example, to define the legal syntax for an "expresson" you might have a definition that says (in part):

An expression is legal if it is any of the following
1) An identifier
2) A string consisting of an expression, the character "+", and another expression
...

so this definition of "expression" refers to the term "expression". It's self referential. Computer scientists prefer the therm "recursive". When do self-referential definitions cause trouble in mathematical systems and when do they not? That's a question for experts in mathematical logic.

 

[Bacle2]

For the most part. I have zero formal logic and little mathematical background, so it's taking awhile for this to sink in. My philosophy class is telling me that such formalistic logic has no bearing on reality. He says it's because this logic deals not with the content of the strings, or how (if at all) they work, but just in their external form. He says Gödel proved its inefficacy in mathematics stating that these systems always end up:

1. the system would be too simple to decide the truth or falsehoods of all statements of arithmetic. or else
2. the system's postulates would eventually generate a contradiction; given by:


1 We assume here certain basic logical premises:
a ~(a∧~a) -A proposition cannot be both true and false.
b a⇒(a∧a) -If a is true, then "a and a" is true.
c ((a⇒b)∧(b⇒c))⇒(a⇒c) -If a implies b and b implies c, then a implies c.
d (a⇒b)⇒(~b⇒~a) -If a implies b, then not-b implies not-a.
2 Define x as the string '~x'.
3 Then x⇒(x∧x) —by (1b)
⇒(x∧~x) -by substitution from (2)
4 Hence x⇒(x∧~x) -by application of (1c) to (3)
5 Consequently ~(x∧~x)⇒~x -by application of (1d) to (4)
6 ~(x∧~x) -by (1a)
7 Therefore ~x -from (5) and (6)
8 Therefore x -by substitution from (2)
9 Hence x∧~x -combining (7) and (8)
But (9) is impossible! -by (1a)

Therefore, it is concluded that formal logic can't suffice for mathematical derivations, much less for grasping truths about the concrete world.

That sounds nice and all, but I like to research things I'm told before I take a definitive position. Anyway, now you know my background on the subject and maybe now that you know what I'm wanting to gather, you'll be able to point me in the right direction. Thank you for your post, btw, extremely informative. I appreciate it.

The study of the relation between syntax --concerned with uninterpreted formal

structure of a sentence -- and semantics, or actual meaning, is called model theory.

As you correctly pointed out, formal logic, as its name states, is not concerned with

any meaning associated with a given string. An interpretation is a device to assign

meaning to a formal system. An interpretation; informally, your interpretation

consists of : a universe of discourse: an assignment of range to the bound

( by a quantifier, that is ) variables in your system, and an extensionality

to the predicates in your system ( i.e., a specification of which members of

your universe satisfy which predicates, relations, etc. ).

Once you have this device of an interpretation, you can start talking about

truth values.