From c73152e74a7704bca7b616712773c8f8a29310ab Mon Sep 17 00:00:00 2001 From: thomasabishop Date: Wed, 21 Dec 2022 05:34:20 +0000 Subject: [PATCH] Autosave: 2022-12-21 05:34:20 --- Logic/General_concepts/Logical_consistency.md | 10 +++++----- .../{Indeterminacy.md => Logical_indeterminacy.md} | 8 +++++--- 2 files changed, 10 insertions(+), 8 deletions(-) rename Logic/General_concepts/{Indeterminacy.md => Logical_indeterminacy.md} (77%) diff --git a/Logic/General_concepts/Logical_consistency.md b/Logic/General_concepts/Logical_consistency.md index ab9b114..ba1fa0a 100644 --- a/Logic/General_concepts/Logical_consistency.md +++ b/Logic/General_concepts/Logical_consistency.md @@ -38,7 +38,7 @@ $$ \{P \lor Q, Q\} $$ -### Truth-table +### Truth table $ \{P, Q\} $ form a consistent set because there is at least one assignment when both propositions are true. In fact there are two (corresponding to each disjunct) but one is sufficient. @@ -52,14 +52,14 @@ F F F F ## Derivation -> In terms of logical derivation, a finite $\Gamma$ of propositions is **inconsistent** in a system of derivation for propositional logic if and only if a proposition of the $P & \sim P$ is derivable from $\Gamma$. It is **consistent** just if this is not the case. +> In terms of logical derivation, a finite $\Gamma$ of propositions is **inconsistent** in a system of derivation for propositional logic if and only if a proposition of the form $P \& \sim P$ is derivable from $\Gamma$. It is **consistent** just if this is not the case. In other terms, if you can derive a contradiction from the set, the set is logically inconsistent. -A [contradiction](Logical%20truth%20and%20falsity.md#logical-falsity) contradiction has very important consequences for reasoning because if a set of propositions is inconsistent, every and all other propositions are derivable from that set. +A [contradiction](/Logic/General_concepts/Logical_truth_and_falsity.md#logical-falsity) has very important consequences for reasoning because if a set of propositions is inconsistent, any other proposition is derivable from it. -![proofs-drawio-Page-5.drawio 3.png](../img/proofs-drawio-Page-5.drawio%203.png) +![](/img/derivation_from_contradiction.png) _A demonstration of the the consequences of deriving a contradiction in a sequence of reasoning._ -Here we want to derive some proposition $Q$. If we can derive a contradiction from its negation as an assumption then, by the [negation elimination](Negation%20Elimination.md) rule, we can assert $Q$. This is why contradictions should be avoided in arguments, they 'prove' everything which, by association, undermines any particular premise you are trying to assert. +Here we want to derive some proposition $Q$. If we can derive a contradiction from its negation as an assumption then, by the [negation elimination](/Logic/Proofs/Negation_Elimination.md)) rule, we can assert $Q$. This is why contradictions should be avoided in arguments, they 'prove' everything which, by association, undermines any particular premise you are trying to assert. diff --git a/Logic/General_concepts/Indeterminacy.md b/Logic/General_concepts/Logical_indeterminacy.md similarity index 77% rename from Logic/General_concepts/Indeterminacy.md rename to Logic/General_concepts/Logical_indeterminacy.md index f44eaca..d189c27 100644 --- a/Logic/General_concepts/Indeterminacy.md +++ b/Logic/General_concepts/Logical_indeterminacy.md @@ -1,10 +1,12 @@ --- categories: - - Mathematics -tags: [logic] + - Logic +tags: [propositional-logic] --- -The vast majority of sentences in natural and formal logical languages are neither [ logically true](Logical%20truth%20and%20falsity.md#logical-truth) or [\| logically false](Logical%20truth%20and%20falsity.md#logical-falsity). This makes sense because sentences of this form are all either tautologies or contradictions and as such do not express information about the state of events in the world. We call sentences that are neither logically true or logically false, logically indeterminate sentences. +# Logical indeterminacy + +The vast majority of sentences in natural and formal logical languages are neither [logically true](/Logic/General_concepts/Logical_truth_and_falsity.md#logical-truth) or [logically false](/Logic/General_concepts/Logical_truth_and_falsity.md#logical-falsity). This makes sense because sentences of this form are all either tautologies or contradictions and as such do not express information about the state of events in the world. We call sentences that are neither logically true or logically false, logically indeterminate sentences. ## Informal definition