Logic can use declarative sentences {statement, logic} {logical statement} with only constants, no variables. Statements are always either true or false.
relations
Statement contains subject and action, state, or relation {predicate}. Statements have. Statement uses basic elements {symbol} with no meaning.
meaning
Statement meaning associates actual objects and events with symbols and relations. For example, statements can describe parts, relations among parts, and spatial axes.
language
Statements are built from pre-language ideas such as number, case, gender, nouns, verbs, modifiers, tenses, gerunds, infinitives, particles, prepositions, and articles. Pre-language ideas include spatial and temporal ideas such as line, group, boundary, figure and background, movement, ascending and descending, association, and attraction and repulsion.
Logic can use declarative sentences {proposition, logic} {formula, logic} with constants and variables. Propositions are true or false, depending on variable substitution. Propositions contain subject and predicate. Propositions allow analogy, perspective, coordinates, space, time, fields, stories, and images.
Propositions can be neither provable nor unprovable {undecidability, logic}, because proof is not finite. Propositional-calculus propositions are decidable. Predicate-calculus theorems have no terminus and are not decidable.
Two opposing statements cannot be true, but both can be false {contrary, logic}. Two propositions can be contraries, such as "All A are B" and "No A are B". Two opposing statements sometimes are not both false {subcontrary}.
Functions {propositional function} can have formulas with variables. Propositional functions are true or false, depending on variable value.
Subjects and predicates with variables {sentential function} can make sentences. Sentential functions have constant connectives.
Premise's negative {contradictory, logic} has opposite truth-value.
Changing subject to negative of predicate and predicate to negative of subject can make new statements {contrapositive}| {transposition, logic} {contraposition}. "If A then B" transposes to "If not B, then not A". If theorem is true, contrapositive is true. If contrapositive is true, theorem is true. "All A are B" transposes to "All not B are not A". "Some A are not B" transposes to "Some B are not A". "No A are B" transposes to "No not B are not A". "Some A are B" transposes to "Some not B are not A". For true statements, contrapositive of "All A are B" and "Some A are not B" are true. For true statements A and B, "No A are B" and "Some A are B" contrapositives are not true.
Exchanging subject and predicate can make new statements {converse, logic}|. For example, "No A is B" has converse "No B is A". "Some A is B" converts to "Some B is A". For categorical statements "No A is B" and "Some A is B", if statement is true, converse is true. All other categorical converse statements must have independent proof. Conversion {conversion per accidens} from "All A are B" to "Some B are A" is valid.
Changing connectives to opposite makes negative statements {denial} {negation}. NOT added to statements denies statement: NOT a or -a. Negation changes statement truth-value. Negative statements only entail similar qualities and are less specific. Negation can result in statements about ambiguous, non-existent, or arbitrary things. In common language, negative particles, word meanings, or inflections negate statements.
Negation can apply to sentences with quantities. NOT every a is b, so Every a is NOT b {equipollence, negation}|.
Changing subject to negative subject {inversion, logic} {inverse, logic} makes new statements. For example, "All A are B" inverts to "All not A are B". "All A are B" implies "Some not A are not B". If inverse is true, converse is true. If converse is true, inverse is true.
Changing predicate to complement or negative and negating the statement can make new statements {obverse}|. Obversions of the four categorical forms are valid. All A are B, so Not (All A are not B). Some A are B, so Not (Some A are not B). Some A are not B, so Not (Some A are B). No A are B, so Not (No A are not B). If converse is true, obverse is true.
Facts, judgments, or expert testimony can provide statements from which to reason {premise}. Arguments start with premises. Premises can be definitions, axioms, postulates, or previously proved theorems.
not contradictory
Premises must have no contradictions.
not dependent
Premises must be independent.
definition
Definition includes how to use word {operational definition, premise}, what words can substitute {synonym, premise}, class and distinction, pointing at object {ostensive definition, premise}, and conceiving example. Definition can eliminate ambiguity and clarify idea. Word has precise meaning {denotation, meaning}, as well as properties, associations, and feelings {connotation, meaning}. Theory can explain definition, which can influence attitudes and increase vocabulary.
axiom
Premises can be about general or fundamental objects or symbols, assumed to be true.
postulates
Premises can be general statements about mathematical or logical objects and symbols. Anything implied by an elementary and true proposition is true. Disjunction of proposition with itself implies proposition: (a | a) -> a. Propositions imply disjunction of themselves and other propositions: a -> (a | b). Disjunction of one proposition with another implies disjunction of other and first {commutation, postulate}: (a | b) -> (b | a). Disjunction of proposition with disjunction of two other propositions implies disjunction of second with disjunction of first and third {association, postulate}: (a | (b | c)) -> (b | (a | c)). Assertion that statement is true and that the statement implies second statement is true is equivalent to assertion that second statement is true {modus ponens, postulate}. a. a -> b. b. Assertion that proposition implies second proposition is equivalent to disjunction of inverse of first proposition and second proposition {implication, postulate}. (a -> b) = (-a | b). These postulates have no proof that they are independent or consistent.
Reasoning wants to obtain true results {conclusion, logic} {theorem}. Conclusions are true or false. Conclusions must be more general than premises.
If theorem is true, other statements {corollary} can be true.
To prove theorems can require first proving another theorem {lemma}.
Statements {analytic statement} can be true because subject and object meanings are the same, or one is part of other. Using general logical laws and definitions, denying analytic statements leads to contradiction, so analytic statements are logically necessary.
Statements {necessary truth}| can be true and cannot be false, like arithmetical equalities.
Statements {tautology}| {valid formula} can be true in themselves, without reference to anything else. Tautologies are true for any term or predicate substitutions. Tautologies have necessary and sufficient conditions. Tautologies depend on accepted definitions. The opposite of tautology is self-contradictory.
Statements {categorical statement} can use All, Some, or No.
Statements {existential import} can imply that something exists, as in "Some A Is B". Existential import can be in statements with "Some" and in statements with "All" or "No" that refer to particular or individual.
Statements {modal judgment} can be about the possible or necessary.
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Date Modified: 2022.0225