Thinking {reasoning} can start with true and complete facts and make logically valid inferences. If reasoning needs testimony, testimony must have no bias. All parties accept all judgments. Causal reasoning can have errors. Use effect as cause. Use something as cause just because it happened first. Use merely contributory cause as the only sufficient cause. Use only one cause, when causes are many.
If two things are alike in some ways, they will be alike in other ways {analogy, reasoning}.
Wholes can divide into two mutually exclusive {disjoint} parts {dichotomy}. Dividing whole into two parts can make non-mutually-exclusive parts. Dividing whole into only two parts can leave out important or necessary third parts.
Laws can cause generalizations {generalization}. Only laws support counterfactual conditionals. Other generalizations are just situations or accidental generalizations. Laws are inductions from instances, but accidents are not. Laws fit into knowledge systems, but accidents do not.
If statements "if A then B" and "B" are true, then A is probably true {heuristic reasoning, logic}|.
If premises are invariant under transformation, so is conclusion {invariance, logic}.
Properties are equivalent {equivalence of property} {property equivalence} if they determine same set. The equivalence property sign is double arrow.
Reasoning can leave out argument and only give premises and conclusions, if logic follows a recognized syllogism type {sorites, logic}.
Traditional logic used relations {square of opposition} between the four proposition forms to show inferences, contradictions, and contraries.
four forms
All a are b. No a are b. Some a are b. Some a are not b.
contraries
"All a are b" and "No a are b" are contraries, because both can be false and both cannot be true {contrary relation}.
"Some a are b" and "Some a are not b" are subcontraries, because both can be true and both cannot be false {subcontrary relation}.
contradiction
"All a are b" and "Some a are not b" are contradictories, because one must be true and one must be false {contradictory relation}. "No a are b" and "Some a are b" are contradictories, because one must be true and one must be false.
subalternation
"All a are b" entails "Some a are b" {subalternation relation}. "No a are b" entails "Some a are not b".
Universal implies particular {subalternation}. "All a are b" entails "Some a are b".
If one thing relates to another thing, second thing relates to first thing {symmetry, logic}.
Starting from statements, logical steps {argument, logic} can prove that statement is true or false. Arguments link statement and proposition constants and variables. Terms can rearrange or substitute.
variables
Propositions can use variables, such as place and time.
syllogism
Changing verbal argument into syllogism can find inconsistencies and incorrect inferences.
fallacy
Arguments can be invalid, argument forms can be invalid, or proofs can be faulty. Argument irrelevance, invalid deduction, or ambiguity can cause fallacies.
categories
People can make category mistakes.
People can state propositions {argument, obligationes}, to which other people {respondent to argument} agree, disagree, or leave open, using relation rules, such as counterfactuals {obligationes}.
Experts or authorities can state that propositions are true or false {argument from authority} {argumentum ad verecundium}, a plausible argument. Showing that proposition is false can show that proposition propounders are not experts or authorities.
Honest and believable people can state that propositions are true or false {argument from ethos}, a plausible argument. Showing that proposition is false can show that proposition propounders are not honest.
If one property happens, second property happens {argument from sign}, a plausible argument.
People can state that propositions not proved true or false are false or true {argumentum ad ignorantium}, a plausible argument. This relates to burden of proof.
Propositions can have support from emotions, such as pity {argumentum ad misericordiam}, a plausible argument. This appeals to secondary effects.
Propositions can have support by mass opinion {argumentum ad populum}, a plausible argument. This relates to peer pressure, emotional ties, or customs and traditions.
Starting from true general statement or statements, logical steps prove conclusion true {deduction}. Deduction is true if premises are true.
Proposition proofs have finite numbers of steps {decision procedure}.
Proofs {existence proof} can try to show that something exists, preliminary to showing what it is like. Disproving non-existence or proving no non-existence cannot establish existence.
Logic {natural deduction} can have only inference rules, with no axioms. It reaches results but is not about truth. Natural deduction uses sequent calculus. Basic sequent statements are premises or conclusions. Statement sequence shows reasoning chain and conclusion. Introduction rules make more-complex formulas from simpler ones. Elimination rules change complex formulas to simpler formulas. Proofs and truth-trees eliminate formulas by reductio ad absurdum {cut elimination theorem, natural deduction}.
Proof methods {reductio ad impossibile} {reductio ad absurdum}| {indirect proof} {method of contradiction} {contradiction method} can assume that negative of theorem is true, and then prove that theorem or its premise is false, establishing contradiction. For any component-statement truth-values, contradictions are always false.
Reasoning can go from true similar things to true general thing or pattern {induction, logic}. Starting from examples, induction can formulate conclusion that is not implicit in premises. Properties of some class members can predict properties of all class members.
complete
Premises can be less general than conclusion, but together they can cover all instances in conclusion {complete induction}.
numerical
If property of number one is also property of number n, then property is also property of n+1 and property of all natural numbers {numerical induction}.
eliminative
Observing many examples can find properties that remain constant or true and causes that have effects and can eliminate properties that are untrue or change and causes have no or different effect {Baconian induction} {eliminative induction, Bacon}.
invalid cases
Induction does not always apply. Valid predictions about the future based on hypothesis do not necessarily confirm the hypothesis. Two independent studies can inductively prove hypothesis, but when combined can disprove hypothesis. Highest event probability is not highest combined-event probability. Pairwise probability choices are not necessarily transitive.
Proof methods {mathematical induction} {first principle of mathematical induction} can be: Prove theorem true for the number one and then, assuming theorem is true for a number, prove it true for any number plus one. Recursive definitions or inductive definitions use mathematical induction.
Proof methods {second induction principle} {second principle of mathematical induction} can be: Theorem can be true for the number one and true for arbitrary number, assuming theorem is true for number minus 1.
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Date Modified: 2022.0225