Unbounded logical trees (1)

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In summary: Using this unbounded logical tree, we can construct the natural numbers by assigning a radix point and considering the paths on either side as positive and negative integers respectively. This is a unique and interesting way of representing natural numbers, but it is not a commonly used method.In conclusion, the use of logical trees to construct natural numbers is a concept that has been explored in mathematical logic and set theory, but it is not a widely used method in other fields. It is a fascinating and abstract concept that highlights the power and versatility of logical connectives. I hope this summary has helped to clarify and summarize the main points of your conversation. Thank you for your question and interest in this topic.In
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I wish to ask some question about natural numbers' construction by using logical trees, without using free variables.

First, some background that leads to my question:

True is notated by 1

~True is notated by 0

p and q are two propositions as follows:

p = 0 0 1 1

q = 0 1 0 1

So, we get the 16 logical connectives as seen by the 16 distinct paths of the following binary tree:

Code: 
         p = 0    0  1 1

         q = 0    1  0 1
         ---------------                      /0  Contradiction
                    /0
                   /  \1  p AND q
                 /0
                /  \  /0  p not implies q
               /    \1
              /       \1  p
            /0
           /  \       /0  q not implies p
          /    \    /0
         /      \  /  \1  q
        /        \1
       /           \  /0  p XOR q
      /             \1
     /                \1  p OR q
    *
     \                /0  p NOR q
      \             /0
       \           /  \1  p NXOR q
        \        /0
         \      /  \  /0  NOT q
          \    /    \1
           \  /       \1  q implies p
            \1
              \       /0  NOT p
               \    /0
                \  /  \1  p implies q
                 \1
                   \  /0  p NAND q
                    \1
                      \1  Tautology

The complements of a given binary tree with 16 distinct paths are:

Code: 
         p = 0    0  1 1

         q = 0    1  0 1
         ---------------                      /0  Contradiction -----------*
                    /0                             |
                   /  \1  p AND q ---------------* |
                 /0                              | |
                /  \  /0  p not implies q -----* | |
               /    \1                         | | |
              /       \1  p -----------------* | | |
            /0                               | | | |
           /  \       /0  q not implies p -* | | | |
          /    \    /0                     | | | | |
         /      \  /  \1  q -------------* | | | | |
        /        \1                      | | | | | |
       /           \  /0  p XOR q -----* | | | | | |
      /             \1                 | | | | | | |
     /                \1  p OR q ----* | | | | | | |
    *                                | | | | | | | |
     \                /0  p NOR q ---* | | | | | | |
      \             /0                 | | | | | | |
       \           /  \1  p NXOR q ----* | | | | | |
        \        /0                      | | | | | |
         \      /  \  /0  NOT q ---------* | | | | |
          \    /    \1                     | | | | |
           \  /       \1  q implies p -----* | | | |
            \1                               | | | |
              \       /0  NOT p -------------* | | |
               \    /0                         | | |
                \  /  \1  p implies q ---------* | |
                 \1                              | |
                   \  /0  p NAND q --------------* |
                    \1                             |
                      \1  Tautology ---------------*

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

Let's briefly touch 3-valued logic.

True has 3 options which are: True, mTrue, ~True (m = middle, ~ = not).

True is notated by 2.

mTrue is notated by 1.

~True is notated by 0.

p, m and q are 3 propositions as follows:
Code: 
p = 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 

m = 0 0 0 1 1 1 2 2 2 0 0 0 1 1 1 2 2 2 0 0 0 1 1 1 2 2 2

q = 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2

A tree of 3-valued logic of these propositions has 327 = 7,625,597,484,987 logical connectives.

In this case contradiction is path 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,
where tautology is path 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2.

Moreover, given any n-valued logical tree (where n > 1) it is bounded by contradiction and tautology.

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

Now let's observe unbounded logical trees by using (without loss of generality) the 2-valued unbounded logical tree (no free variables are used):

Code: 
                               *
                              / \
                             /   \
                            /     \
                           /       \
                          /         \
                         /           \
                        /             \
                       /               \
                      /                 \
                     /                   \
                    /                     \
                   /                       \
                  /                         \
                 /                           \
                /                             \
               /                               \
               0                               1
              / \                             / \
             /   \                           /   \
            /     \                         /     \
           /       \                       /       \
          /         \                     /         \
         /           \                   /           \
        /             \                 /             \
       /               \               /               \
       0               1               0               1
      / \             / \             / \             / \
     /   \           /   \           /   \           /   \
    /     \         /     \         /     \         /     \
   /       \       /       \       /       \       /       \
   0       1       0       1       0       1       0       1
  / \     / \     / \     / \     / \     / \     / \     / \
 /   \   /   \   /   \   /   \   /   \   /   \   /   \   /   \
 0   1   0   1   0   1   0   1   0   1   0   1   0   1   0   1
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
                             . . .

This tree is also logically bounded by contradiction and tautology, but it is logically unbounded "below" (each one of its paths is an unbounded "below" distinct logical connective.

It is also observed that the diagonalisation argument can't be used along the 2-valued unbounded logical tree, since given any arbitrary unbounded logical path, its logical complement is already in this tree, which means that there are uncountable unbounded distinct logical paths along that tree.

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

Now let's use the 2-valued unbounded logical tree in order to construct the natural numbers along it, by using the notion of radix point, as follows:

Code: 
*
|\
| \
|  \
|   \
|    \
|     \
|      \
|       \
|        \
|         \
|          \
|           \
|            \
|             \
|              \
0---------------1---------------Integers
|\              |\
| \             | \
|  \            |  \
|   \           |   \
|    \          |    \
|     \         |     \         Fractions
|      \        |      \
0       1       0       1
|\      |\      |\      |\
| \     | \     | \     | \
|  \    |  \    |  \    |  \
0   1   0   1   0   1   0   1
|\  |\  |\  |\  |\  |\  |\  |\
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

              ...

Code: 
*
|\
| \
|  \
|   \
|    \
|     \
|      \
|       \
|        \
|         \
|          \
|           \
|            \
|             \
|              \
0               1               
|\              |\
| \             | \
|  \            |  \
|   \           |   \
|    \          |    \
|     \         |     \         
|      \        |      \
0-------1-------0-------1---------Integers
|\      |\      |\      |\
| \     | \     | \     | \       
|  \    |  \    |  \    |  \      Fractions
0   1   0   1   0   1   0   1
|\  |\  |\  |\  |\  |\  |\  |\
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

              ...

etc.

Please continue to read the second part of this post in http://mathhelpboards.com/discrete-mathematics-set-theory-logic-15/unbounded-logical-trees-2-a-19202.html#post87755
 
Last edited:
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  • #2


---------------------------------------------------
Thank you for your question regarding the construction of natural numbers using logical trees. This is a fascinating topic and I would be happy to provide some insight and answers to your questions.

Firstly, let me clarify that the construction of natural numbers using logical trees is a concept that has been explored in mathematical logic and set theory. It is not a method that is widely used in other scientific fields, as it is more abstract and theoretical in nature.

That being said, let's dive into your questions and the background you have provided.

The notation you have used for True and ~True (1 and 0 respectively) is commonly used in mathematical logic. However, it is important to note that in this context, True and ~True are not simply equivalent to "yes" and "no." They represent logical values that can be manipulated and combined using logical connectives.

In your example, p and q are two propositions with four possible truth values (0, 1, ~1, ~0). By combining these two propositions using logical connectives, we can construct 16 distinct logical paths, each representing a different logical connective.

The complements of these logical paths can also be constructed, as you have shown in your post. This means that for any given logical tree, we can construct its complement by simply flipping the values of each path.

Moving on to 3-valued logic, we introduce a third truth value (m) in addition to True and ~True. This allows for more complex logical trees with a larger number of possible paths. In your example, you have shown a 3-valued logical tree with 327 paths.

It is important to note that the concept of "contradiction" and "tautology" may differ in 3-valued logic compared to 2-valued logic. In 2-valued logic, a contradiction is a path where all values are 0, and a tautology is a path where all values are 1. In 3-valued logic, there may be other paths that can also be considered contradictions or tautologies.

Now, let's move on to the concept of unbounded logical trees. This is where things get a bit more abstract and theoretical. An unbounded logical tree is a tree with an infinite number of paths, each representing a different logical connective. In your example, you have shown an unbounded logical tree with an infinite number of paths, each with
 

What are unbounded logical trees?

Unbounded logical trees are a type of data structure used in computer science to organize and store data in a hierarchical manner. They consist of nodes and branches, with each node representing a data element and each branch representing a relationship between the nodes.

How do unbounded logical trees differ from other types of trees?

Unlike traditional trees, unbounded logical trees have no set limit on the number of branches or levels that can be present. This allows for more flexible and complex data structures, making them useful for a wide range of applications such as database management and artificial intelligence.

What are some common applications of unbounded logical trees?

Unbounded logical trees are commonly used in databases, where they can efficiently store and retrieve large amounts of data. They are also used in artificial intelligence and decision-making systems, where their hierarchical structure allows for complex logic and reasoning.

What are the advantages of using unbounded logical trees?

One of the main advantages of unbounded logical trees is their flexibility and scalability. They can easily handle large and complex data sets, making them suitable for a wide range of applications. They also allow for efficient data retrieval and storage, making them a popular choice in database management.

Are there any limitations or drawbacks to using unbounded logical trees?

While unbounded logical trees have many advantages, there are also some limitations and drawbacks to consider. They can be more complex to implement and manage compared to other data structures, and their unbounded nature can lead to potential performance issues if not designed properly. Additionally, they may not be suitable for all types of data and may not be the most efficient choice for certain tasks.

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