Logic: Fundamentals and Applications

By Fouad Sabry

What Is Logic

The discipline of logic is the study of sound reasoning. Both formal and informal logic are included in its scope. The study of inferences and conclusions that can be justified by deduction is known as formal logic. It investigates how inferences can be drawn from premises notwithstanding the subjects and topics of those premises. Informal logical fallacies, critical thinking, and argumentation theory are often discussed in conjunction with informal logic. In contrast to formal logic, which employs formal language, informal logic investigates arguments given in everyday language. The term "a logic" refers to a logical formal system that articulates a proof system when it is employed in the sense of a countable noun. The use of logic is essential to the study of a wide variety of subjects, including mathematics, linguistics, computer technology, and philosophy.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Logic

Chapter 2: Deductive reasoning

Chapter 3: Inductive reasoning

Chapter 4: Logical reasoning

Chapter 5: Argument

Chapter 6: Modus ponens

Chapter 7: Soundness

Chapter 8: Validity (logic)

Chapter 9: Philosophy of logic

Chapter 10: Logic and rationality

(II) Answering the public top questions about logic.

(III) Real world examples for the usage of logic in many fields.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of logic.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 24, 2023

Book preview

Chapter 1: Logic

The discipline of logic is the study of sound reasoning. Both formal and informal logic are included in its scope. The study of inferences and conclusions that can be justified by deduction is known as formal logic. It investigates how inferences can be drawn from premises notwithstanding the subjects and topics of those premises. Informal logical fallacies, critical thinking, and argumentation theory are often discussed in conjunction with informal logic. In contrast to formal logic, which employs formal language, informal logic investigates arguments given in everyday language. The term a logic refers to a logical formal system that articulates a proof system when it is employed in the sense of a countable noun. The use of logic is essential to the study of a wide variety of subjects, including mathematics, linguistics, computer technology, and philosophy.

Logic studies arguments, They are made up of a series of premises along with a verdict or conclusion.

A good illustration of this would be the line of reasoning that leads from the premises it's Sunday and if it's Sunday then I don't have to work to the conclusion that I don't have to work..

The premises, as well as the conclusions express propositions.

The internal structure of propositions is an important aspect of their composition.

For example, complex propositions are made up of simpler propositions linked by logical vocabulary like \land (and) or \to (if...then).

Even the most elementary arguments contain component pieces, in the example, Sunday or work, for instance.

In most cases, the veracity of a proposition is dependent on the meanings of all of its constituent elements.

However, This is not the case for assertions that are true according to logic.

They are only valid due to the fact that their logical structure is sound, which is independent of their other components.

Arguments can be right or wrong, depending on the circumstances. If the premises of an argument support the conclusion, then the argument is valid. The most compelling type of support is found in deductive arguments, which assert that if their premises are true, then their conclusion must also be true. This is not the case with elaborative arguments, which arrive to information that is in fact brand new and cannot be found in the premises. A significant number of the arguments that are made in common language as well as in the sciences are ampliative arguments. Inductive and deductive lines of reasoning are distinguished among them. Statistics are used to support generalizations made by inductive arguments. Arguments that are abductive draw conclusions to the most plausible explanation. Arguments that do not meet the criteria for valid reasoning are referred to as fallacies. The evaluation of the validity of arguments can be done using theoretical structures known as systems of logic..

Since ancient times, people have been interested in studying logic. Aristotelian logic, Stoic logic, Nyaya, and Mohism are all examples of early schools of thought. Reasoning expressed in the form of syllogisms is at the heart of Aristotelian logic. It was thought to be the most important method of logic in the Western world until it was superseded by modern formal logic, which can trace its origins back to the work of mathematicians from the late 19th century such as Gottlob Frege. Classical logic is currently the most popular type of reasoning. Both propositional logic and first-order logic make up this type of logic. Only the logical connections between complete statements are taken into account by propositional logic. Additionally, first-order logic takes into consideration the constituents of propositions that are located on the inside, such as predicates and quantifiers. Extended logics acknowledge the fundamental intuitions that underpin classical logic and then apply those intuitions to a wider variety of subject areas, including metaphysics, ethics, and epistemology. On the other hand, deviant logics are characterized by their rejection of particular classical intuitions and their provision of alternative explanations for the fundamental principles of logic.

The term logic comes from the Greek word logos, which can be translated into a number of different English words, including reason, discourse, and language..

The field of mathematical logic makes extensive use of formal logic, which is also referred to as symbolic logic. It does this by exchanging concrete statements for abstract symbols in order to analyze the logical form of arguments independent of the content that they really contain. This is called a formal approach to the study of reasoning. In this respect, it is not concerned with any particular subject matter because its focus is solely on the conceptual framework of arguments rather than their specific content.

Formal logic involves the use of formal languages for the purpose of analyzing arguments and expressing ideas.

When understood in a wide sense, Formal logic and informal logic are both included in the scope of logic.

With this in mind, the justification that birds fly, etc.

Tweety is a winged creature.

Therefore, The phrase Tweety flies is a part of natural language and is investigated using informal logic.

But the formal translation (1) {\displaystyle \forall x(Bird(x)\to Flys(x))} ; (2) {\displaystyle Bird(Tweety)} ; (3) {\displaystyle Flys(Tweety)} is studied by formal logic.

Given that inferences and arguments are built on the foundation of premises and conclusions, logic places a significant emphasis on these two components. When an inference is sound or an argument is sound, the conclusion will follow logically from the premises, or, to put it another way, the premises will provide support for the conclusion.

Both the premises and the conclusions have a structure within themselves. In the form of propositions or statements, they can either be straightforward or convoluted.

Certain complicated assertions can be proven correct despite the fact that the literal interpretations of its constituent pieces are irrelevant to their validity.

Validity tables are a useful tool for demonstrating how logical connectives function or how the truth of complex statements is dependent on the elements that make up those claims.

They feature a column for every variable that can be input.

Each row represents a unique combination of the truth values that these variables are able to have.

In the remaining columns, the truth values of the appropriate expressions are displayed according to the supplied values.

For example, the expression {\displaystyle p\land q} uses the logical connective \land (and).

A sentence such as yesterday was Sunday, and the weather was good could be expressed using this expression.

Only if both of its input variables are true does it have any validity, p (yesterday was Sunday) and q (the weather was good), are true.

In every other circumstance, The entirety of the statement is deceptive.

Other important logical connectives are \lor (or), \to (if...then), and \lnot (not).

Additionally, truth tables can be built for more sophisticated expressions that use many propositional connectives.

For example, given the conditional proposition p\to q , one can form truth tables of its inverse ( {\displaystyle \lnot p\to \lnot q} ), and its contraposition ( {\displaystyle \lnot q\to \lnot p} ).

The study of the validity of arguments or inferences is a popular definition of logic, which can be expressed in terms of arguments or inferences.

Arguments and conclusions can either be right on the