Autosave: 2022-12-21 09:00:01

Logic/General_concepts/Logical_equivalence.md

@@ -1,7 +1,7 @@
 ---
 categories:
-  - Mathematics
-tags: [logic]
+  - Logic
+tags: [propositional-logic]
 ---
 
 # Logical equivalence

Logic/General_concepts/Logical_possibility_and_necessity.md

@@ -1,12 +1,14 @@
 ---
 categories:
-  - Mathematics
-tags: [logic]
+  - Logic
+tags: [propositional-logic]
 ---
 
+# Logical possibility and necessity
+
 ## 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.
+In distinguishing the properties of [logical consistency](/Logic/General_concepts/Logical_consistency.md) and [validity](/Logic/General_concepts/Validity_and_entailment.md) 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.
 
 It is evident from the case of arguments that are valid but not sound that logic operates with a specialised notion of possibility. For example it has to be the case that the proposition _Every woman can levitate_ is logically possible since the following argument is valid:
 
@@ -32,6 +34,6 @@ From this we can derive the following property of logical possibility:
 
 ## Logical necessity
 
-A sentence is _logically necessary_ if it is true in every logically possible circumstance which is to say: true on every possible truth functional assignment. Necessity and [ logical truth](Logical%20truth%20and%20falsity.md#logical-truth) are therefore synonyms: anything that is logically true (a tautology) is true by necessity (could not be otherwise.)
+A sentence is _logically necessary_ if it is true in every logically possible circumstance which is to say: true on every possible truth functional assignment. Necessity and [logical truth](/Logic/General_concepts/Logical_truth_and_falsity.md#logical-truth) are therefore synonyms: anything that is logically true (a tautology) is true by necessity (could not be otherwise.)
 
 Further, every logical truth is logically possible but not everything that is logically possible is logically true. It is possible that it is raining but this is not logically necessary - it could be otherwise, i.e not raining. However it is not possible that it could be both raining and not raining.

Logic/General_concepts/Logical_truth_and_falsity.md

@@ -1,9 +1,11 @@
 ---
 categories:
-  - Mathematics
-tags: [logic]
+  - Logic
+tags: [propositional-logic]
 ---
 
+# Logical truth and falsity
+
 We say of certain sentences that they are logically true or logically false.
 
 ## Logical falsity
@@ -19,7 +21,7 @@ There is a country that is not a country.
 Apples are fruits and apples are not fruits
 ```
 
-Neither sentence can be true because the truth of the first clause is contradicted by the second. By the principle of [consistency](Consistency.md), it is not possible for both clauses to be true at once therefore the sentence, overall has the truth value of falsity
+Neither sentence can be true because the truth of the first clause is contradicted by the second. By the principle of [consistency](/Logic/General_concepts/Logical_consistency.md), it is not possible for both clauses to be true at once therefore the sentence, overall has the truth value of falsity
 
 The examples above are simple sentences but logical falsity also applies to compound sentences and it is actually easier to see the logical principle at work with compound sentences since once simple sentence of the compound contradicts the other such that the overall sentence cannot be consistently asserted:
 
@@ -33,24 +35,20 @@ It is raining and it is not raining.
 
 ### Formal expression
 
-```
-P & ~ P
-```
+$$ P \& \sim P $$
 
 ### Truth-table
 
-```
-    Can
-P				P	&	~	P
-T					F
-F					F
-```
+| $P$ | $P \& \sim P$ |
+| --- | ------------- |
+| T   | F             |
+| T   | F             |
 
 ## Logical truth
 
 ### Informal definition
 
-A sentence is logically true if and only if it is not possible for the sentence to be false. The sentence itself cannot be [consistently](Consistency.md) denied.
+A sentence is logically true if and only if it is not possible for the sentence to be false. The sentence itself cannot be consistently denied.
 
 **Demonstration**
 
@@ -71,17 +69,14 @@ It is Monday and Monday is a day of the week.
 
 > A sentence P is truth-functionally true if and only if P is true on every truth-value assignment
 
-```
-P v ~P
-```
+$$ P \lor \sim P$$
 
 ### Truth-table
 
-```
-P				P	∨	¬	P
-T					T
-F					T
-```
+| $P$ | $P \lor \sim P$ |
+| --- | --------------- |
+| T   | T               |
+| F   | T               |
 
 ### Consequences
 

Logic/General_concepts/Object_language_and_meta-language.md

@@ -1,10 +1,10 @@
 ---
 categories:
-  - Mathematics
-tags: [logic]
+  - Logic
+tags: [propositional-logic]
 ---
 
-## Object and metalanguages
+# 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.