This website uses cookies to ensure you have the best experience. Learn more

Introduction To Logic Essay

2695 words - 11 pages

Predicate Logic

All men are mortal.
Socrates is a man.
··· Socrates is mortal.
Note: We need logic laws that work for statements involving quantities like “some” and “all”.
In English, the predicate is the part of the sentence that tells you
something about the subject.


More on predicates
Example: Nate is a student at UT.
What is the subject? What is the predicate?
Example: We can form two different predicates.
Let P(x) be “x is a student at UT”.
Let Q(x, y) be “x is a student at y”.
Definition: A predicate is a property that a variable or a finite
collection of variables can have. A predicate becomes a proposition
when specific values are assigned to the variables. P(x1, x2, ..., ...view middle of the document...

Ways to read ∀xP (x):
For every x, P(x)
For every x, P(x) is true
For all x, P(x)


More on the universal quantifier
Definition: A counterexample for ∀xP (x) is any t ∈ U , where
U is the universe, such that P(t) is false.
Some Examples
Example: P(x, y): x + y = 8
Assign x to be 1, and y to be 7. We get proposition P(1, 7) which is
Proposition P(2, 5) is false since 2 + 5 = 8.
Example: ∀x[x ≥ 0]
U = N (non-negative integers)
We could re-write this proposition as: ∀x ∈ N, x ≥ 0
Is the proposition true?
What if the universe is R?
Example: ∀x∀y[x + y > x]
Is this proposition true if:
1. If U = N?
2. If U = R?
Example: ∀x∀y[x > y]
True if:
universe for x = the non-negative integers
universe for y = the non-positive integers


The Existential Quantifier: ∃
Definition: The symbol ∃ is call the existential quantifier and
represents the phrase “there exists” or “for some”. The existential
quantification of P(x) is the statement “P(x) for some values x in
the universe”, or equivalently, “There exists a value for x such that
P(x) is true”, which is written ∃xP (x).
Note: If P(x) is true for at least one element in the domain, then
∃xP (x) is true. Otherwise it is false.
Note: Let P(x) be a predicate and c ∈ U (U = domain).
The following implications are true:
∀xP (x) → P (c)
P (c) → ∃xP (x)
Example: ∃x [x is prime] where U = Z
Is this proposition true or false?
Example: ∃x[x2 < 0] where U = R
True or false?
Exercises: True or false? Prove your answer.
1. ∃n[n2 = n] where U = Z.
2. ∃n[n2 = n] where U = {4, 5, 6, 7}.


Translating Quantified Statements
Translate the following into English.
1. ∀x[x2 ≥ 0] where U = R.
2. ∃t[(t > 3) ∧ (t3 > 27)] where U = R.
3. ∀x[(2|x) ∨ (2 |x)] where U = N
Translate the following into logic statements.
1. There is an integer whose square is twice itself.
2. No school buses are purple.
3. If a real number is even, then its square is even.
Note: Let U = {1, 2, 3}.
Proposition ∀xP (x) is equivalent to P (1) ∧ P (2) ∧ P (3).
Proposition ∃xP (x) is equivalent to P (1) ∨ P (2) ∨ P (3).


Bound and Free Variables
Definition: All variables in a predicate must be bound to turn a
predicate into a proposition. We bind a variable by assigning it a
value or quantifying it. Variables which are not bound are free.
Note: If we bind one variable in a predicate P (x, y, z) with 3
variables, say by setting z = 4, we get a predicate with 2 variables:
P (x, y, 4).
Example: Let U = N.
P (x, y, z) : x + y = z ← 3 free variables
Let Q(y, z) = P (2, y, z) : 2 + y = z ← 2 free variables


Examples with Quantifiers
Example: U = Z
N(x): x is a non-negative integer
E(x): x is even
O(x): x is odd
P(x): x is prime
Translate into logical notation.
1. There exists an even integer.
2. Every integer is even or odd.
3. All prime integers are non-negative.
4. The only even prime is 2.
5. Not all integers are odd.
6. Not all primes are odd.
7. If an integer is not odd, then it is even.


Examples with Nested Quantifiers
Note about nested...

Other Papers Like Introduction To Logic-

Life and Teaching of Aristotle Essay

946 words - 4 pages , 1999). With the help of this component, Aristotle also established both practical and theoretical reasoning. The second component of the Aristotle’s system is logic. This is among the influential components of Aristotle because it influences decision making in many areas. According to Aristotle, logic was an important component in his study because it was instrumental in the acquisition of knowledge. This component dealt with definitions

Introduction To Philosophy Essay

651 words - 3 pages Introduction to Philosophy Φιλοσοφια BRANCHES OF PHILOSOPHY: Epistemology, Metaphysics, Logic, Ethics * Epistemology - Studies how human beings acquire knowledge Proponents: John Locke: we came into this world naked both physically and intellectually Empiricism – All knowledge comes from experience which is made possible through sense perceptions Example: I have an idea of an apple because I see an apple Immanuel Kant

Rapid Control Prototyping Approach to Fuzzy Speed Control of Brushless Dc Motor

1973 words - 8 pages RAPID CONTROL PROTOTYPING APPROACH TO FUZZY SPEED CONTROL OF BRUSHLESS DC MOTOR Abstract This study involves the fuzzy logic controlling of brushless DC (BLDC) motor with rapid control prototyping method. Initially a MATLAB/Simulink model of the six pulse BLDC motor and its power electronic driver is established. A fuzzy controller is developed by using MATLAB Fuzzy-Logic Toolbox and then it is inserted into the Simulink model. This

Web Design

2415 words - 10 pages . INTRODUCTION Web sites have quickly evolved from simple, static pages to complex Web applications performing critical operations for many businesses. These applications involve dynamic data from multiple sources; ever changing and various features for e-commerce, personalization and many more. At the same time, customers and internal users have understandably come to expect and demand more and more sophistication in the Web-based


1229 words - 5 pages : Bauzon, Priciliano (2002) Logic for Filipinos. 2nd ed. Mandaluyong City: National Bookstore. (ASL, 160, B328, 2002) Copi, Irving M. and Carl Cohen. (2005) Introduction to Logic. 12th ed. USAA: Prentice Hall. (ASL, 160, C791, 2005) Hurley, Patrick. (2000) A Concise Introduction to Logic. 7th ed. USA: Wadsworth. (ASL, 160, H939, 2000) Meer, Thelma Q. (2004) Basics of Logic. Philippines: Trinitas Pub. Inc. (ASL, 160, M471, 2004

Fuzzy Logic-Based State-of-Charge Meter

4096 words - 17 pages room temperature the battery pack was initially able to deliver 42 pulses early in the cycle life whereas at 0 °C the battery-pack is only able to initially deliver 12 pulses. The voltage recovery profile upon removal of the 10 A load has been used both in the time domain and frequency domain to develop fuzzy logic models to estimate the number of remaining pulses that the battery-pack can deliver. Accurate models are being developed to estimate

Organizatinional Intelligence

1071 words - 5 pages Intelligence, Business Model Logic, and Bloom’s Taxonomy. Introduction Organizational Intelligence (OI) is applied in all organizations to enhance the understanding of managerial processes and resources within the business. Organizations need to understand the barriers that surround them and be able to take the necessary actions to remove them. OI can be definitely, measured, and used to successfully achieve performance improvement when applied. OI

Java Programming

925 words - 4 pages Credit Units 20 0 20 200 Level Course Code 1st Semester GST 201 CIT 211 GST 203 CIT 213 CIT 215 CIT 237 MTH 211 MTH 213 MTH 241 MTH 281 Course Titles Unit(s) Status Nigerian Peoples and Cultures Introduction to Operating Systems Introduction to Philosophy and Logic Elementary Data Processing Introduction to Programming Languages Programming & Algorithms Introduction to Set Theory and Abstract Algebra Numerical Analysis I Introduction to

How to Write an Essay

649 words - 3 pages . Look for weaknesses of logic, and also strengths. Learning how to write an essay begins by learning how to analyze essays written by others. 3. Brainstorming: Your essay will require insight of your own, genuine essay-writing brilliance. Ask yourself a dozen questions and answer them. Meditate with a pen in your hand. Take walks and think and think until you come up with original insights to write about. 4. Thesis: Pick your best idea and pin

How To Write An Essay

612 words - 3 pages the structure of your argument, and make sure each paragraph is unified. 6. Introduction: Now sit down and write the essay. The introduction should grab the reader's attention, set up the issue, and lead in to your thesis. Your intro is merely a buildup of the issue, a stage of bringing your reader into the essay's argument. (Note: The title and first paragraph are probably the most important elements in your essay. This is an essay

Circuit Design

2454 words - 10 pages 144 CHAPTER 6 NOVEL DESIGN OF ALU 6.1 INTRODUCTION The computation unit is the main unit of any technology, which performs different arithmetic operations like as addition, subtraction and multiplication etc. and logical operations like AND, OR, invert, XOR etc. which is a dominant feature in the digital domain based applications. The arithmetic and logic unit (ALU) perform all the arithmetic operations (addition, subtraction

Related Essays

Back Log Of Data Essay

1437 words - 6 pages ENGI 241 Experiment 5 Basic Logic Gates OBJECTIVE This experiment will examine the operation of the AND, NAND, OR, and NOR logic gates and compare the expected outputs to the truth tables for these devices. The NOT function will be implemented using NAND and NOR gates. EQUIPMENT AND PARTS REQUIRED 1 Protoboard 1 DVM 1 Logic Probe 1 74LS00, 74LS02, 74LS08, 74LS32 2 each 1kΩ resistors INTRODUCTION The NOT circuit or inverter performs the basic

Phi 103 Essay

317 words - 2 pages Critical Thinking Journal Brien Derrick PHI 103: Informal Logic Prof. Michael Rings December 22, 2014 Initially after reviewing the week 1 requirements and the introduction to the “Critical Thinking Quiz” I was confident of my knowledge, that confidence was quickly removed after taking the quiz and missing 8 of the 25. I was a little down on myself and will admit that the quiz was much tougher than I initially thought. After taking

Samba Server Essay

562 words - 3 pages SYNOPSIS TOPIC: ROLE OF NANOWIRE IN FUTURE STOREAGE TECHNOLOGY: SUBMITTED TO: SUBMITTED BY: Ms. Kuljeet kaur Vanita B40 INTRODUCTION A nanowire is an extremely thin wire with a diameter on the order of a few nanometres (nm) or less, where 1 nm = 10 -9 meters. Two processes in nanotechnology by which nanowires can be

An Analysis Of Paul Celan

2066 words - 9 pages , this is only possible if one presupposes a fundamental rift between people. The labyrinthine difficulty is how subjectivities, which are in essence isolated, could ever come together. In the "labyrinth sea," mourning would be a means, through empathy, for people to relate to one another, a means to bridge the rift, yet for Celan, in the introduction of the alternative of the "wellspring," mourning as a means of relating what is