What is Logic?

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

The reasoning is what logic involves. A legal opinion or mathematical justification may be used to support the reasoning. In mathematics, certain logics are used. Negation, conjunction, and disjunction are the three basic mathematical logic. The three logical operators used in Mathematics are (NOT), (AND), (OR). Symbols for the same are, '~' for negation, '^' for conjunction and 'v' for disjunction.

Statements

A mathematical statement, or simply a statement, is a meaningful set of terms that can be considered true or false.

The truth value of a statement is T or F, depending on whether the statement is true or false e.g., the truth value of the statement "1 plus 1 equals to two" is T, while the truth value of the statement "1 plus 1 equals three" is F. Knowing the true value of a statement helps one to substitute it with another (similar) statement (s).

Note: A statement can't be considered a sentence when

• It's an exclamation
• It's an appeal or an order
• It's a question
• It uses words like "yesterday," "'today'," and "tomorrow".
• It includes various areas, such as 'there,' 'here,' 'everywhere,' and so on.
• It includes pronouns like 'he,' 'she,' and 'they,' among others.

Let see a few examples of statements.

1. "In India, New Delhi is located" is true. So, it is a statement.
2. "Each rectangle is square" is false. So, it is a statement.
3. "It is not possible to allocate true or false to close the door"(In fact, it is a command). So, it cannot be called a statement.
4. "How old are you? You know?" It is not possible to assign true or false (In fact, it is a question). So, it is not a statement.
5. The truth or falsehood of the expression "x is a natural number" depends on the value of x. So, this is not treated as a statement. However, in some books, it is called an open statement.

Note: Truth and falsity of a statement is called its truth value.

Let us consider some examples of statements that can be interpreted mathematically.

1. Consider the following set of statements and note which of them are statements agreed mathematically:
2. In the east, the sun rises.
3. New Delhi is a country.
4. The red rose is nicer than the yellow rose.

Solution: We can automatically assume when we read the first statement that the first statement is undoubtedly true, and the second one is false. As far as the third statement is considered, it can rely on different people's perceptions. It may, therefore, be true for some individuals and false for others at the same time. But for logic in mathematics, such ambiguous statements are not appropriate.

Therefore, a statement is only mathematically appropriate when it is either true or false, but not both at the same time. So, statements 1 and 2 are mathematically accepted, while statement 3 is not mathematically accepted.

• The sum of the three x, y, and z natural numbers is always negative.

Solution: This statement is reasonable. Since all-natural numbers are greater than zero, it can never be true, so the sum of positive numbers can never be negative.

• The product of the three valid x, y and z numbers is always zero.

Solution: We cannot find out in this provided statement whether the statement is true or false. Such a sentence is not suitable mathematically for reasoning.

There are Six Types of Logical Statements to Know

Simple Logical Statement

Simple statements, as their name suggests, are, well, simple. They posit the reality of one idea, and they use one basic phrase to do so. Let's take a look at a few examples:

• Antarctica is the southernmost continent on the globe.
• Penguins cannot fly.
• The key is in the room.
• The dog's asleep.

Compound Statement

A compound statement comprises at least one basic statement and a logical operator (or connectives).

There are several ways to combine simple statements to create new ones. Connectives are words that combine or modify simple statements to produce new or compound statements.

A logical connective (also known as a logical operator, sentential connective, or sentential operator) is a logical  constant used only to  link two or more formulas together in logic.

Notice that the above phrases differ in duration, but each of them communicates one basic idea at its core. They're each making one point, and it's easy to understand the claim. We have to consider what they are saying when evaluating the reasoning of basic logical claims and decide whether that argument is true or false.

Conjunction

The conjunction is another type of logical statement. Conjunctions are compound words, consisting of two or more basic phrases joined by the word "and" (or any of the other conjunctions such as "but," "yet," "so," and so on...) Conjunctions convey two meanings and are slightly more complicated than simple logical statements. Each of the separate independent clauses in the conjunction is a "conjunct." Let's take a look at a few examples of logical conjunctive statements.

• The dog is asleep, but the cat is awake.
• I tried hard, yet I failed.

Note that the two thoughts are complicating each other. The conjunction's logic depends not only on the concepts found in both of the conjuncts but also on the relation between the two concepts. For instance, the word "and" will indicate a logical relationship different from the word "but," which will indicate a logical relationship different from the word "so." Please pay careful attention to each of the conjuncts' context and pay attention to the link its elf's the logical relationship. "Continuers," adding detail, "contradictors," showing contrast or counterpoint, and "cause and effect," which shows a relationship of cause and effect, are the three broad types of conjunctions.

Disjunction

Disjunctions are compound sentences that consist of two simple sentences linked to the words "either/or," "or," or "unless." The logic conveyed by disjunctions is different from the logic conveyed by conjunctions, so pay attention to the speaker's use of conjunctions to pursue his or her logic. Disjunctions imply two exclusive possibilities; that is to say, they indicate that one of the propositions is valid, but not both.

Let us look at a few examples of disjunctions:

• Either it will rain, or it won't.
• Either you support Brazil, or you support Argentina.

Each of the different independent clauses in a disjunction is called a "disjunct."

Note that, according to the reasoning expressed in the disjunction's conjunction, either of the disjuncts may be true, but not both.

Conditional Statements

Conditional statements allow ties between concepts, are a great way to illustrate the logic. The statements are in form of “if and then”. Conditional statements are, by their essence, hypothetical. When determining whether they are logically valid, pay careful attention to the arguments that conditional statements produce.

For example:

• If you want to be a YouTube then you have to struggle a lot.

Note how one part of the statement relies on the other to be true to make it true. You must pay attention to the condition and the condition's consequence. The two elements of the conditional sentence are called the "antecedent" and the "consequent." The "if" clause is the antecedent, and the "then" clause is the consequent.

Bi-Conditional

The bi-conditional statements are more strict and emphatic than the conditional statements. As its name indicates, the biconditional argument includes two conditionals, but one of those conditionals implies and suggests the other. Biconditionals typically take the form of declarations of "if and only if". In bi-conditional statements not only we have “if and only if”, along with this “implies and is implied by” are into the business.

For example:

• The World Test Championship will take place if and only if there are no raise of COVID-19 cases.
• The fact that Brazil has won the match implies and is implied by the fact that they scores three goals without considering a single goal.

Negation

The last kind of sentence that we're going to look at today is negation. It is nothing but the opposite form of the affirmative sentenaces. A negation expressly Rejects that a statement is valid, thus negating it instead of asserting a proposal's validity. There are several ways in which you can use negation to refute a proposition, such as using the words "not," "never," "it's false," and so on.

For example:

• The world isn't flat.

Technicalities Involved

It is sufficient to accept propositional variables (sometimes called sentential variables), usually capital letters in the centre of the alphabet, to evaluate logical connections: P, Q, R, S. We think of these as representing (usually atomic) statements, but the variables can achieve only two values: true or false.

We have symbols for the logical connectives: ∧, ∨, →, ↔, ¬.

• P∧Q represents conjunction of "P and Q".
• P∨Q represents "P or Q" and called a disjunction.
• P→Q represents "if P then Q" and called an implication or conditional.
• P↔Q represents "P if and only if Q" and called a biconditional.
• ¬P represents the negation of P or “not p”.

Truth Conditions

• P∧Q is true when both P and Q are true.
• P∨Q is true when P or Q or both are true.
• P→Q is true when P is false, or Q is true or both.
• P↔Q is true when P and Q are both true and both false.
• ¬P is true when P is false.

Implications

A molecular declaration of the form is an implication or conditional.

P→Q

where, P and Q are statements

We say that

• P is the hypothesis (or antecedent).
• Q is the conclusion (or consequent).

If Q is true or both P and Q are false then an implication will be evaluated to true.

If and only if, P↔Q is logically equivalent to (P→Q) ∧ (Q→P).

Example: It is true that n is even if and only if n2 is even, given an integer n. If n is equal, then n2 is equal, much as the other way around: if n2 is equal, then n is equal.

Necessary and Sufficient

• "P is necessary for Q" means Q→P.
• "P is sufficient for Q" means P→Q.
• If P is necessary and sufficient for Q, then P↔Q.

Common Mistakes

• Students should remember when we apply negation to the negation of any statement, the result will be the statement itself.
• Students should be aware with two most commonly used mathematical terms “for every” and “there exist” and students should be able to determine the difference between these two.

Context & Applications

The topic has a high impact on the education. It helps to build logical knowledge. Especially it is studied in,

• 10th, 11th, 12th Standard.
• Engineering Students.
• Bachelors in Mathematics.
• Binary operations
• AND OR GATE
• Logical operations
• Logical Gates

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