Symbolic Logic

Symbolic Logic explores the principles of formal reasoning and the use of symbols to represent logical expressions.

Propositions and Logical ConnectivesTranslating Statements into SymbolsTruth Tables and Logical Equivalence
Predicate Logic and QuantifiersRules of InferenceLogical Fallacies and Invalid Reasoning
Logic in Computer ScienceAnalyzing Arguments in Everyday LifeLogic in Law and Ethics
Building a Logic Study RoutineTackling Complex Arguments Step by Step
Propositions and Log...Translating Statemen...Truth Tables and Log...Predicate Logic and ...Rules of InferenceLogical Fallacies an...Logic in Computer Sc...Analyzing Arguments ...Logic in Law and Eth...Building a Logic Stu...Tackling Complex Arg...
Advanced Topics

Predicate Logic and Quantifiers

Beyond Simple Statements

Predicate logic lets us talk about properties of objects and relationships between them, not just whole statements.

Predicates

A predicate says something about an object. For example, \( P(x) \) could mean "x is a dog". You can apply it to different objects: \( P(Fido) \), \( P(Bella) \), etc.

Quantifiers

  • Universal quantifier (\( \forall \)): Means "for all". Example: \( \forall x, P(x) \) means "Everything is a dog".
  • Existential quantifier (\( \exists \)): Means "there exists". Example: \( \exists x, P(x) \) means "There is at least one dog".

Symbolizing Statements

Quantifiers make it possible to express things like "Everyone loves pizza" (\( \forall x, L(x, \text{pizza}) \)) or "Someone is sad" (\( \exists x, S(x) \)).

Why Is This Useful?

Predicate logic allows for much more detailed and flexible reasoning, especially in mathematics and computer science.

Key Formula

\[\forall x, P(x) \rightarrow Q(x)\]

Examples

  • Using \( \forall x, (C(x) \rightarrow M(x)) \) to say 'All cats are mammals'.

  • Expressing 'Some students are happy' as \( \exists x, (S(x) \land H(x)) \).

In a Nutshell

Predicate logic uses quantifiers and predicates to express complex ideas about objects and their properties.

Key Terms

Predicate
A property or relation that can be true or false about an object.
Quantifier
A symbol that indicates 'for all' (\( \forall \)) or 'there exists' (\( \exists \)).
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Rules of Inference