CST 329 - Week 6

This was our sixth week in CST 329 - Reasoning with Logic.

Chapter 13: Reasoning With Quantifiers

α: Any variable
 
β: Any symbolic term 
 
Universal Instantiation: Taking a universal claim and applying it to a specific instance. 
 
Universal Instantiation

Proof using Universal Instantiation

Existential Generalization: Taking an instance and generalizing it to a general claim. 
 
Existential Generalization
 
Proof using Existential Generalization

Existential Instantiation: Taking a general claim for ∃ and applying it to a indefinite instance that does not appear in an existing open proof.
 
 

Existential Instantiation 
 
Proof using Existential Instantiation

Chapter 14: Universal Derivation

Arbitrary Instance: A variable representing any or all of a type of thing. For example, arbitrary instance could be all squares. An apostrophe is appended to arbitrary instances, so a', b', c'... 
 
Arbitrary Term: Refers to an arbitrary instance or refers to what each instance in said arbitrary instance has in common. An apostrophe is appended to arbitrary terms, so x', y', z'...
 
Universal Derivation: Proving that something is true for an arbritrary instance, meaning that thing is true for all instances of that type. Also called Universal Proof. The arbitrary instance/term cannot appear before the beginning of the universal proof.
 
Basic form of a Universal Proof
 
Proof using Universal Proof
 
Theorem: A sentence in First Order Logic that must be true. Also called Logical Sentences. Unlike theorems in Propositional Logic, a truth table cannot be used to show that they are always true.
  • ∀x(Fx v ¬Fx) 
  • (∃xFx ↔ ¬∀x¬Fx)
  • (∀xFx ↔ ¬∃x¬Fx)
  • ((∀x (Fx → Gx) ^ ∀x (Fx → Hx))→ ∀x (Fx → (Gx ^Hx)) 
Illustrating Invalidity: To show that an argument is invalid in First Order Logic, we assign a meaning to each predicate and variable (often using natural numbers), and see if the interpretation ends up being obviously invalid.
 

 
Along with the previous image, an invalid argument and why it is invalid
 
 
 

Comments

Post a Comment

Popular posts from this blog

Week 26

CST-334 Week 2 This week was our second week in CST-334, also known as Operating Systems. Operating Systems: Three Easy Pieces This week we read chapters 4-8 of our course textbook. Chapter 4: Processes This chapter discusses processes, which are running programs, and how the OS virtualizes the CPU to run multiple processes. Key Takeaways: The OS runs multiple processes by stopping and starting them during context switches. All OSs have APIs that processes use to interact with resources and the OS. To create a process executable code of a program must be loaded from storage onto memory. Processes can be running, ready, or blocked. The OS tracks data for each process.  Chapter 5: Process API This chapter discusses certain important aspects in Process APIs. Key Takeaways: The fork() system call creates a new process. The new process is identical to its parent process when it made the system call. The wait() system call allows a process to wait for another process, or its child proces...
Read more

Week 15/16

 CST-338 Week 7/8 We are in our seventh and eight weeks of CST-338, also known as Software Engineering. Markov Revisit This week we have been asked to discuss how we would approach the Markov assignment differently if we were to do it now, after having gone through CST-338. If there is a prompt for us to follow, like before, then I would spend less time writing TODOs and detailed comments before starting to write the code. In other words, I would start writing the code sooner. The first time we did the Markov assignment, I feel like I wasted some time writing comments before starting the code, and I believe that I could have been more efficient. If there is not a prompt for us to follow, I would begin by trying to understand the basic idea of the assignment. I would ask myself what the Markov program is supposed to do, what some example use cases might be, and what the program might require based on that. Then I would write pseudo code for the program, until I felt like I had a sol...
Read more

Week 22

Image
 CST-363 Week 6 This is our sixth week in CST-363, also known as Introduction to Database Systems. Summary This week we read about integrating SQL into programs and other languages. We learned about some of  the differences between SQL and other languages and some of the ways to integrate SQL into a program, such as Embedded SQL and APIs. Then, we learned how to use an SQL API for Java called JDBC. Vocabulary Programming Languages Imperative Languages:  Contain control flow statements that determine the execution order of program steps. Two popular types of imperative languages are: Procedural Languages:  Composed of procedures, also called functions or subroutines. Most languages developed prior to 1990 are procedural. Ex: C and COBOL. Object-Oriented Languages:  Organize code into classes. A class combines variables and procedures into a single construct. Most languages developed since 1990 are object-oriented. Ex: Java, Python, and C++. Declarative Languages:...
Read more