[DEFINE] LOGIC: Terminologies Flashcards
Logic
a system of reasoning that allows inferences to be drawn from facts.
Proposition
a declarative sentence that is either true or false, but not both.
Conjunction
‘AND’ statement in mathematics. A conjunction of two statements is true only when both statements are true.
Disjunction
“OR” statement in mathematics. A disjunction is false if and only if both statements are false; otherwise it is true.
Negation
“NOT” statement in mathematics. It is a statement with the opposite truth value. If a statement is true, then its negation is false, and vice versa.
Conditional (or implication)
a logical compound statement in which p implies q. The statement is only false when p is true but q is false.
Biconditional
True when both have the same truth value.
Exclusive-or
The statement is only true if it has differnent truth values. P and q cannot be the same
EX: I am a dog or I am a cat
Truth Values of proposition
The truth value of a proposition is a determination of whether the proposition is true (T) or false (F). In classical logic, a proposition can only have one of these two truth values.
Proposition: “The sun rises in the east.”
This statement is true. Hence, its truth value is T (true).
Conditional Propositions
an “if-then” expression. It is false only when p is true and q is false, and is true in all other situations.
Converse of p-q
A conditional proposition in which the statement “p implies q” is “q implies p”. It is created by swapping the order of the hypothesis and conclusion.
Inverse of p-q
A conditional proposition in which the statement “p implies q” is “not p implies not q.” The order of the hypothesis and the conclusion remains the same, but they are both negated.
Contrapositive
A conditional proposition in which the statement “p implies q” is “not q implies not p.” It is formed by negating and swapping the hypothesis and the conclusion
Tautology
a proposition that is always true
Contradiction
a proposition that is always false
Contingency
It is neither a tautology nor a contradiction. In other words, it is a proposition that is neither true nor false.
Proof
a logical argument that shows a mathematical statement is true.
Premises
AKA the cause. a statement that is assumed to be true and is used as the basis for an argument.
Conclusion
AKA the effect. a statement that is reached by applying logical rules to a set of premises.
Number Theory
branch of pure mathematics mainly to study natural numbers and integers.
Counting
a subset of whole numbers that do not include zero
Whole
a subset of whole numbers that includes the number zero
Integers
a whole number (not a fractional number) that can be positive, negative, or zero.
Rational Numbers
are numbers that can be written as a fraction or a ratio.
Irrational Numbers
are numbers that can’t be written as a fraction or ratio.
Real Numbers
are numbers that include rational and irrational numbers.
Imaginary numbers
it is the product of a real number and the imaginary unit i. When squared, imaginary numbers also give a negative result.
Complex numbers
these are numbers that have two parts: a real part and an imaginary part. Imaginary numbers are denoted by i.
*** it comes in the form a + ib
Odd numbers
these numbers cannot be equally divided into pairs. They always end with the last digits 1, 3, 5, 7, 9.
Even Numbers
any number divisible by 2. They always end with the last digits 0, 2, 4, 6, 8.
Prime numbers
a number that can only be divided by itself and 1 without remainders. It is not a product of two smaller natural numbers.
Composite Numbers
numbers that have more than two factors. It can be formed by multiplying two smaller positive integers.
Rule of Inference
a logical form or guide consisting of premises (or hypotheses) and draws a conclusion.
Conjecture
a mathematical statement which appears to be true, but has not yet been rigorously proved.
Axiom (or postulate)
a statement that is assumed to be true without the need for proof.
Theorem
a statement that has been or can be proven to be true based on known facts and mathematical operations
Lemma (or pre-theorem)
a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem
Corollary (or post-theorem)
a result in which the proof relies heavily on a given theorem
Proof
a logical argument that shows a mathematical statement is true.
Divisibility
this means a number can be divided without leaving a remainder
Greatest Common Divisor
is the greatest value that can divide two integers evenly.
Least Common Multiple
the smallest positive multiple that two or more numbers share
Relatively Prime Integers
Two integers that have no common factors other than 1. This means that no other integer could divide both numbers evenly. Two integers a,b are called relatively prime to each other if gcd(a,b)=1
Direct Proof
A method that involves proving a statement by assuming the hypothesis is true and demonstrating that the conclusion logically follows.
Proof of Contrapositive (or proof of transposition)
uses the contrapositive of a conditional statement to prove the statement itself
Proof of contradiction
It is when we assume a proposition is not true and show that it leads to a contradiction, which proves that the proposition is true.
Proof of Exhaustion
To prove a statement, you must break the statement into a finite amount of cases.
EXAMPLE:
To prove If p then q:
p₁ then q
p₂ then q
Proof of Counterexamples
Providing a single example where a statement is false to disprove it
Mathematical Induction
a technique of proving a statement, theorem, or formula which is thought to be true, for each and every natural number n.
Appeal to the Authority
occurs when we accept a claim merely because someone tells us that an authority figure supports that claim.
Appeal to Force
argumentation using force or the threat of force to convince others to accept an argument’s conclusion
Appeal to Ignorance
occurs when you argue that your conclusion must be true, because there is no evidence against it
Appeal to Pity
an attempt to persuade others by exploiting and provoking feelings of guilt or pity instead of presenting factual evidence.
Appeal to people
argues that a claim is true simply because that’s what a large number of people believe.
Argumentum Ad Hominem
occurs when, instead of addressing someone’s argument or position, you irrelevantly attack the person or some aspect of the person who is making the argument.
Circular Argument
a logical fallacy wherein it tries to prove itself using its conclusion as evidence.
Equivocation
happens when a word or phrase in an argument has more than one meaning, and the person switches between those meanings to make their point seem more logical or convincing than it really is.
Fallacy of Division
This happens when a person believes that what is true for the entire group or thing also holds true for each of its component pieces
False Dilemma
logical fallacy that presents only two options or sides to an issue when there are actually more complexities.
Hasty Generalization
happens when someone makes a decision or forms a conclusion without having enough evidence or by only looking at one side of the story.
Red Herring
occurs when someone introduces an irrelevant point or topic to divert attention from the original issue.
Slippery Slope (or snowball/domino theory)
Also known as Domino Fallacy, It is a logical fallacy that assumes a cause-and-effect relationship between events without evidence.
Strawman Fallacy
the distortion of someone else’s argument to make it easier to attack or refute.
