LOGIC AND PREDICATES APPLICATIONS
- 1. Guided By: PROF. HIREN BHATT
- 2. 91600103143 – SHYAM HINDOCHA 91600103108 – AJAY POKAR 91600103126 – DHARMARAJSINH RANA 91600103110 – MOHIT VASOYA 91600103123 – JENISH GAJERA 91600103107 – GHANSHYAM CHAUHAN 91600103112 – AYUSH JAIN 91600103114 – PRATIK SHINGALA 91600103144 – RAHUL SAVASARIYA 91600103118 – RAJAN KHUNT
- 4. INTRODUCTION & HISTORY “Logic” is a language for reasoning. It is a collection of rules we use when doing logical reasoning. Human reasoning has been observed over centuries from at least the times of Greeks, and patterns appearing in reasoning have been extracted, abstracted, and streamlined.
- 5. Charles Sanders Peirce Gottlob Frege Two founding fathers : Logic is a streamlined version of a “LANGUAGE OF THOUGHT” that was proposed in 1878 by the German philosopher and mathematician Gottlob Frege (1848 – 1925). The experience of a century of work with this language is that, in principle, it can write all of mathematics as we know it today. Peirce’s interest was general reasoning in science and daily life, and his ideas are still inspirational to modern areas philosophers, semioticists, and researchers in Artificial Intelligence.
- 7. Suppose we use B for the property of being a bicycle,M for the property of being a man, and R for the relation of riding, then we can say that the following formula describes the following salient fact about the situation in the picture: (Mx ∧By ∧Rxy) Of course, you can also write statements about the trees.
- 8. Aristotle's writings on the general subject of logic and reasoning was the chief preparatory instrument of scientific investigation. Aristotle himself, however, uses the term "LOGIC" as equivalent to verbal reasoning. The Categories of Aristotle are classifications of individual words (as opposed to sentences or propositions), and include the following ten: substance, quantity, quality, relation, place, time, situation, condition, action, passion. Notions when isolated do not in themselves express either truth or falsehood: it is only with the combination of ideas in a proposition that truth and falsity are possible. REAL INVENTOR OF LOGIC
- 9. DEFINATION: Logic is derived from greek word “logos” which means – study, reason or discourse. Logic is science and art of correct thinking.
- 10. LOGIC LOGIC •The whole world depends on logic. •New discoveries and inventions must follow logic.
- 11. •Working of Processors and Integrated Circuits (IC) LOGIC
- 13. LOGIC •Computer Programming •Designing of Chipsets
- 14. Suppose we have: “All men are mortal.” “Socrates is a man”. Does it follow that “Socrates is mortal” ? This cannot be expressed in propositional logic. We need a language to talk about objects, their properties and their relations. Extend propositional logic??? WHY? Extend propositional logic by the following new features. Logic: ∧,V,<,>,->...... Variables: x, y, z, . . . Predicates (i.e., propositional functions): P(x), Q(x), R(y), M(x, y), . . . . Quantifiers: ∀, ∃. Propositional functions are a generalization of propositions. Can contain variables and predicates, e.g., P(x). Variables stand for (and can be replaced by) elements from their domain.
- 16. INTRODUCTION & HISTORY In this section we will introduce a more powerful type of logic called predicate logic. We will see how predicate logic can be used to express the meaning of a wide range of statements in mathematics and computer science in ways that permit us to reason and explore relationships between objects. To understand predicate logic, we first need to introduce the concept of a LOGIC. Afterward, we will introduce the notion of quantifiers, which enable us to reason with statements that assert that a certain property holds for all objects of a certain type and with statements that assert the existence of an object with a particular property.
- 17. CHARLES SANDERS PEIRCE (1839–1914) He is noted as the preeminent system-building philosopher competent and productive in Predicate logic , mathematics, and a wide range of sciences. He was encouraged by his father, Benjamin Peirce, a professor of mathematics and natural philosophy at Harvard, to pursue a career in science. Instead, he decided to study logic and scientific methodology. Peirce attended Harvard (1855–1859) and received a Harvard master of arts degree (1862) and an advanced degree in chemistry from the Lawrence Scientific School (1863).
- 18. PREDICATES •Search Engine Optimization (SEO) - the process of maximizing the number of visitors to a particular website by ensuring that the site appears high on the list of results returned by a search engine.
- 20. FUN WITH LOGIC.... Applications which will help you to know why is Logic so INTRESTING and IMPORTANT.
- 23. Next we rewrite the clues inside these symbols, and logical operators: 1. s -> c 2. b V a 3. b -> y 4. –c (- = NEGATION) 5. o -> u 6. a ->s From 1. and 4., we deduce that -s. From 6. and -s, we deduce -a. Once we have -a, with 2., we deduce b. Once we have b, with 3., we deduce y, that is, the knife is in the yellow bin!
- 24. Suppose that two people play a game taking turns removing one, two, or three stones at a time from a pile that begins with 15 stones. The person who removes the last stone wins the game. Show that the first player can win the game no matter what the second player does.
- 25. Solution: To prove that the first player can always win the game, we work backward. At the last step, the first player can win if this player is left with a pile containing one, two, or three stones. The second player will be forced to leave one, two, or three stones if this player has to remove stones from a pile containing four stones. Consequently, one way for the first person to win is to leave four stones for the second player on the next-to-last move. The first person can leave four stones when there are five, six, or seven stones left at the beginning of this player’s move, which happens when the second player has to remove stones from a pile with eight stones. Consequently, to force the second player to leave five, six, or seven stones, the first player should leave eight stones for the second player at the second-to-last move for the first player. This means that there are nine, ten, or eleven stones when the first player makes this move. Similarly, the first player should leave twelve stones when this player makes the first move. We can reverse this argument to show that the first player can always make moves so that this player wins the game no matter what the second player does. These moves successively leave twelve, eight, and four stones for the second player.
- 26. THANK YOU!!!