Reasoning in Cognitive Science: Deductive Validity & Argument Forms - Prof. Bram Van Heuve, Study notes of Computer Science

Download Reasoning in Cognitive Science: Deductive Validity & Argument Forms - Prof. Bram Van Heuve and more Study notes Computer Science in PDF only on Docsity! AI: Reasoning Introduction to Cognitive Science Normative and Descriptive Theories of Reasoning • Psychology of reasoning is a scientific study of how humans reason: – What do humans infer from what? – What is the mechanism behind human reasoning? • As such, psychologists (and cognitive scientists) come up with descriptive theories of reasoning: hypotheses as to how humans reason based on empirical studies. • Logicians (philosophers), however, try to come up with normative theories of reasoning: – What actually follows from what? Argument Forms • “If I win the lottery, then I am poor. I win the lottery. Hence, I am poor.” • This argument has the following abstract structure or form: “If P then Q. P. Hence, Q” • Any argument of the above form is valid, including “If flubbers are gook, then trugs are brig. Flubbers are gook. Hence, trugs are brig.”! • Hence, we can look at the abstract form of an argument, and tell whether it is valid without even knowing what the argument is about!! Formal Logic • Formal logic studies the validity of arguments by looking at the abstract form of arguments. • Formal logic thus works in 2 steps: – Step 1: Use certain symbols to express the abstract form of premises and conclusion. – Step 2: Use a certain procedure to figure out whether the conclusion follows from the premises based on their symbolized form alone. Example • “Either the housemaid or the butler killed Mr. X. However, if the housemaid would have done it, the alarm would have gone off, and the alarm did not go off. Therefore, the butler did it.” Truth-Tables P ¬P T T F F P P ∧ Q T T F F Q T FF F F F T T P P ∨ Q T T F T Q T FF T F F T T Housemaid, Butler, and Alarm Sa ea Uae era File Edit Table Window Help Av oa ee Tet Srrall Leftot SareCol Adjoir Delete Column) Verify Row abede ft Cube Medium = RightOf SameRow Betwe Build Rer cols) Verify Tab Wee et) 4 Dodec Large FrontOf Smaller Samesh 2°] ————$—! ty x Y ZU ¥ WwW Samesize BackOf Larger Fill Ref Cols | Verify Assess Correct? Complete? Assessment (mone given) F2= a= a Q@) @) qh) H |B JA |JH¥ BJH>A]-A/B manana saas 71 7am a mamanatnta mas 4494 Sa 4gnTm4aTGH a74an74aaq47 apa a Be ae m4 De Morgan’s Laws P ¬(P ∧ Q) T T F F Q T FF F F F T T T F T T ¬P ∨ ¬Q T F T F F F T T T F T T ↑ ↑ ¬(P ∨ Q) ¬P ∧ ¬Q T T F T F F T F T F T F F F T T F F T F ↑ ↑ So: ⇔ ¬(P ∧ Q) ⇔ ¬P ∨ ¬Q ¬(P ∨ Q) ⇔ ¬P ∧ ¬Q Modus Ponens P P → Q T T F T Q T TF F F F T T P → Q P PQ Q T T F FF F T T ← We see that whenever the premises (P → Q and P) are true, the conclusion (Q) is also true. Hence, given the premises, the conclusion is necessarily true as well. So, this argument is deductively valid. Some Valid Inferences P → Q ¬Q ¬P Modus Tollens P → Q Q P Modus Ponens P → Q Q → R P → R Hypothetical Syllogism P ∨ Q ¬P Q Disjunctive Syllogism P → Q (P ∧ R) → Q P → (Q ∧ R) P → Q Strengthening the Antecedent Weakening the Consequent Some Invalid |Inferences ory ory WV 718 © a Affirming the Denying the Consequent Antecedent Fitch Rules P ∧ Q P  P ∧ Q Q P ∨ Q P  P ∨ Q P  S Q  S S ∧E ∨E ∨I ∧I P ¬¬P  ¬E P ¬P  ⊥ ¬I P ⊥  ⊥E P ¬P  ⊥  P ⊥I P P → Q  Q P P → Q  Q P P ↔ Q  Q P P ↔ Q  Q Q  P ↔E →E ↔I →I Proof by Cases Proof by Contradiction Housemaid, Butler, and Alarm elas Eitan ni File Edit Proof Goal Window Help - L h ae 7. Tet ‘Strvall LeftOt SameCal Adioir Check Ste | abedef Cube Medium: RightOf SameRow Betwe ; —, ¥ad 241 4] Dodec Large FrontO? Smaller Samesh > verity praot_| Cif x ye uw Samesize = BackOf Larger Goal Constraints| @HYB wh BHA Baa FH a4 vo¥ > Elim al vo 71 Intro 3B v7 1 Elim we 3B vo y Elim Goals | be ¢ The goal checks out J Some Formal Logic Axioms for Set Theory • Identity: – ∀x ∀y (x = y ↔ ∀z (z ∈ x ↔ z ∈ y)) • Subset: – ∀x ∀y (x ⊆ y ↔ ∀z (z ∈ x → z ∈ y)) • Intersection: – ∀x ∀y ∀z (z ∈ x ∩ y ↔ (z ∈ x ∧ z ∈ y)) • Union: – ∀x ∀y ∀z (z ∈ x ∪ y ↔ (z ∈ x ∨ z ∈ y)) Automated Theorem Proving • Formal proofs seem perfect for automation: proofs require tediously many applications of precisely defined rules: just something a computer would be good at! • Problem: the rules of logic are like the rules of chess: they tell you what you can do, but not what you must do. • In Automated Theorem Proving (a branch of Artificial Intelligence) researchers try and come up with algorithms to create formal proofs. How good are Automated Theorem Provers? • Well, pretty disappointing, really! • In 1956, things looked promising: the Logic Theorist was able to prove a theorem from Russell and Whitehead’s book Principia Mathematica using a shorter proof than Russell and Whitehead themselves had found. • This, by the way, is often seen as the birth of AI. • However, 50 years later, the best ATP’s around still can’t prove that P(∅) = {∅} given basic set theory axioms. • Some researchers see this as evidence that human thought cannot be captured through computation (i.e. that AI is a pipe dream), but others say it’s too early to tell. Automated Theorem Proving Demo Prolog Example H ⇒ H Putting into Prolog: H → E ⇒ E :- H. H → D ⇒ D :- H. (E ∧ M) → R ⇒ R :- E, M. (D ∧ E) → R ⇒ R :- D, E. Query: R? {R} {E, M} {H, M} {M} {D, E} {H, E} {E} {H} {} ‘Yes’! Prolog Demo Summary • Reasoning can be studied by logic – Deductive reasoning can be studied through formal logic • In addition to being a tool for analysis, formal logic can be used as a tool for synthesis: use logic to do reasoning – Formal logic can be automated -> Automated Theorem Proving! • However: – Formal proofs are long, and can be hard to construct. Indeed, Automated Theorem Provers still struggle with pretty elementary theorems (at least in context of mathematics) – Automated Theorem Proving = Automated Theorem Verification ≠ Automated Proof Generation ≠ Automated Theorem Generation – What about non-deductive reasoning?