From ffcd803a0a457dae7b62380472e58059dc628417 Mon Sep 17 00:00:00 2001 From: Grigoriy Beziuk Date: Mon, 4 Mar 2024 00:52:56 +0300 Subject: [PATCH 1/2] typo fix 05_logic.md --- _chapters/05_logic.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_chapters/05_logic.md b/_chapters/05_logic.md index 640218e..e05fe68 100644 --- a/_chapters/05_logic.md +++ b/_chapters/05_logic.md @@ -435,7 +435,7 @@ By now you probably realized that the *and* and *or* operations are the bread an ![Order meet and joing](../05_logic/lattice_meet_join.svg) -Here comes the first criteria for an order to represent a logical system accurately - *it has to have $meet$ and $join$ operations for all elements*. Having two elements without a meet would mean that you would have a logical system where there are propositions for which you cannot say that one or the other is true. And this not how logic works, so our order has to have meets and joins for all elements. Incidentally we already know how such orders are called - they are called *lattices*. +Here comes the first criteria for an order to represent a logical system accurately - *it has to have $meet$ and $join$ operations for all elements*. Having two elements without a meet would mean that you would have a logical system where there are propositions for which you cannot say that one or the other is true. And this is not how logic works, so our order has to have meets and joins for all elements. Incidentally we already know how such orders are called - they are called *lattices*. One important law of the *and* and *or* operations, that is not always present in the *meet*-s and *join*-s concerns the connection between the two, i.e. way that they distribute, over one another. From 044c0743c9c418696371e0822a15ac9ac40b9cfa Mon Sep 17 00:00:00 2001 From: Grigoriy Beziuk Date: Mon, 4 Mar 2024 01:01:03 +0300 Subject: [PATCH 2/2] typo fix 05_logic.md --- _chapters/05_logic.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_chapters/05_logic.md b/_chapters/05_logic.md index e05fe68..f19b269 100644 --- a/_chapters/05_logic.md +++ b/_chapters/05_logic.md @@ -497,7 +497,7 @@ So this is the final condition for an order/lattice to be a representation of lo Without being too formal, let's try to test if this definition captures the concept correctly by examining a few special cases. -For example, let's take $A$ and $B$ to be the same object. In this case, ($A → B$) (or ($A → A$) if you want to be pedantic) would be the topmost object $X$ for which the criteria given by the formula $A ∧ X → A$ is satisfied. But in this case the formula is *always satisfied* as the *meet* of $A$ and any other object would always be below $A$. So this formula is always for all $X$. The topmost object that fits it is, then, the topmost object out there i.e. $True$. +For example, let's take $A$ and $B$ to be the same object. In this case, ($A → B$) (or ($A → A$) if you want to be pedantic) would be the topmost object $X$ for which the criteria given by the formula $A ∧ X → A$ is satisfied. But in this case the formula is *always satisfied* as the *meet* of $A$ and any other object would always be below $A$. So this formula is always satisfied for all $X$. The topmost object that fits it is, then, the topmost object out there i.e. $True$. ![Implies identity](../05_logic/implies_identity.svg)