Approaches to logic are logicist, intuitionist, and formalist {logic foundations}.
Universally accepted logical principles plus simple formal systems can establish logic and formal-system consistency {formalism, logic}. Formalism tries to establish arithmetic, number theory, and logic consistency and foundations, without set theory.
formula
Formalism uses symbolic expressions {formula} for logical relations. Formulas connect symbols using logic rules {well-formed formula} {wff} and therefore have syntax. Formulas have truth-values. Formulas have meaning though they have no words.
schema
Sentence or formula can use term or clause placeholders {schema, logic}. Schema is not true or false, until terms or clauses substitute for placeholders.
Logic comes from mathematics {intuitionism}. Whole numbers come from time intuition. The only proof of existence is to make something exist. Truth is about provability or assertibility. In intuitionism, all definitions and proofs are constructive. Excluded-middle law can only work for proofs with finite numbers of steps. Double negation is not equivalent to original statement. Kripke trees can formalize intuitionist logic.
Statements can be true for observer, be false for observer, have later decision, or never have decision {intuitionist logic} {topos theory}. Observer actions can access different information and can affect truth. Shared observations have same truth. The same information always gives same truth-value.
Logic can be an axiomatic system {logicism}. Undefined terms are elementary proposition, propositional function, elementary-proposition truth assertion, proposition negation, and proposition disjunction {inclusive OR, logicism}. Syllogism rules are theorems.
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Date Modified: 2022.0225