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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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---
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---
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Sentences or propositions (we will use 'sentences' for consistency) are expressions **that have truth values**, either true or false.
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Sentences or propositions (we will use 'sentences' for consistency) are expressions **that have truth values**, either true or false.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- derivation-rules
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---
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---
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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*.
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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*.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- derivation-rules
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---
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---
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If two conjuncts have each been independently derived then they can be conjoined. Also known more simply as *Conjunction*
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If two conjuncts have each been independently derived then they can be conjoined. Also known more simply as *Conjunction*
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- consistency
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---
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---
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## Informal definition
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## Informal definition
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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.
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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.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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---
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---
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## Corresponding material conditional to show validity
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## Corresponding material conditional to show validity
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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.
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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.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- derivation-rules
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- theorems-axioms-laws
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---
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---
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DeMorgan's laws express some fundamental equivalences that obtain between the Boolean [connectives](Truth-functional%20connectives.md):
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DeMorgan's laws express some fundamental equivalences that obtain between the Boolean [connectives](Truth-functional%20connectives.md):
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- derivation-rules
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---
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---
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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.
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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.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- derivation-rules
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---
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---
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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.
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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.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- proofs
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---
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---
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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.
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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.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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---
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---
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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.
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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.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- theorems-axioms-laws
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---
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---
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>
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> A proposition cannot be true and false at the same time.
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> A proposition cannot be true and false at the same time.
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> $$
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> $$
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> \\sim (P & \sim P)
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> \\sim (P & \sim P)
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- theorems-axioms-laws
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---
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---
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>
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>
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> Every proposition has to be either true or false. There can be no middle ground.
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> Every proposition has to be either true or false. There can be no middle ground.
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> $$
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> $$
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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---
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---
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>
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>
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> 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
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> 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
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- Philosophy
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tags: [propositional_logic]
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- propositional-logic
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- modality
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---
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---
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## Logical possibility
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## Logical possibility
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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.
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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.
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tags:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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---
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We say of certain sentences that they are logically true or logically false.
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We say of certain sentences that they are logically true or logically false.
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tags:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- derivation-rules
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tags:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- derivation-rules
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---
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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.
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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.
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tags:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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---
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---
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## Object and metalanguages
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## Object and metalanguages
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When we talk about a language we call that language the **object language**. A **metalanguage** is a language used to describe some object language.
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When we talk about a language we call that language the **object language**. A **metalanguage** is a language used to describe some object language.
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- derivation-rules
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---
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---
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**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.
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**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.
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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---
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---
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### Soundness
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### Soundness
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- proofs
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---
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---
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## General strategy
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## General strategy
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---
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tags:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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---
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---
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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.
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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.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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---
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---
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## Syntax of formal languages versus semantics
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## Syntax of formal languages versus semantics
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>
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>
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- proofs
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- theorems-axioms-laws
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---
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---
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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.
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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.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- truth-tables
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---
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---
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## Truth-functional connectives
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## Truth-functional connectives
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Sentences generated from other (simple) sentences by means of sentential connectives are [compound sentences](Atomic%20and%20molecular%20sentences.md).
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Sentences generated from other (simple) sentences by means of sentential connectives are [compound sentences](Atomic%20and%20molecular%20sentences.md).
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- recursion
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- truth-tables
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---
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---
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# Truth-tables
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# Truth-tables
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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---
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---
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## Rationale
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## Rationale
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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.
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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.
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---
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---
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tags:
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categories:
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- Logic
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- Logic
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- propositional-logic
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tags: [propositional_logic]
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- validity
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- entailment
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---
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---
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## Validity
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## Validity
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### Informal definition
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### Informal definition
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---
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---
|
||||||
tags:
|
categories:
|
||||||
- Mathematics
|
- Mathematics
|
||||||
- Algebra
|
tags: [algebra]
|
||||||
---
|
---
|
||||||
|
|
||||||
* **Variable**
|
* **Variable**
|
||||||
|
|
|
||||||
|
|
@ -1,9 +1,8 @@
|
||||||
---
|
---
|
||||||
tags:
|
categories:
|
||||||
- Mathematics
|
- Mathematics
|
||||||
- Algebra
|
tags: [algebra]
|
||||||
---
|
---
|
||||||
|
|
||||||
## Equivalent equations
|
## Equivalent equations
|
||||||
|
|
||||||
>
|
>
|
||||||
|
|
|
||||||
|
|
@ -1,10 +1,8 @@
|
||||||
---
|
---
|
||||||
tags:
|
categories:
|
||||||
- Mathematics
|
- Mathematics
|
||||||
- Algebra
|
tags: [algebra, exponents]
|
||||||
- exponents
|
|
||||||
---
|
---
|
||||||
|
|
||||||
## Equivalent equations
|
## Equivalent equations
|
||||||
|
|
||||||
>
|
>
|
||||||
|
|
|
||||||
|
|
@ -1,10 +1,8 @@
|
||||||
---
|
---
|
||||||
tags:
|
categories:
|
||||||
- Mathematics
|
- Mathematics
|
||||||
- Algebra
|
tags: [algebra]
|
||||||
- logarithms
|
|
||||||
---
|
---
|
||||||
|
|
||||||
Most simply a logarithm is a way of answering the question:
|
Most simply a logarithm is a way of answering the question:
|
||||||
|
|
||||||
>
|
>
|
||||||
|
|
|
||||||
|
|
@ -1,8 +1,7 @@
|
||||||
---
|
---
|
||||||
tags:
|
categories:
|
||||||
- Mathematics
|
- Mathematics
|
||||||
- Algebra
|
tags: [algebra, exponents]
|
||||||
- exponents
|
|
||||||
---
|
---
|
||||||
|
|
||||||
When calculating the exponents of a negative number the answer will always will be positive:
|
When calculating the exponents of a negative number the answer will always will be positive:
|
||||||
|
|
|
||||||
|
|
@ -1,10 +1,8 @@
|
||||||
---
|
---
|
||||||
tags:
|
categories:
|
||||||
- Mathematics
|
- Mathematics
|
||||||
- Algebra
|
tags: [algebra]
|
||||||
- operators
|
|
||||||
---
|
---
|
||||||
|
|
||||||
## Use inversion of operators
|
## 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.
|
When solving equations we frequently make use of the [ operator inversion rules](../Prealgebra/Inversion%20of%20operators.md) to find the solutions.
|
||||||
|
|
|
||||||
|
|
@ -2,7 +2,7 @@
|
||||||
tags:
|
tags:
|
||||||
- Programming_Languages
|
- Programming_Languages
|
||||||
- shell
|
- shell
|
||||||
- abra
|
- abracadabra
|
||||||
---
|
---
|
||||||
# Cron
|
# Cron
|
||||||
|
|
||||||
|
|
|
||||||
Loading…
Add table
Reference in a new issue