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Exercise 33 - Chapter 2

Let \( \text{Prop}^{\mathbb{N}} \) denote the set of all mathematical statements one can make about natural numbers. For example “n is prime” is an element of \( \text{Prop}^{\mathbb{N}} \) , as is "\( n=2 \)” and "\( n \ge 11 \)”.

Given \( P, Q \in \text{Prop}^{\mathbb{N}} \) , we say \( P \le Q \) if for all \( n \in \mathbb{N} \) , whenever \( P(n) \) is true, so is \( Q(n) \).

Define a monoidal unit and a monoidal product on \( \text{Prop}^{\mathbb{N}} \) that satisfy the conditions of Definition 2.2.

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  • 1.
    edited May 2018

    The monoidal unit is the statement that n is a natural number. The monoidal product is the logical "and". For the demonstration of property a) of definition 2.1 I created an Excel spreadsheet with columns a, b, c, d, a implies c, b implies d, (a implies c) and (b implies d), a and b, c and d, (a and b) implies (c and d). Consider a, b, c, and d to be elements of \( \text{Prop}^{\mathbb{N}} \). Now construct 16 rows with all possible combinations of true and false for a, b, c and d. Calculate the remaining columns using logical functions (note for implication use OR(NOT(x), y)).

    Property a) assumes (a implies c) and (b implies d), so only those rows for which this column is true should be considered. There are nine such rows. Now examine the final column. All nine rows are also true. Since this an exhaustive set of possibilities and we have a tautology so property a) is verified. Properties b) through d) are obvious.

    Comment Source:The monoidal unit is the statement that n is a natural number. The monoidal product is the logical "and". For the demonstration of property a) of definition 2.1 I created an Excel spreadsheet with columns a, b, c, d, a implies c, b implies d, (a implies c) and (b implies d), a and b, c and d, (a and b) implies (c and d). Consider a, b, c, and d to be elements of \\( \text{Prop}^{\mathbb{N}} \\). Now construct 16 rows with all possible combinations of true and false for a, b, c and d. Calculate the remaining columns using logical functions (note for implication use OR(NOT(x), y)). Property a) assumes (a implies c) and (b implies d), so only those rows for which this column is true should be considered. There are nine such rows. Now examine the final column. All nine rows are also true. Since this an exhaustive set of possibilities and we have a tautology so property a) is verified. Properties b) through d) are obvious.
  • 2.
    edited May 2018

    It would seem that I don't understand how to use this editor. I tried to copy the formatting in Fredrick's problem statement post and it didn't work. Also there is no way to back out of a comment you wish to delete. Arg!

    Comment Source:It would seem that I don't understand how to use this editor. I tried to copy the formatting in Fredrick's problem statement post and it didn't work. Also there is no way to back out of a comment you wish to delete. Arg!
  • 3.
    edited May 2018

    You need to select the gear, in the upper right corner, and then 'view source'. You can copy that. Although you cannot delete a comment you can 'edit' it, again via the 'gear'.

    Comment Source:You need to select the gear, in the upper right corner, and then 'view source'. You can copy that. Although you cannot delete a comment you can 'edit' it, again via the 'gear'.
  • 4.

    Wow, that is odd. I did exactly that and then when I rendered it using the save comment button I still got all of the formatting characters. Just now after refreshing the page it rendered properly. Ok, I guess it is just a timing issue.

    Thanks for responding to my my post.

    Comment Source:Wow, that is odd. I did exactly that and then when I rendered it using the save comment button I still got all of the formatting characters. Just now after refreshing the page it rendered properly. Ok, I guess it is just a timing issue. Thanks for responding to my my post.
  • 5.

    @PeterGates Oh, It is a MathJax thing. At least part of the rendering happens locally. It takes some time to download the relevant javascript libraries. So, you will see a basic text representation followed by the properly rendered version. Once you have the libraries rendering will be snappier.

    Comment Source:@PeterGates Oh, It is a MathJax thing. At least part of the rendering happens locally. It takes some time to download the relevant javascript libraries. So, you will see a basic text representation followed by the properly rendered version. Once you have the libraries rendering will be snappier.
  • 6.
    edited June 2018

    I found two symmetric monoidal structures on \( (\textrm{Prop}^{\mathbb{N}}, \leq) \).

    One is $$ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is a natural number"}, \land) $$ and the other one is

    $$ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is not a natural number"}, \lor) $$ If that looks similar to monoidal structures on \( \mathbb{B} \), it's because it is. Given a natural number \( n \in \mathbb{N} \), there's a monotone map \( f_n: (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}}) \rightarrow (\mathbb{B}, \leq_{\mathbb{B}}) \) given by assigning true to \( f_A(x) \) whenever \( x \) holds for \( n \). It's strict monoidal and gives us: $$ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is a natural number"}, \land) \rightarrow (\mathbb{B}, \leq_\mathbb{B}, \textrm{true}, \land) $$ and

    $$ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is not a natural number"}, \lor) \rightarrow (\mathbb{B}, \leq_\mathbb{B}, \textrm{false}, \lor) $$ We can check the validity of conditions for monoidal monotone maps from Definition 2.38. In case of condition (a) we get:

    $$ \textrm{true} = f_n(\textrm{"n is a natural number''}) $$ and $$ \textrm{false} = f_n(\textrm{"n is not a natural number''}) $$ both of which hold. It holds that it's true that \( n \) is a natural number. It also holds that it's false that \( n \) is not a natural number.

    In the case of condition (b) of definition 2.38, it holds that the conjunction of truth values of statements about natural numbers is the same as the truth value of of the conjunction of statements about natural numbers. Similar argument goes for disjunction.

    Comment Source:I found two symmetric monoidal structures on \\( (\textrm{Prop}^{\mathbb{N}}, \leq) \\). One is \[ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is a natural number"}, \land) \] and the other one is \[ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is not a natural number"}, \lor) \] If that looks similar to monoidal structures on \\( \mathbb{B} \\), it's because it is. Given a natural number \\( n \in \mathbb{N} \\), there's a monotone map \\( f_n: (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}}) \rightarrow (\mathbb{B}, \leq_{\mathbb{B}}) \\) given by assigning true to \\( f_A(x) \\) whenever \\( x \\) holds for \\( n \\). It's strict monoidal and gives us: \[ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is a natural number"}, \land) \rightarrow (\mathbb{B}, \leq_\mathbb{B}, \textrm{true}, \land) \] and \[ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is not a natural number"}, \lor) \rightarrow (\mathbb{B}, \leq_\mathbb{B}, \textrm{false}, \lor) \] We can check the validity of conditions for monoidal monotone maps from Definition 2.38. In case of condition (a) we get: \[ \textrm{true} = f_n(\textrm{"n is a natural number''}) \] and \[ \textrm{false} = f_n(\textrm{"n is not a natural number''}) \] both of which hold. It holds that it's true that \\( n \\) is a natural number. It also holds that it's false that \\( n \\) is not a natural number. In the case of condition (b) of definition 2.38, it holds that the conjunction of truth values of statements about natural numbers is the same as the truth value of of the conjunction of statements about natural numbers. Similar argument goes for disjunction.
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