January 12
... Such limitations of Aristotelian logic give Frege criteria for the adequacy of a proper logical notation, or Begriffsschrift. Such a notation must be able to do 2 things: A. express all (and only) propositions, i.e., all and only things that are either true or false; and B. state all logical relati ...
... Such limitations of Aristotelian logic give Frege criteria for the adequacy of a proper logical notation, or Begriffsschrift. Such a notation must be able to do 2 things: A. express all (and only) propositions, i.e., all and only things that are either true or false; and B. state all logical relati ...
Gresham Ideas - Gresham College
... thinking, for example, about the Riemann Hypothesis, establishing the truth of which is an unsolved mathematical problem. But if you are unsure, then the proposition in that case is true, so you have established it and you aren’t unsure. Therefore the proposition must be true: and yet as far as any ...
... thinking, for example, about the Riemann Hypothesis, establishing the truth of which is an unsolved mathematical problem. But if you are unsure, then the proposition in that case is true, so you have established it and you aren’t unsure. Therefore the proposition must be true: and yet as far as any ...
Informal proofs
... Methods of proving theorems Basic methods to prove the theorems: • Direct proof – p q is proved by showing that if p is true then q follows • Indirect proof – Show the contrapositive ¬q ¬p. If ¬q holds then ¬p follows • Proof by contradiction – Show that (p ¬ q) contradicts the assumptions • P ...
... Methods of proving theorems Basic methods to prove the theorems: • Direct proof – p q is proved by showing that if p is true then q follows • Indirect proof – Show the contrapositive ¬q ¬p. If ¬q holds then ¬p follows • Proof by contradiction – Show that (p ¬ q) contradicts the assumptions • P ...
Chapter1_Parts2
... observations and the knowledge base are consistent (i.e., satisfiable).! The augmented knowledge base is clearly not consistent if the assumables are all true. The switches are both up, but the lights are not lit. Some of the assumables must then be false. This is the basis for the method to diagnos ...
... observations and the knowledge base are consistent (i.e., satisfiable).! The augmented knowledge base is clearly not consistent if the assumables are all true. The switches are both up, but the lights are not lit. Some of the assumables must then be false. This is the basis for the method to diagnos ...
Justifying Underlying Desires for Argument
... their individual problems on negotiation. However, determining agent’s underlying desires behind the given goals or proposals, or revising them is outside the scope of these literatures. In [8–10], the authors give mechanisms for generating knowledge-dependent and context-dependent desires in practi ...
... their individual problems on negotiation. However, determining agent’s underlying desires behind the given goals or proposals, or revising them is outside the scope of these literatures. In [8–10], the authors give mechanisms for generating knowledge-dependent and context-dependent desires in practi ...
Logic, Sets, and Proofs
... Direct Proof. The simplest way to prove A ⇒ B is to assume A (the “hypothesis”) and prove B (the “conclusion”). See Proof 2 in Section 5 for a direct proof of n is even ⇒ n2 is even. Proof by Contradiction. One way to prove A ⇒ B is to assume that A is true and B is false. In other words, you assume ...
... Direct Proof. The simplest way to prove A ⇒ B is to assume A (the “hypothesis”) and prove B (the “conclusion”). See Proof 2 in Section 5 for a direct proof of n is even ⇒ n2 is even. Proof by Contradiction. One way to prove A ⇒ B is to assume that A is true and B is false. In other words, you assume ...
![overhead 12/proofs in predicate logic [ov]](http://s1.studyres.com/store/data/011612593_1-f0437b45ca3df4f633ddcc6197d427a6-300x300.png)
overhead 12/proofs in predicate logic [ov]
... universal statement WAS derived from universal statements, we require that this instance is derived within a "flagged subproof" and apply the following restrictions: R1 A letter being flagged must be new to the proof, that is, it may not appear, either in a formula or as a letter being flagged, prev ...
... universal statement WAS derived from universal statements, we require that this instance is derived within a "flagged subproof" and apply the following restrictions: R1 A letter being flagged must be new to the proof, that is, it may not appear, either in a formula or as a letter being flagged, prev ...
PHIL12A Section answers, 9 February 2011
... true and ¬Small(b) is true. But then Small(a) is false and so is Small(b). This just means that a is either medium or large, and b is either medium or large. Since neither a nor b is small, Small(a)∨Small(b) is false, since neither disjunct is true. But then ¬Small(a)∨Small(b) must be true. This sho ...
... true and ¬Small(b) is true. But then Small(a) is false and so is Small(b). This just means that a is either medium or large, and b is either medium or large. Since neither a nor b is small, Small(a)∨Small(b) is false, since neither disjunct is true. But then ¬Small(a)∨Small(b) must be true. This sho ...
Document
... Quantifiers and First Order Logic Formulas in Predicate Logic All statement formulas are considered formulas Each n, n =1,2,...,n-place predicate P( x1 , x2 , ... , xn ) containing the variables x1 , x2 , ... , xn is a formula. If A and B are formulas, then the expressions ~A, (A∧B), (A∨B) , A ...
... Quantifiers and First Order Logic Formulas in Predicate Logic All statement formulas are considered formulas Each n, n =1,2,...,n-place predicate P( x1 , x2 , ... , xn ) containing the variables x1 , x2 , ... , xn is a formula. If A and B are formulas, then the expressions ~A, (A∧B), (A∨B) , A ...
Predicate Logic
... • Representing knowledge using logic is appealing because you can derive new knowledge from old mathematical deduction. • In this formalism you can conclude that a new statement is true if by proving that it follows from the statement that are already known. • It provides a way of deducing new state ...
... • Representing knowledge using logic is appealing because you can derive new knowledge from old mathematical deduction. • In this formalism you can conclude that a new statement is true if by proving that it follows from the statement that are already known. • It provides a way of deducing new state ...
CA208ex1 - DCU School of Computing
... Intutively, are the inferences above logically valid (i.e. is the conclusion true in all situations where the premises are true)? Is the following inference logically valid? ...
... Intutively, are the inferences above logically valid (i.e. is the conclusion true in all situations where the premises are true)? Is the following inference logically valid? ...
p q
... • Basic idea is to assume that the opposite of what you are trying to prove is true and show that it results in a violation of one of your initial assumptions. • In the previous proof we showed that assuming that the sum of a rational number and an irrational number is rational and showed that it re ...
... • Basic idea is to assume that the opposite of what you are trying to prove is true and show that it results in a violation of one of your initial assumptions. • In the previous proof we showed that assuming that the sum of a rational number and an irrational number is rational and showed that it re ...
1 Proof of set properties, concluded
... We can use logic to describe set properties in interesting ways, by associating statements with membership of a named object x in the various sets. Let us consider giving specific names to statements asserting membership, i.e. for sets A and B, let P be the statement x ∈ A, and Q be the statement x ...
... We can use logic to describe set properties in interesting ways, by associating statements with membership of a named object x in the various sets. Let us consider giving specific names to statements asserting membership, i.e. for sets A and B, let P be the statement x ∈ A, and Q be the statement x ...
Chapter 1 Section 2
... observations and the knowledge base are consistent (i.e., satisfiable). The augmented knowledge base is clearly not consistent if the assumables are all true. The switches are both up, but the lights are not lit. Some of the assumables must then be false. This is the basis for the method to diagnose ...
... observations and the knowledge base are consistent (i.e., satisfiable). The augmented knowledge base is clearly not consistent if the assumables are all true. The switches are both up, but the lights are not lit. Some of the assumables must then be false. This is the basis for the method to diagnose ...
pdf
... Is tactics for building proofs up When all thru this stuff you have churned Prop logic the students have learned To you they’ll direct Their thanks and respect Respect that indeed you have earned Now show some techniques to apply That do on the logic rely ...
... Is tactics for building proofs up When all thru this stuff you have churned Prop logic the students have learned To you they’ll direct Their thanks and respect Respect that indeed you have earned Now show some techniques to apply That do on the logic rely ...
Uninformed Search
... • We can use symbols P, Q, and R to denote the three propositions, but this leads us to nowhere because knowledge important to infer R from P and Q (i.e., relationship between being a human and mortality, and the membership relation between Confucius and human class) is not expressed in a way that c ...
... • We can use symbols P, Q, and R to denote the three propositions, but this leads us to nowhere because knowledge important to infer R from P and Q (i.e., relationship between being a human and mortality, and the membership relation between Confucius and human class) is not expressed in a way that c ...
Knowledge Representation
... • Like in propositional logic, it is all about determining whether something is true or false. • X P(X) means that P(X) must be true for every object X in the domain of interest. • X P(X) means that P(X) must be true for at least one object X in the domain of interest. • So if we have a domain o ...
... • Like in propositional logic, it is all about determining whether something is true or false. • X P(X) means that P(X) must be true for every object X in the domain of interest. • X P(X) means that P(X) must be true for at least one object X in the domain of interest. • So if we have a domain o ...
A Primer on Mathematical Proof
... is not enough to simply plug in a few numbers for x and check in those cases. Playing around with specific numbers may help you discover the proof, but it is not sufficient for the final proof write-up. In contrast, you can disprove a statement by finding a single example where the hypotheses hold b ...
... is not enough to simply plug in a few numbers for x and check in those cases. Playing around with specific numbers may help you discover the proof, but it is not sufficient for the final proof write-up. In contrast, you can disprove a statement by finding a single example where the hypotheses hold b ...
Truth, Conservativeness and Provability
... consistency of S, and by Gödel’s second incompleteness theorem S by itself doesn’t prove that. What can the deflationist do? In Ketland’s opinion there is only one strategy available to him: he should deny that (GR) should follow from his theory of truth and at the same time offer some non-truth-the ...
... consistency of S, and by Gödel’s second incompleteness theorem S by itself doesn’t prove that. What can the deflationist do? In Ketland’s opinion there is only one strategy available to him: he should deny that (GR) should follow from his theory of truth and at the same time offer some non-truth-the ...
WUMPUS
... • This game appears to have been the first to use a non-random graph-structured map (as opposed to a rectangular grid like the even older Star Trek games). • In this respect, as in the dungeon-like setting and its terse, amusing messages, it prefigured ADVENT and Zork. •It was directly ancestral to ...
... • This game appears to have been the first to use a non-random graph-structured map (as opposed to a rectangular grid like the even older Star Trek games). • In this respect, as in the dungeon-like setting and its terse, amusing messages, it prefigured ADVENT and Zork. •It was directly ancestral to ...
p and q
... • Basic idea is to assume that the opposite of what you are trying to prove is true and show that it results in a violation of one of your initial assumptions. • In the previous proof we showed that assuming that the sum of a rational number and an irrational number is rational and showed that it re ...
... • Basic idea is to assume that the opposite of what you are trying to prove is true and show that it results in a violation of one of your initial assumptions. • In the previous proof we showed that assuming that the sum of a rational number and an irrational number is rational and showed that it re ...
ch1_1
... 2. Show that ~p is also true. 3. Then we have that p ^ (~p) is true. 4. But this is impossible, since the statement p ^ (~p) is always false. There is a contradiction! 5. So, q cannot be false and therefore it is true. ...
... 2. Show that ~p is also true. 3. Then we have that p ^ (~p) is true. 4. But this is impossible, since the statement p ^ (~p) is always false. There is a contradiction! 5. So, q cannot be false and therefore it is true. ...
slides
... Want a way to prove partial correctness statements valid... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional ...
... Want a way to prove partial correctness statements valid... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional ...
1 TRUTH AND MEANING Ian Rumfitt C.E.M. Joad`s catchphrase—`It
... Two initial observations about this derivation: ...
... Two initial observations about this derivation: ...
Solutions
... Problem 8: Use direct method to prove the following theorem: Theorem: If ∀a, b, c ∈ Z , If ab c, then a c Solution: (12pt) Theorem: If ∀a, b, c ∈ Z , If ab c, then a c Proof: Since ab c , by definition of divisible we have that (ab)q = c where q is some integer. By commutative law we have a(bq) = c ...
... Problem 8: Use direct method to prove the following theorem: Theorem: If ∀a, b, c ∈ Z , If ab c, then a c Solution: (12pt) Theorem: If ∀a, b, c ∈ Z , If ab c, then a c Proof: Since ab c , by definition of divisible we have that (ab)q = c where q is some integer. By commutative law we have a(bq) = c ...