Formation Rule - Propositional and Predicate Logic

Propositional and Predicate Logic

The formation rules of a propositional calculus may, for instance, take a form such that;

if we take Φ to be a propositional formula we can also take ¬Φ to be a formula;
if we take Φ and Ψ to be a propositional formulas we can also take (Φ & Ψ), (Φ → Ψ), (Φ Ψ) and (Φ Ψ) to also be formulas.

A predicate calculus will usually include all the same rules as a propositional calculus, with the addition of quantifiers such that if we take Φ to be a formula of propositional logic and α as a variable then we can take (α)Φ and (α)Φ each to be formulas of our predicate calculus.

Logic

Overview

Academic areas	Argumentation theory Axiology Critical thinking Computability theory Formal semantics History of logic Informal logic Logic in computer science Mathematical logic Mathematics Metalogic Metamathematics Model theory Philosophical logic Philosophy Philosophy of logic Philosophy of mathematics Proof theory Set theory

Foundations	Abduction Analytic truth Antinomy A priori Deduction Definition Description Induction Inference Logical form Logical consequence Logical truth Name Necessity Meaning Paradox Possible world Presupposition Probability Reason Reasoning Reference Semantics Statement Strict implication Substitution Syntax Truth Truth value Validity

Philosophical logic

Critical thinking Informal logic	Analysis Ambiguity Argument Belief Bias Credibility Evidence Explanation Explanatory power Fact Fallacy Inquiry Opinion Parsimony Premise Propaganda Prudence Reasoning Relevance Rhetoric Rigor Vagueness

Theories of deduction	Constructivism Dialetheism Fictionalism Finitism Formalism Intuitionism Logical atomism Logicism Nominalism Platonic realism Pragmatism Realism

Metalogic Metamathematics

Cantor's theorem Church's theorem Church's thesis Consistency Effective method Foundations of mathematics Gödel's completeness theorem Gödel's incompleteness theorems Soundness Completeness Decidability Interpretation Löwenheim–Skolem theorem Metatheorem Satisfiability Independence Type–token distinction Use–mention distinction

Mathematical logic

General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory

Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables

Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Set theory	Set Empty set Enumeration Extensionality Finite set Function Subset Power set Countable set Recursive set Domain Range Ordered pair Uncountable set

Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity

Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax

Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

Non-classical logic

Modal logic	Alethic Axiologic Deontic Doxastic Epistemic Temporal

Intuitionism	Intuitionistic logic Constructive analysis Heyting arithmetic Intuitionistic type theory Constructive set theory

Fuzzy logic	Degree of truth Fuzzy rule Fuzzy set Fuzzy finite element Fuzzy set operations

Substructural logic	Structural rule Relevance logic Linear logic

Paraconsistent logic	Dialetheism

Description logic	Ontology Ontology language

Logicians

Anderson Aristotle Averroes Avicenna Bain Barwise Bernays Boole Boolos Cantor Carnap Church Chrysippus Curry De Morgan Frege Geach Gentzen Gödel Hilbert Kleene Kripke Leibniz Löwenheim Peano Peirce Putnam Quine Russell Schröder Scotus Skolem Smullyan Tarski Turing Whitehead William of Ockham Wittgenstein Zermelo

Lists

Topics	Index of logic articles Mathematical logic Boolean algebra Set theory

Other	Logicians Rules of inference Paradoxes Fallacies Logic symbols

Common logical symbols

& ∨ ¬ ~ → ⊃ ≡ \| ∀ ∃ ⊤ ⊥ ⊢ ⊨ ∴ ∵

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