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mahesaaaprilia · 5 years ago

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Mathematical Logic

“Logic and Language”

What is Logic? According to Charles Pierce, “Nearly a hundred definition of it have been given, it will, however, generally be conceded that its central problem is the classification of arguments, so that all those that are bad are thrown into one division, and those which are good into another”. According to me logic is the way of thinking or reasoning. Mathematical Logic / Symbolic Logic is a Logic that uses the language of Mathematics, that is by using symbols.

The study of logic will help one to distinguish between correct and incorrect arguments, and it will do so in several ways.

1. The proper study of logic will approace it as an art as well as a science, and the student will do exercise will help to make perfect.

2. The study of logic, especially simbolyc logic, like the studi of any other exact scince, will tend to increase proficiency in reasoning.

3. The study of logic will give students certain techniques for testing the validity of all arguments, including their own

Does the conclusion tht is reached follow from the premises used assumed? If the premises provide adequate grounds for accepting the conclusion, if asserting the premises to be true warrants asserting the conclution to be true also, then the reasoning is correct. Otherwise, the reasoning is incorrect.

Inferring is an activity in which one proportion is affirmed on the basis of one or more other propositions that are accepted as the starting point of the process.

Proposition are either true or false, and in this they differ from questions, commands, and exclations. Proposition is that the same sentences may be uttered in different contexts to assert different proposition. The same sort of distinction can be drawn between sentences and statements. The same statement can be made using different words, and the same sentence can be uttered in different contexts to make differents statements. The terms “Proposition” and “Statement” are not exact synonyms, but in the writings of logicians they are used in much the same sense.

Corresponding to every possible inference is an argument, and it is with these arguments that logic is chiefly concerned. In ordinary usage, the word “argument” also has other meanings, but in logic it has the technical sense explained.

Every argument has structure, in the analysis of which the terms “Premiss” and “Conclusion” are usually employed. The conclusion of an argument is that proposition which is affirmed on the basis of the other proposition of the argument.

Truth and falsehood characterize proposition or statements and may also be said to characterize the declarative sentences in which they are formulated. There is a connection between the validity or invalidity of an argument and the truth or falsehood of its premises and conclusion, but the connection is by no means a simple one.

“Arguments Containing Compound Statements”

All statements can be divided into two kinds, simple and compound. A simple statement is one that does not contain any other statement as a component part, whereas every compound statement does contain another statement as a component part. For example.

“Atmospheric testing of nuclear weapons will be discontinued of this planet will become uninhabitable”

That’s is a compound statement that contains, as its components, the two simple statements “atmospheric testing of nuclear weapons will be discontinued” and “This planet will become uninhabitable” the component parts of compound statement may themselves be compound, of course.

As the term “component” is used in logic, not every statement that is part of a larger statement is a component of the larger statement. For a part of a statement to be a component of a larger statement, two condition must be satisfied.

1. The part must be a statement in its own right

2. If the part is replace in the larger statement by any other statement, the result of that replacement must be meaningful.

To determine the validity or invalidity of an argument form, we must examine all possible substitution instances of it to see if any of them have true premises and false conclusions. Possible substitution instances can be set fourth most conveniently in a truth table, with an initial or guide column for each distinct statement variable that appears in the argument form.

Truth table provide a purely mechanical or effective method of deciding the validity or invalidity of any argument of the general type here considered.

“Connectives”

In the previous section we m1ade clear what mathematical statement is. In this section we talk about how mathematical statements can be combined to make more complex statements. This is done using what are called 'logical connectives' or 'logical operators'. You can think of these as functions of one or more variables, where the variables can be either True or False and the value of the function can be either True or False. The logical connectives commonly used in mathematics are negation, conjunction, disjunction, implication, and equivalence, which are fancy words for things you encounter in everyday English.

Types of connectives

1. Negation

2. Conjunction

3. Disjunction

4. Conditional

5. Biconditional

1. Negation

The negation (or denial or contradictory) of a statement P {\displaystyle P} is the statement that P {\displaystyle P} is not true. The negation of a true statement is false and the negation of false statement is true. The logical symbol for negation is " ¬ {\displaystyle \lnot } " or “ called a curl (or a tilde). We can take the following truth table as defining the curl symbol:

P {\displaystyle P}

P {\displaystyle P} P

P {\displaystyle P}

2. Conjunction

Conjuction, a compound statement formed by interesting the word “and” between two statements. Two statement so combined are called conjuncts. The word “and” has other uses, however, as in the statement, which is not compound, but a simple statement asserting a relationship. The logical symbol for conjunction is " ∧ {\displaystyle \land } ", so you can write P ∧ Q {\displaystyle P\land Q} for P {\displaystyle P} P and Q {\displaystyle Q} Q.

P {\displaystyle P} P

Q {\displaystyle Q} Q

P {\displaystyle P} Q {\displaystyle Q} And

True

False

True

False

3. Disjunction

The disjunction of two statements P {\displaystyle P} P and Q {\displaystyle Q} Q is the statement that at least one of P {\displaystyle P} P and Q {\displaystyle Q} Q are True. When two statements are combined disjunctively by interesting the word “or” between them, the resulting compound statement is disjunction (or alternation), and the two statement so combined are called disjuncts (or alternatives).

In Latin, the word “vel” expresses the inclusive sense of the word “or” and the word “aut” expresses the exclusive sense. The symbol “∨”, called a wegde (or a vee), is a truth-functional connective, so you can write P ∨ Q {\displaystyle P\lor Q} for P {\displaystyle P} P and Q {\displaystyle Q} Q and is defined by the following truth table :

P {\displaystyle P} P

Q {\displaystyle Q} Q

P {\displaystyle P} P or Q {\displaystyle Q} Q

True

False

True

False

True

False

4. Conditional

A conditional (or a hypothetical, an implication, or an implicative statement). The component between the “if” and the “then” is called the antecedent (or the implicans or protasis). The logical symbol for implication is "⟹ ⟹ {\displaystyle \implies } ", though "⊃ ⊃ {\displaystyle \supset } " is sometimes seen instead. so you can write P ⟹ Q {\displaystyle P\implies Q} for P {\displaystyle P} P implies Q {\displaystyle Q} Q. The implication P {\displaystyle P} P implies Q {\displaystyle Q} Q is True when P {\displaystyle P} P is false. It's also True when Q {\displaystyle Q} Q is True and only false when P {\displaystyle P} P is True and Q {\displaystyle Q} Q is False. In tabular form:

P {\displaystyle P} P

Q {\displaystyle Q} Q

P {\displaystyle P} P ⊃ Q {\displaystyle Q} Q

5. Biconditional

Two statements are saidThe equivalence of two statements are said to be logically equivalent when the biconditional that expresses their material equivalence is a tautology. P {\displaystyle P} P and Q {\displaystyle Q} Q is the statement is that P {\displaystyle P} P and Q {\displaystyle Q} Q have the same truth value. Another way of say this is that P {\displaystyle P} P implies Q {\displaystyle Q} Q and Q {\displaystyle Q} Q implies P {\displaystyle P} P. The logical symbol for implication is "⟺ ⟺ {\displaystyle \iff } ", so you can write P ⟺ Q {\displaystyle P\iff Q} for P {\displaystyle P} P iff Q {\displaystyle Q} Q.

The equivalence P {\displaystyle P} P iff Q {\displaystyle Q} Q is True when P {\displaystyle P} P and Q {\displaystyle Q} Q have the same truth values, and False when they have different truth values. In other words P {\displaystyle P} P iff Q {\displaystyle Q} Q is True when P {\displaystyle P} P and Q {\displaystyle Q} Q are both True or both False, and P {\displaystyle P} P iff Q {\displaystyle Q} Q is False is one of P {\displaystyle P} P and Q {\displaystyle Q} Q is True while the other is false. In tabular form:

P {\displaystyle P} P

Q {\displaystyle Q} Q

P {\displaystyle P} P Q {\displaystyle Q} Q

True

False

True

False

True

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