Autosave: 2022-12-23 13:00:06

Logic/General_concepts/Logical_consistency.md

@@ -51,7 +51,7 @@ $ \{P, Q\} $ form a consistent set because there is at least one assignment when
 
 ## 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 form $P \& \sim P$ 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 $P \& \lnot P$ 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.
 

Logic/General_concepts/Logical_equivalence.md

@@ -17,17 +17,17 @@ Q: The pavement is not wet unless it is raining.
 ### Formal expression
 
 $$
-P \supset Q \equiv \sim P \lor Q
+(P \rightarrow Q) \Leftrightarrow (\lnot P \lor Q)
 $$
 
 ### Truth-tables
 
-| $P$ | $Q$ | $ P \supset Q $ | $ \sim P \lor Q$ |
-| --- | --- | --------------- | ---------------- |
-| T   | T   | T               | T                |
-| T   | F   | T               | F                |
-| F   | T   | T               | T                |
-| F   | F   | F               | T                |
+| $P$ | $Q$ | $ P \rightarrow Q $ | $ \lnot P \lor Q$ |
+| --- | --- | ------------------- | ----------------- |
+| T   | T   | T                   | T                 |
+| T   | F   | T                   | F                 |
+| F   | T   | T                   | T                 |
+| F   | F   | F                   | T                 |
 
 ### Derivation
 

Logic/General_concepts/Logical_truth_and_falsity.md

@@ -33,14 +33,14 @@ It is raining and it is not raining.
 
 ### Formal expression
 
-$$ P \& \sim P $$
+$$ P \land \lnot P $$
 
 ### Truth-table
 
-| $P$ | $P \& \sim P$ |
-| --- | ------------- |
-| T   | F             |
-| T   | F             |
+| $P$ | $P \land \lnot P$ |
+| --- | ----------------- |
+| T   | F                 |
+| T   | F                 |
 
 ## Logical truth
 
@@ -55,32 +55,32 @@ A rose is a rose.
 Today is Tuesday unless today is not Tuesday.
 ```
 
-Regardless of any facts obtaining in the world, these propositions cannot be false.
+Regardless of any facts obtaining in the world, these propositions cannot be false.should be avoided in arguments, they 'prove' everything whi
 
 As with logically false propositions, logical truth can also apply to compound propositions:
 
 ```
-It is Monday and Monday is a day of the week.
+A rose is a rose and a shoe is a shoe
 ```
 
 ### Formal definition
 
 > A proposition P is truth-functionally true if and only if P is true on every truth-value assignment
 
-$$ P \lor \sim P$$
+$$ P \lor \lnot P$$
 
 ### Truth-table
 
-| $P$ | $P \lor \sim P$ |
-| --- | --------------- |
-| T   | T               |
-| F   | T               |
+| $P$ | $P \lor \lnot P$ |
+| --- | ---------------- |
+| T   | T                |
+| F   | T                |
 
 ### Consequences
 
 The existence of logically false and logically true propositions affects the validity and soundness of arguments in which they are used. These are technicalities that have philosophically interesting consequences.
 
-- If an argument contains premises which are logically false than this argument will perforce be valid. This is because one cannot consistently assert the premises and deny the conclusion which is the definition of validity. However the _reason_ why one cannot consistently assert the premises and deny the conclusions is because one cannot consistently assert the premises - they conflict with each other. Furthermore as the argument contains false premises, it cannot be sound.
+- If an argument contains premises which are logically false than this argument will perforce be valid. This is because one cannot consistently assert the premises and deny the conclusion which is the definition of [validity](/Logic/General_concepts/Validity_and_entailment.md). However the _reason_ why one cannot consistently assert the premises and deny the conclusions is because one cannot consistently assert the premises - they conflict with each other. Furthermore as the argument contains false premises, it cannot be sound.
 
   ```
   (P1) Russia is a country.

Logic/General_concepts/Validity_and_entailment.md

@@ -39,23 +39,21 @@ The test for a strong inductive argument is not whether the conclusion is true,
 
 > An argument is truth-functionally valid if and only if there is no truth-assignment on which all the premises are true and the conclusion is false.
 
-Linking this to [derivation](Formal%20proofs%20in%20propositional%20logic.md), we say:
+Linking this to derivation, we say:
 
 > In a system of derivation in propositional logic, an argument is valid if the conclusion of the argument is derivable within the system of derivation from the set consisting of the premises, and invalid otherwise.
 
 ### Demonstration
 
-The inference from the set ${P, P \supset Q}$ to $Q$ is valid
+The inference from the set ${P, P \rightarrow Q}$ to $Q$ is valid
 
 ### Truth-table
 
-```
-P	Q				P	⊃	Q     P      Q
-T	T				    T		  T      T     *
-T	F					F		  T      F
-F	T					T		  F      T
-F	F					T         F      F
-```
+| $P$ | $Q$ | $P \rightarrow Q$ | $P$ | $Q$ | Assessment |
+| --- | --- | ----------------- | --- | --- | ---------- |
+| T   | T   | T                 | T   | T   | Valid      |
+| T   | F   | F                 | T   | F   |            |
+| F   | T   | T                 | F   | T   |            |
 
 ## Entailment