| Next revision | Previous revision |
| public:t-713-mers:mers-24:reasoning-intro [2024/09/24 12:33] – created thorisson | public:t-713-mers:mers-24:reasoning-intro [2024/11/06 10:32] (current) – [Well-Known Syllogisms] thorisson |
|---|
| |
| =====Syllogisms===== | =====Syllogisms===== |
| | \\ What is it? | A form of deductive argument/reasoning in which a conclusion is drawn from 2 given or assumed propositions (premises / statements). The premises and the conclusion are simple declarative statements constructed using only three simple terms between them, each term appearing twice (as a subject and as a predicate) \\ E.g. **all dogs are animals; \\ all animals have four legs; \\ therefore all dogs have four legs.** \\ The argument in such syllogisms is valid by virtue of the fact that it would not be possible to assert the premises and to deny the conclusion without contradicting oneself. \\ (Based on Oxford Dictionary and Encyclopedia Britannica) | | | \\ What is it? | A form of deductive argument/reasoning in which a conclusion is drawn from 2 given or assumed propositions (premises / statements). The premises and the conclusion are simple declarative statements constructed using only three simple terms between them, each term appearing twice (as a subject and as a predicate) \\ E.g. **all dogs are animals; \\ all animals have four legs; \\ therefore all dogs have four legs.** \\ The argument in such syllogisms is valid by virtue of the fact that it would not be possible to assert the premises and to deny the conclusion without contradicting oneself. \\ (Based on Oxford Dictionary and Encyclopedia Britannica) || |
| | 3 types | Categorical, conditional and disjunctive. | | | 3 types: | Categorical, conditional and disjunctive. || |
| | \\ Categorical | The traditional type is the categorical syllogism in which both premises and the conclusion are simple declarative statements that are constructed using only three simple terms between them, each term appearing twice. \\ Assumes all premises are true. | | | \\ Categorical | The traditional type is the categorical syllogism in which both premises and the conclusion are simple declarative statements that are constructed using as few as three simple terms between them, each term appearing twice. \\ Assumes all premises are true. || |
| | \\ Conditional | Implies an "''if''" in the premises. \\ E.g. ** You are injured. \\ I am qualified to assist with injuries. \\ I can heal you.** | | | | \\ Example | All men are mortal. \\ No gods are mortal. \\ Thus, no men are gods. | |
| | \\ Disjunctive | Uses an either/or premise. \\ If it is known that at least one of two statements is true, and that it is not the former that is true; we can infer that it has to be the latter that is true. | | | Conditional | Implies an (unspoken) "''if''" in its premises. || |
| | | | \\ Example | If you are injured. \\ I am qualified to assist with injuries. \\ Thus, I can heal you (-- //given// that you accept my help). | |
| | | \\ Disjunctive | Uses an either/or premise. \\ If it is known that at least one of two statements is true, and that it is not the former that is true; we can infer that it has to be the latter that is true. || |
| | | | \\ Example | A solid thing cannot be in two places at the same time. \\ X is a solid thing currently at position P1 now. \\ Thus, it CANNOT currently be at position P2. | |
| \\ | \\ |
| |
| |
| =====Well-Known Syllogisms===== | =====Well-Known Syllogisms===== |
| | \\ Moduls Ponens | If a conditional statement \\ **if P then Q** \\ is accepted, and the antecedent P holds, then the consequent Q may be inferred. \\ E.g. **If it's raining then its cloudy. \\ It is raining. \\ Then it's cloudy. ** | | | \\ Moduls Ponens | If a conditional statement \\ **if P, then Q.** \\ is accepted, and its stated antecedent P holds, then the consequent Q must be rightly inferred. \\ E.g. **If it's raining then its cloudy. \\ It is raining. \\ Then it's cloudy. ** | |
| | \\ Moduls Tollens | A mixed syllogism that takes the form of \\ **If P, then Q. \\ Not Q. \\ Therefore, not P.** \\ Application of the general truth that if a statement is true, then so is its contrapositive ("if not-B then not-A" is the contrapositive of "if A then B"). | | | \\ Moduls Tollens | A mixed syllogism that takes the form of \\ **If P, then Q. \\ Not Q. \\ Therefore, not P.** \\ Application of the general truth that if a statement is true, then so is its contrapositive ("if not-B then not-A" is the contrapositive of "if A then B"). | |
| |
| | \\ Analogy | Figuring out how things are similar or different. \\ Making inferences about how something X may be (or is) through a comparison to something else Y, where X and Y share some observed properties. | | | \\ Analogy | Figuring out how things are similar or different. \\ Making inferences about how something X may be (or is) through a comparison to something else Y, where X and Y share some observed properties. | |
| |
| | \\ |
| | |
| | =====Non-Axiomatic Reasoning===== |
| | | \\ NAL | Distinguishes itself from other reasoning languages in that it is intended for knowledge in worlds where the axioms are unknown, not guaranteed, and/or fallible. \\ NAL is itself axiomatic, but it is designed for //domains// that are non-axiomatic. | |
| | | NAL Features | Instead of being either {T,F}, statements have a degree of truth to them, represented by a value between 0 and 1. \\ NAL uses term logic, which is different from propositional logic in the way it expresses statements. | |
| | | Evidence | w<sup>+</sup> is positive evidence; w<sup>-</sup> is negative evidence. | |
| | | \\ Uncertainty | Frequency: f = w<sup>+</sup> / w, where w = w<sup>+</sup> + w<sup>-</sup> (total evidence). \\ Confidence: c = w/(w + k), where k ≥ 1. \\ Ignorance: i = k/(w + k). | |
| | | \\ Deduction | The **premises** are given. \\ Figuring out the implication of facts (or predicting what may come). Producing implications from premises. \\ E.g. "The last domino will fall when all the other dominos between the first and the last have fallen". | |
| | | \\ Abduction | A particular **outcome X** is given. \\ Figuring out how things came to be the way they are (or how particular outcomes could be made to come about, or how particular outcomes could be prevented). \\ E.g. Sherlock Holmes, who is a genius abducer. | |
| | | \\ Induction | A **small set of examples** is given. \\ Figuring out the general case. Making general rules from a (small) set of examples. \\ E.g. "The sun has risen in the East every morning up until now, hence, the sun will also rise in the East tomorrow". | |
| | | \\ Analogy | A set of **two (or more) things** is given. \\ Figuring out how things are similar or different. Making inferences about how something X may be (or is) through a comparison to something else Y, where X and Y share some observed properties. \\ E.g. "What does a pen have in common with an arrow?" "What is the difference between a rock and a ball?" | |
| | | | <sup>Author of the Non-Axiomatic Reasoning covered here: Pei Wang </sup> | |
| | |
| | |
| | \\ |
| \\ | \\ |
| \\ | \\ |
