Propositional Calculus
... Logic helps to clarify the meanings of descriptions written, for example, in English. After all, one reason for our use of logic is to state precisely the requirements of computer systems. Descriptions in natural languages can be imprecise and ambiguous. An ambiguous sentence can have more than one ...
... Logic helps to clarify the meanings of descriptions written, for example, in English. After all, one reason for our use of logic is to state precisely the requirements of computer systems. Descriptions in natural languages can be imprecise and ambiguous. An ambiguous sentence can have more than one ...
Revisiting Preferences and Argumentation
... antecedents ϕ1 , . . . , ϕn hold, then without exception, respectively presumably, the consequent ϕ holds’. There are two ways to use these rules: they could encode domain-specific information (as in e.g. default logic) but they could also express general laws of reasoning. For example, the defeasib ...
... antecedents ϕ1 , . . . , ϕn hold, then without exception, respectively presumably, the consequent ϕ holds’. There are two ways to use these rules: they could encode domain-specific information (as in e.g. default logic) but they could also express general laws of reasoning. For example, the defeasib ...
P - Bakers Math Class
... Example: Show that if x and y are integers and both x∙y and x+y are even, then both x and y are even. Proof: Use a proof by contraposition. Suppose x and y are not both even. Then, one or both are odd. Without loss of generality, assume that x is odd. Then x = 2m + 1 for some integer m. Case 1: y is ...
... Example: Show that if x and y are integers and both x∙y and x+y are even, then both x and y are even. Proof: Use a proof by contraposition. Suppose x and y are not both even. Then, one or both are odd. Without loss of generality, assume that x is odd. Then x = 2m + 1 for some integer m. Case 1: y is ...
Slides
... The action for x A is something like, “pick any x from A you want” Since a “for all” must work on everything, it doesn’t matter which you pick The action for y B is something like, “find some y from B” Since a “there exists” only needs one to work, you should try to find the one that matches ...
... The action for x A is something like, “pick any x from A you want” Since a “for all” must work on everything, it doesn’t matter which you pick The action for y B is something like, “find some y from B” Since a “there exists” only needs one to work, you should try to find the one that matches ...
x - Koc Lab
... Example: Show that if x and y are integers and both x∙y and x+y are even, then both x and y are even. Proof: Use a proof by contraposition. Suppose x and y are not both even. Then, one or both are odd. Without loss of generality, assume that x is odd. Then x = 2m + 1 for some integer k. Case 1: y is ...
... Example: Show that if x and y are integers and both x∙y and x+y are even, then both x and y are even. Proof: Use a proof by contraposition. Suppose x and y are not both even. Then, one or both are odd. Without loss of generality, assume that x is odd. Then x = 2m + 1 for some integer k. Case 1: y is ...
Review - Gerry O nolan
... or, for that matter, the rest of the book. And, in contrast to the rest of the chapter (which, by Stove's own admission (127), can only charitably be described as radical), Section (ii) is a much needed discussion of the question of whether there can ever exist purely formal judgements of invalidity ...
... or, for that matter, the rest of the book. And, in contrast to the rest of the chapter (which, by Stove's own admission (127), can only charitably be described as radical), Section (ii) is a much needed discussion of the question of whether there can ever exist purely formal judgements of invalidity ...
![overhead 7/conditional proof [ov]](http://s1.studyres.com/store/data/001382039_1-0b1da7da92f361d09e7b75df5e92d0f1-300x300.png)
overhead 7/conditional proof [ov]
... animals. Conclusion All whales are warm blooded animals. we need to represent the logical structure INTERNAL to simple sentences (REMEMBER: a simple sentence is one that does not contain any other sentence as a component--for example, dictionary entries used for symbolization in sentential logic are ...
... animals. Conclusion All whales are warm blooded animals. we need to represent the logical structure INTERNAL to simple sentences (REMEMBER: a simple sentence is one that does not contain any other sentence as a component--for example, dictionary entries used for symbolization in sentential logic are ...
On writing proofs
... contradiction are rigorous methods of proof and are logically equivalent. But . . . , they are listed in order of elegance and desirability. It can be very difficult to follow a proof by contradiction and really understand why the implication is true. However, with some problems (e.g. to show there ...
... contradiction are rigorous methods of proof and are logically equivalent. But . . . , they are listed in order of elegance and desirability. It can be very difficult to follow a proof by contradiction and really understand why the implication is true. However, with some problems (e.g. to show there ...
Lesson 1 Contents - Headlee's Math Mansion
... • Axiom – or a postulate, is a statement that describes a fundamental relationship between the basic terms of geometry ...
... • Axiom – or a postulate, is a statement that describes a fundamental relationship between the basic terms of geometry ...
Document
... remember: we’re always stating a truth about the world, hence every wff is something that has a Boolean value (it is either a true or a false statement about the world) ...
... remember: we’re always stating a truth about the world, hence every wff is something that has a Boolean value (it is either a true or a false statement about the world) ...
Chapter 2, Logic
... Extracting from the third diagram just the information about gardeners and pipe smokers, we have the last diagram, which corresponds to the conclusion. Does Logic just state the obvious? Perusal of examples provided by logicians often suggests that their validity is so obvious that it is hardly wort ...
... Extracting from the third diagram just the information about gardeners and pipe smokers, we have the last diagram, which corresponds to the conclusion. Does Logic just state the obvious? Perusal of examples provided by logicians often suggests that their validity is so obvious that it is hardly wort ...
Predicate logic. Formal and 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 • ...
... 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 • ...
ppt - Purdue College of Engineering
... Examples where propositional logic fails Every positive number is greater than zero. Five is a positive number. Therefore, five is greater than zero. Minimal statements? A = Every positive number is greater than zero. B = Five is a positive number. C = Five is greater than zero. Hypotheses: A, B. C ...
... Examples where propositional logic fails Every positive number is greater than zero. Five is a positive number. Therefore, five is greater than zero. Minimal statements? A = Every positive number is greater than zero. B = Five is a positive number. C = Five is greater than zero. Hypotheses: A, B. C ...
Introduction to Theoretical Computer Science, lesson 3
... then determine the truth-values of atomic formulas, and finally, determine the truth-value of the (composed) formula Evaluation of terms: Let v be a valuation that associates each variable x with an element of the universe: v(x) U. By evaluation e of terms induced by v we obtain an element e(x) of ...
... then determine the truth-values of atomic formulas, and finally, determine the truth-value of the (composed) formula Evaluation of terms: Let v be a valuation that associates each variable x with an element of the universe: v(x) U. By evaluation e of terms induced by v we obtain an element e(x) of ...
Syllogisms
... of only the few. Government is for the common good. Ergo, government is not for the good of only the few. Valid [ ] Invalid [ ] Reason: 25. If life exists on Mars, then Mars has an atmosphere. Mars has an atmosphere. Ergo, life exists on Mars. Valid [ ] Invalid [ ] Reason: 26. If li ...
... of only the few. Government is for the common good. Ergo, government is not for the good of only the few. Valid [ ] Invalid [ ] Reason: 25. If life exists on Mars, then Mars has an atmosphere. Mars has an atmosphere. Ergo, life exists on Mars. Valid [ ] Invalid [ ] Reason: 26. If li ...
Presenter Handout - MENA Common Core Conference
... Integer Wars In groups of 2, take 5 cards each from the pack. Number cards are valued as stated. Aces are 1, picture cards are 10. No Jokers. At the same time, turn your top card over, add the values of the cards together. First player to correctly state sum wins cards. ...
... Integer Wars In groups of 2, take 5 cards each from the pack. Number cards are valued as stated. Aces are 1, picture cards are 10. No Jokers. At the same time, turn your top card over, add the values of the cards together. First player to correctly state sum wins cards. ...
IntroToLogic - Department of Computer Science
... Theorem-proving would be mechanical. No intuition would be involved. Strict syntax and formal rules of inference. ...
... Theorem-proving would be mechanical. No intuition would be involved. Strict syntax and formal rules of inference. ...
9. “… if and only if …”
... The philosopher David Hume is remembered for being a brilliant skeptical empiricist. A person is a skeptic about a topic if that person both has very strict standards for what constitutes knowledge about that topic and also believes we cannot meet those strict standards. Empiricism is the view that ...
... The philosopher David Hume is remembered for being a brilliant skeptical empiricist. A person is a skeptic about a topic if that person both has very strict standards for what constitutes knowledge about that topic and also believes we cannot meet those strict standards. Empiricism is the view that ...
first order logic
... In this example, let domain be integers, Q(z) be true if z is an even number, i.e. Q(z)=even(z) P(z) be true if z is an odd number, i.e. P(z)=odd(z) Then z [Q(z) P(z)] is true, because every number is either even or odd. But x.Q(x) is not true, since not every number is an even number. Similarly ...
... In this example, let domain be integers, Q(z) be true if z is an even number, i.e. Q(z)=even(z) P(z) be true if z is an odd number, i.e. P(z)=odd(z) Then z [Q(z) P(z)] is true, because every number is either even or odd. But x.Q(x) is not true, since not every number is an even number. Similarly ...
Predicate logic, motivation
... --if there is a tilde, the quantifier applies to the tilde and to whatever the tilde applies to --if there is a parenthesis (or bracket), the quantifier applies to everything in that pair of parentheses (or brackets) A variable is bound if and only if it is within the scope of a quantifier that con ...
... --if there is a tilde, the quantifier applies to the tilde and to whatever the tilde applies to --if there is a parenthesis (or bracket), the quantifier applies to everything in that pair of parentheses (or brackets) A variable is bound if and only if it is within the scope of a quantifier that con ...
On Dummett`s Pragmatist Justification Procedure
... Here, Dummett assumes that the minor premiss A is atomic. This assumption is essential, because, according to Dummett’s original definition, a canonical argument must have an atomic conclusion. Prawitz’s counterexample is basically the same (Dummett’s example being an instance thereof) with the dif ...
... Here, Dummett assumes that the minor premiss A is atomic. This assumption is essential, because, according to Dummett’s original definition, a canonical argument must have an atomic conclusion. Prawitz’s counterexample is basically the same (Dummett’s example being an instance thereof) with the dif ...
Propositional/First
... – Q is entailed by KB (a set of premises or assumptions) if and only if there is no logically possible world in which Q is false while all the premises in KB are true. – Or, stated positively, Q is entailed by KB if and only if the conclusion is true in every logically possible world in which all th ...
... – Q is entailed by KB (a set of premises or assumptions) if and only if there is no logically possible world in which Q is false while all the premises in KB are true. – Or, stated positively, Q is entailed by KB if and only if the conclusion is true in every logically possible world in which all th ...