From a18fcb7e4a3a1005268188713a9bfe462c1daad6 Mon Sep 17 00:00:00 2001 From: thomasabishop Date: Fri, 23 Dec 2022 15:00:06 +0000 Subject: [PATCH] Autosave: 2022-12-23 15:00:06 --- .../Law_of_Non-Contradiction.md | 14 ++++++++------ .../Law_of_the_Excluded_Middle.md | 14 ++++++++------ .../Theorems_and_empty_sets.md | 11 ++++++----- 3 files changed, 22 insertions(+), 17 deletions(-) diff --git a/Logic/Laws_and_theorems.md/Law_of_Non-Contradiction.md b/Logic/Laws_and_theorems.md/Law_of_Non-Contradiction.md index cdde0bb..106cd69 100644 --- a/Logic/Laws_and_theorems.md/Law_of_Non-Contradiction.md +++ b/Logic/Laws_and_theorems.md/Law_of_Non-Contradiction.md @@ -1,11 +1,13 @@ --- categories: - - Mathematics -tags: [logic, theorems] + - Logic +tags: [propositional-logic] --- +# Law of Non-Contradiction + > A proposition cannot be true and false at the same time. -> -> $$ -> \\sim (P & \sim P) -> $$ + +$$ +\lnot (P \land \lnot P) +$$ diff --git a/Logic/Laws_and_theorems.md/Law_of_the_Excluded_Middle.md b/Logic/Laws_and_theorems.md/Law_of_the_Excluded_Middle.md index 2a3c909..6a954b3 100644 --- a/Logic/Laws_and_theorems.md/Law_of_the_Excluded_Middle.md +++ b/Logic/Laws_and_theorems.md/Law_of_the_Excluded_Middle.md @@ -1,11 +1,13 @@ --- categories: - - Mathematics -tags: [theorems, logic] + - Logic +tags: [propositional-logic] --- +# Law of the Excluded Middle + > Every proposition has to be either true or false. There can be no middle ground. -> -> $$ -> P \lor \sim P -> $$ + +$$ +P \lor \sim P +$$ diff --git a/Logic/Laws_and_theorems.md/Theorems_and_empty_sets.md b/Logic/Laws_and_theorems.md/Theorems_and_empty_sets.md index 6d42776..c629de4 100644 --- a/Logic/Laws_and_theorems.md/Theorems_and_empty_sets.md +++ b/Logic/Laws_and_theorems.md/Theorems_and_empty_sets.md @@ -1,13 +1,14 @@ --- categories: - - Mathematics -tags: [logic] + - Logic +tags: [propositional-logic] --- -We know that when we construct a [derivation](Formal%20proofs%20in%20propositional%20logic.md#constructing-proofs) we start from a set of assumptions and then attempt to reach a proposition that is a consequence of the starting assumptions. However it does not always have to be the case that the starting set contains members. The set can in fact be empty. +We know that when we construct a [derivation](/Logic/Proofs/Formal_proofs_in_propositional_logic.md#derivation-rules) we start from a set of assumptions and then attempt to reach a proposition that is a consequence of the starting assumptions. However it does not always have to be the case that the starting set contains members. The set can in fact be empty. _Demonstration_ -![proofs-drawio-Page-5.drawio 2.png](../img/proofs-drawio-Page-5.drawio%202.png) + +![](/img/proofs-drawio-Page-5.drawio_2.png) We see in this example that there is no starting set and thus no primary assumptions. Instead we start with nothing other than the proposition we wish to derive. The proposition is effectively derived from itself. In these scenarios we say that we are constructing a derivation from an **empty set**. @@ -18,7 +19,7 @@ Propositions which possess this property are called theorems: We represent a theorem as: $$ -\\vdash P +\vdash P $$ (There is no preceding $\Gamma$ as the set is empty. )