diff --git a/Logic/Atomic_and_molecular_sentences.md b/Logic/Atomic_and_molecular_sentences.md index bbe2607..f919c43 100644 --- a/Logic/Atomic_and_molecular_sentences.md +++ b/Logic/Atomic_and_molecular_sentences.md @@ -1,7 +1,7 @@ --- -tags: - - Logic - - propositional-logic +categories: + - Logic +tags: [propositional_logic] --- Sentences or propositions (we will use 'sentences' for consistency) are expressions **that have truth values**, either true or false. diff --git a/Logic/Conjunction_Elimination.md b/Logic/Conjunction_Elimination.md index 06d9626..7050835 100644 --- a/Logic/Conjunction_Elimination.md +++ b/Logic/Conjunction_Elimination.md @@ -1,10 +1,10 @@ --- -tags: - - Logic - - propositional-logic - - derivation-rules +categories: + - Logic +tags: [propositional_logic] --- + If a conjunction exists, it means that both conjuncts are the case; therefore we can legitimately extract either one of them. Also known as *Simplification*. ![conjunc-elim.png](../img/conjunc-elim.png) diff --git a/Logic/Conjunction_Introduction.md b/Logic/Conjunction_Introduction.md index ffa9085..114e69e 100644 --- a/Logic/Conjunction_Introduction.md +++ b/Logic/Conjunction_Introduction.md @@ -1,8 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - derivation-rules +categories: + - Logic +tags: [propositional_logic] --- If two conjuncts have each been independently derived then they can be conjoined. Also known more simply as *Conjunction* diff --git a/Logic/Consistency.md b/Logic/Consistency.md index 353fef2..d4ea87e 100644 --- a/Logic/Consistency.md +++ b/Logic/Consistency.md @@ -1,10 +1,8 @@ --- -tags: - - Logic - - propositional-logic - - consistency +categories: + - Logic +tags: [propositional_logic] --- - ## Informal definition A set of sentences is consistent if and only if **it is possible for all the members of the set to be true at the same time**. A set of sentences is inconsistent if and only if it is not consistent. diff --git a/Logic/Corresponding_material_and_biconditional.md b/Logic/Corresponding_material_and_biconditional.md index 1c0aa8e..5039116 100644 --- a/Logic/Corresponding_material_and_biconditional.md +++ b/Logic/Corresponding_material_and_biconditional.md @@ -1,9 +1,8 @@ --- -tags: - - Logic - - propositional-logic +categories: + - Logic +tags: [propositional_logic] --- - ## Corresponding material conditional to show validity To demonstrate *truth-functional validity* we have to construct a truth-table which contains each of the premises and the conclusion and then review each row to see if there is an assignment where both the premises and the conclusion are true. diff --git a/Logic/DeMorgan's_Laws.md b/Logic/DeMorgan's_Laws.md index 1e48864..9ef3c7b 100644 --- a/Logic/DeMorgan's_Laws.md +++ b/Logic/DeMorgan's_Laws.md @@ -1,9 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - derivation-rules - - theorems-axioms-laws +categories: + - Logic +tags: [propositional_logic] --- DeMorgan's laws express some fundamental equivalences that obtain between the Boolean [connectives](Truth-functional%20connectives.md): diff --git a/Logic/Disjunction_Elimination.md b/Logic/Disjunction_Elimination.md index ae0a6c5..3380cd2 100644 --- a/Logic/Disjunction_Elimination.md +++ b/Logic/Disjunction_Elimination.md @@ -1,8 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - derivation-rules +categories: + - Logic +tags: [propositional_logic] --- This rule is sometimes also referred to as *Constructive Dilemma*. This can be a bit tricky to understand because the goal is to derive or *introduce* a new proposition separate from the disjunction you start out with. This may be disjunction, a single proposition or a proposition containing any other logical connective. You do this by constructing two sub-proofs, one for each of the disjuncts comprising the disjunction you start out with. If you can derive your target proposition as the conclusion of each subproof then you may invoke the conclusion in the main proof and take it to be derived. diff --git a/Logic/Disjunction_Introduction.md b/Logic/Disjunction_Introduction.md index 1712bb0..4bf8882 100644 --- a/Logic/Disjunction_Introduction.md +++ b/Logic/Disjunction_Introduction.md @@ -1,8 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - derivation-rules +categories: + - Logic +tags: [propositional_logic] --- This rule can seem a little odd: like we are randomly introducing an additional proposition without giving any justification. However this is just a consequence of the fact if $P$ is true, so is $P \lor Q$ since disjunction is not the same as conjunction: only one disjunct needs to be true for the compound disjunction to be true. This is represented in the context of [truth-trees](Truth-trees.md#disjunction-decomposition) by the fact that truth can pass up via either branch of a disjunction pattern. diff --git a/Logic/Formal_proofs_in_propositional_logic.md b/Logic/Formal_proofs_in_propositional_logic.md index dd082ae..e22db96 100644 --- a/Logic/Formal_proofs_in_propositional_logic.md +++ b/Logic/Formal_proofs_in_propositional_logic.md @@ -1,8 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - proofs +categories: + - Logic +tags: [propositional_logic] --- When we construct a formal proof in logic we are seeking to show that a certain proposition is **derivable** from other propositions. We use the words *derivation* and *proof* interchangeably. diff --git a/Logic/Indeterminacy.md b/Logic/Indeterminacy.md index 292d78b..442f2b4 100644 --- a/Logic/Indeterminacy.md +++ b/Logic/Indeterminacy.md @@ -1,7 +1,7 @@ --- -tags: - - Logic - - propositional-logic +categories: + - 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. diff --git a/Logic/Law_of_Non-Contradiction.md b/Logic/Law_of_Non-Contradiction.md index 4b1fb25..718d762 100644 --- a/Logic/Law_of_Non-Contradiction.md +++ b/Logic/Law_of_Non-Contradiction.md @@ -1,12 +1,10 @@ --- -tags: - - Logic - - propositional-logic - - theorems-axioms-laws +categories: + - Logic +tags: [propositional_logic] --- - > > A proposition cannot be true and false at the same time. > $$ > \\sim (P & \sim P) diff --git a/Logic/Law_of_the_Excluded_Middle.md b/Logic/Law_of_the_Excluded_Middle.md index fb338a9..c0e7d0a 100644 --- a/Logic/Law_of_the_Excluded_Middle.md +++ b/Logic/Law_of_the_Excluded_Middle.md @@ -1,11 +1,9 @@ --- -tags: - - Logic - - propositional-logic - - theorems-axioms-laws +categories: + - Logic +tags: [propositional_logic] --- - > > Every proposition has to be either true or false. There can be no middle ground. > $$ diff --git a/Logic/Logical_equivalence.md b/Logic/Logical_equivalence.md index ca67b85..9b9e8f5 100644 --- a/Logic/Logical_equivalence.md +++ b/Logic/Logical_equivalence.md @@ -1,10 +1,9 @@ --- -tags: - - Logic - - propositional-logic +categories: + - Logic +tags: [propositional_logic] --- - > > Two sentences, P and Q, are truth-functionally equivalent if and only if there is no truth assignment in which P is true and Q is false diff --git a/Logic/Logical_possibility_and_necessity.md b/Logic/Logical_possibility_and_necessity.md index 02a7dd7..cc9cfac 100644 --- a/Logic/Logical_possibility_and_necessity.md +++ b/Logic/Logical_possibility_and_necessity.md @@ -1,11 +1,8 @@ --- -tags: - - Logic - - Philosophy - - propositional-logic - - modality +categories: + - Logic +tags: [propositional_logic] --- - ## Logical possibility In distinguishing the properties of [logical consistency](Consistency.md) and [validity](Validity%20and%20entailment.md#validity) we make tacit use of the notion of **possibility**. This is because when we consider the validity of an argument we are assessing truth-conditions and this consists in asking ourselves what could or could not be the case: were it such that *P*, then it would be the case that *Q*. It is important to understand what possibility means in the context of logic and how it differs from what we might mean ordinarily when we use the term. diff --git a/Logic/Logical_truth_and_falsity.md b/Logic/Logical_truth_and_falsity.md index a95e30d..2b182b9 100644 --- a/Logic/Logical_truth_and_falsity.md +++ b/Logic/Logical_truth_and_falsity.md @@ -1,7 +1,7 @@ --- -tags: - - Logic - - propositional-logic +categories: + - Logic +tags: [propositional_logic] --- We say of certain sentences that they are logically true or logically false. diff --git a/Logic/Negation_Elimination.md b/Logic/Negation_Elimination.md index 5b2f8e8..ed382d9 100644 --- a/Logic/Negation_Elimination.md +++ b/Logic/Negation_Elimination.md @@ -1,8 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - derivation-rules +categories: + - Logic +tags: [propositional_logic] --- ![negate-elim 1.png](../img/negate-elim%201.png) diff --git a/Logic/Negation_Introduction.md b/Logic/Negation_Introduction.md index c23c2e4..f47afab 100644 --- a/Logic/Negation_Introduction.md +++ b/Logic/Negation_Introduction.md @@ -1,8 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - derivation-rules +categories: + - Logic +tags: [propositional_logic] --- This is also known as *proof by contradiction*. You start with an assumption declared in a subproof. If you can derive a contradiction from this assumption (typically from the introduction of another proposition and its negation), then you are permitted to derive the negation of the auxiliary assumption in the main proof. diff --git a/Logic/Object_language_and_meta-language.md b/Logic/Object_language_and_meta-language.md index b73550e..df23862 100644 --- a/Logic/Object_language_and_meta-language.md +++ b/Logic/Object_language_and_meta-language.md @@ -1,9 +1,10 @@ --- -tags: - - Logic - - propositional-logic +categories: + - Logic +tags: [propositional_logic] --- + ## Object and metalanguages When we talk about a language we call that language the **object language**. A **metalanguage** is a language used to describe some object language. diff --git a/Logic/Reiteration.md b/Logic/Reiteration.md index efe4ed8..03406cf 100644 --- a/Logic/Reiteration.md +++ b/Logic/Reiteration.md @@ -1,8 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - derivation-rules +categories: + - Logic +tags: [propositional_logic] --- **Reiteration (R)** allows us to restate any proposition already in the proof within the main proof or a more deeply nested subproof. Reiteration allows us to reuse any assumptions, or propositions derived from assumptions, without having to introduce a new dependency with another assumption. diff --git a/Logic/Soundness.md b/Logic/Soundness.md index 963cd5c..27691bc 100644 --- a/Logic/Soundness.md +++ b/Logic/Soundness.md @@ -1,7 +1,7 @@ --- -tags: - - Logic - - propositional-logic +categories: + - Logic +tags: [propositional_logic] --- ### Soundness diff --git a/Logic/Strategies_for_constructing_proofs.md b/Logic/Strategies_for_constructing_proofs.md index ac586ed..a8a9798 100644 --- a/Logic/Strategies_for_constructing_proofs.md +++ b/Logic/Strategies_for_constructing_proofs.md @@ -1,8 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - proofs +categories: + - Logic +tags: [propositional_logic] --- ## General strategy diff --git a/Logic/Syllogism.md b/Logic/Syllogism.md index 137073b..1030e3a 100644 --- a/Logic/Syllogism.md +++ b/Logic/Syllogism.md @@ -1,7 +1,7 @@ --- -tags: - - Logic - - propositional-logic +categories: + - Logic +tags: [propositional_logic] --- In order to make assertions about the relative [consistency](Consistency.md) or inconsistency of a set of propositions we advance arguments. Consider everyday life: if we are having an argument with someone, we believe that they are wrong. A more logical way to say this is that we believe that their beliefs are inconsistent. In order to change their viewpoint or point out why they are wrong we advance an argument intended to show that belief A conflicts with belief B. Or if C is true, then you cannot believe that D. diff --git a/Logic/Syntax_of_sentential_logic.md b/Logic/Syntax_of_sentential_logic.md index d541ee6..c979603 100644 --- a/Logic/Syntax_of_sentential_logic.md +++ b/Logic/Syntax_of_sentential_logic.md @@ -1,9 +1,8 @@ --- -tags: - - Logic - - propositional-logic +categories: + - Logic +tags: [propositional_logic] --- - ## Syntax of formal languages versus semantics > diff --git a/Logic/Theorems_and_empty_sets.md b/Logic/Theorems_and_empty_sets.md index 2d1bc44..0de1b32 100644 --- a/Logic/Theorems_and_empty_sets.md +++ b/Logic/Theorems_and_empty_sets.md @@ -1,9 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - proofs - - theorems-axioms-laws +categories: + - 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. diff --git a/Logic/Truth-functional_connectives.md b/Logic/Truth-functional_connectives.md index da548d0..352f3c5 100644 --- a/Logic/Truth-functional_connectives.md +++ b/Logic/Truth-functional_connectives.md @@ -1,10 +1,8 @@ --- -tags: - - Logic - - propositional-logic - - truth-tables +categories: + - Logic +tags: [propositional_logic] --- - ## Truth-functional connectives Sentences generated from other (simple) sentences by means of sentential connectives are [compound sentences](Atomic%20and%20molecular%20sentences.md). diff --git a/Logic/Truth-tables.md b/Logic/Truth-tables.md index 20ff84c..ab4316d 100644 --- a/Logic/Truth-tables.md +++ b/Logic/Truth-tables.md @@ -1,9 +1,7 @@ --- -tags: - - Logic - - propositional-logic - - recursion - - truth-tables +categories: + - Logic +tags: [propositional_logic] --- # Truth-tables diff --git a/Logic/Truth-trees.md b/Logic/Truth-trees.md index 965df27..508cce9 100644 --- a/Logic/Truth-trees.md +++ b/Logic/Truth-trees.md @@ -1,9 +1,8 @@ --- -tags: - - Logic - - propositional-logic +categories: + - Logic +tags: [propositional_logic] --- - ## Rationale Like [truth-tables](Truth-tables.md), truth-trees are a means of graphically representing the logical relationships that may obtain between propositions. Truth-trees and truth-tables complement each other and which method you choose depends on which logical property you are seeking to derive. diff --git a/Logic/Validity_and_entailment.md b/Logic/Validity_and_entailment.md index 6a98ab2..d0f136b 100644 --- a/Logic/Validity_and_entailment.md +++ b/Logic/Validity_and_entailment.md @@ -1,11 +1,8 @@ --- -tags: - - Logic - - propositional-logic - - validity - - entailment +categories: + - Logic +tags: [propositional_logic] --- - ## Validity ### Informal definition diff --git a/Mathematics/Algebra/Algebra_key_terms.md b/Mathematics/Algebra/Algebra_key_terms.md index ddde9c9..15038b8 100644 --- a/Mathematics/Algebra/Algebra_key_terms.md +++ b/Mathematics/Algebra/Algebra_key_terms.md @@ -1,7 +1,7 @@ --- -tags: - - Mathematics - - Algebra +categories: + - Mathematics +tags: [algebra] --- * **Variable** diff --git a/Mathematics/Algebra/Equivalent_equations.md b/Mathematics/Algebra/Equivalent_equations.md index 8d06f30..8caef52 100644 --- a/Mathematics/Algebra/Equivalent_equations.md +++ b/Mathematics/Algebra/Equivalent_equations.md @@ -1,9 +1,8 @@ --- -tags: - - Mathematics - - Algebra +categories: + - Mathematics +tags: [algebra] --- - ## Equivalent equations > diff --git a/Mathematics/Algebra/Exponents.md b/Mathematics/Algebra/Exponents.md index 76a7f13..21856d1 100644 --- a/Mathematics/Algebra/Exponents.md +++ b/Mathematics/Algebra/Exponents.md @@ -1,10 +1,8 @@ --- -tags: - - Mathematics - - Algebra - - exponents +categories: + - Mathematics +tags: [algebra, exponents] --- - ## Equivalent equations > diff --git a/Mathematics/Algebra/Logarithms.md b/Mathematics/Algebra/Logarithms.md index 6f31965..15ed530 100644 --- a/Mathematics/Algebra/Logarithms.md +++ b/Mathematics/Algebra/Logarithms.md @@ -1,10 +1,8 @@ --- -tags: - - Mathematics - - Algebra - - logarithms +categories: + - Mathematics +tags: [algebra] --- - Most simply a logarithm is a way of answering the question: > diff --git a/Mathematics/Algebra/Negative_exponents.md b/Mathematics/Algebra/Negative_exponents.md index 4ca1741..5bfd9c6 100644 --- a/Mathematics/Algebra/Negative_exponents.md +++ b/Mathematics/Algebra/Negative_exponents.md @@ -1,8 +1,7 @@ --- -tags: - - Mathematics - - Algebra - - exponents +categories: + - Mathematics +tags: [algebra, exponents] --- When calculating the exponents of a negative number the answer will always will be positive: diff --git a/Mathematics/Algebra/Solving_equations.md b/Mathematics/Algebra/Solving_equations.md index f969c37..87a0533 100644 --- a/Mathematics/Algebra/Solving_equations.md +++ b/Mathematics/Algebra/Solving_equations.md @@ -1,10 +1,8 @@ --- -tags: - - Mathematics - - Algebra - - operators +categories: + - Mathematics +tags: [algebra] --- - ## Use inversion of operators When solving equations we frequently make use of the [ operator inversion rules](../Prealgebra/Inversion%20of%20operators.md) to find the solutions. diff --git a/Programming_Languages/Shell_Scripting/Cron.md b/Programming_Languages/Shell_Scripting/Cron.md index 6330df9..9ebe7bb 100644 --- a/Programming_Languages/Shell_Scripting/Cron.md +++ b/Programming_Languages/Shell_Scripting/Cron.md @@ -2,7 +2,7 @@ tags: - Programming_Languages - shell - - abra + - abracadabra --- # Cron