# Logic

## Introduction

This site is intended as a resource for those interested in learning Logic. While the author of this page will take a scholarly approach to the subject, this page will approach the subject in a more user friendly manner than available at Wikipedia, at times even using colloquial language. The site will offer a full Course in Logic 101, based on two of the most reliable texts on logic: Copi and Cohen's Introduction to Logic (11th Edition) and Hurely's A Concise Introduction to Logic (7th Edition). The course will cover the basics of Logic, Classical Logic, Propositional Logic, Predicate Logic, Modal Logic and Inductive Logic including a discussion of Bayesian Theory. Those interested in taking the course can consult that page for the proper order of reading the pages of this site.

Now, let's begin.

Logic, (from the Greek word λόγος (logos), originally meaning the word, but also referring to speech or reason) is the science of evaluating the reasoning within arguments.

## Nature of logic

Logic arose from a concern with correctness of argumentation (See Aristotle). Logic is generally accepted and understood by logicians as a set of rules that tell us when an argument's premises support their conclusion. Logic specifically deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference. (Hofweber 2004).

Therefore, a further understanding of just what logic is, can be enhanced by delineating it from what it is not:

• Logic is not the 'groundness of being' - that's metaphysics.
• Logic is not a set of laws that governs the universe - that's physics.
• Logic is not an immaterial "entity" that transcends reality - that's speculative theology.
• Logic is no a method for 'studying the world' - that's science.
• Logic is not the method for assessing axioms - that's a matter of pure reason.
• Logic is not a way of evaluating 'truth' - that's philosophy.
• Logic is not a set of laws that governs human behavior - that's psychology.
• Logic is not even a study of how people reason - Fortunately there is more to human reason than just logic.

Immanuel Kant introduced an alternative idea as to what logic is. He argued that logic should be conceived as the science of judgment, an idea taken up by Gottlob Frege. However, modern psychologists have demonstrated that a significant percentage of human judgment is clearly non logical.

### Relation to other sciences

Closely related to logic is semantics, or the philosophy of language, which concerns the meaning of the words and sentences; epistemology, or the theory of knowledge, which concerns the conditions under which assertions are true; and the psychology of reasoning, which concerns the mental processes involved in reasoning. Logic, however, is generally understood to describe reasoning in a prescriptive manner (i.e. it describes how reasoning ought to take place), whereas psychology is descriptive and therefore more inclusive vis-a-vis the various methods of judgment humans actually use.

### Definition of an Argument

An argument is made up of a group of statements we call propositions. We use the term proposition instead of sentence because a proposition is more than just a sentence, it is a declarative sentence that contains a truth value. In the case of Classical Logic, which is based on the Law of the Excluded Middle, this truth value must either be "true" or false" .

There are two types of Propositions. The first, the Premise, makes a commitment to truth, and is used as evidence to support the second type of proposition, the Conclusion, which is the claim the arguer wants to prove. An argument must at least imply one of each. The study of logic, therefore, is the effort to determine the conditions under which one is justified in passing from the premises to the conclusion that logically must follow them.

### Examining Arguments: Informal, formal, and symbolic logic

Arguments can be examined through examining their premises, the form of the argument, and by looking at them abstractly. We use the terms "informal", "formal" and "symbolic" to capture these meanings.

• Informal Logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic.
• Formal Logic is the study of inference with purely formal content, where that content is made explicit. (An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. We will see later that on many definitions of logic, logical inference and inference with purely formal content are the same thing. This does not render the notion of informal logic vacuous, since one may wish to investigate logic without committing to a particular formal analysis.)
• Symbolic Logic is closely related to formal logic: it the study of symbolic abstractions that capture the formal features of logical inference.

### Consistency, soundness, and completeness

There are three valuable properties that formal systems can have:

• Consistency, which means that none of the theorems of the system contradict each other.
• Soundness, which means that the system's rules of derivation will never let you infer anything false, so long as you start with only true premises. So if a system is sound (and its axioms, if any, are true), then the theorems of a sound formal system are the truths. All of the theorems of a system that has no axioms are its truths and sometimes the truths of such a system are called 'logical truths.' (Note that if a system is not consistent, it cannot be sound. This is because a contradiction is always false, so if two theorems contradict at least one is false.)
• Completeness, which means that there are no true sentences in the system that cannot, at least in principle, be proved using the derivation rules (and axioms, if any) of the system.

Not all systems achieve all three virtues. It has been proven by Kurt Gödel that a system with enough axioms and/or rules of derivation to derive the principles of arithmetic cannot be both consistent and complete. This is called Gödel's Incompleteness Theorem.

### Deductive and inductive reasoning

Originally, logic consisted only of deductive reasoning which concerns what follows universally from given premises. However, it is important to note that inductive reasoning—the study of deriving a reliable generalization from observations—has sometimes been included in the study of logic. Correspondingly, we must distinguish between deductive validity and inductive validity. An inference is deductively valid if and only if there is no possible situation in which all the premises are true and the conclusion false. The notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understood notions of semantics. Inductive validity on the other hand requires us to define a reliable generalization of some set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some of these definitions may use mathematical models of probability. For the most part this discussion of logic deals only with deductive logic.

## Types of logic

Formal logic encompasses a wide variety of logical systems. Various systems of logic include Classical or Syllogistic logic, Predicate Logic, Propositional Logic, and Modal Logic, and formal systems are indispensable in all branches of mathematical logic. The table of logic symbols describes various widely used notations in symbolic logic.

### Syllogistic or Classical Logic

The Organon was Aristotle's body of work on logic, with the Prior Analytics constituting the first explicit work in formal logic, introducing to the world the syllogism. Syllogistic Logic, Term logic, Aristotelean Logic or Classical Logic are all references to the logical form of the syllogism. Syllogistic logic is the analysis of the judgments into propositions consisting of two terms that are related by one of a fixed number of relations, and the expression of inferences by means of syllogisms that consisted of two propositions sharing a common term as premise, and a conclusion which was a proposition involving the two unrelated terms from the premises.

Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out system. It was not alone: the Stoics proposed a system of Propositional Logic that was studied by medieval logicians; nor was the perfection of Aristotle's system undisputed; for example the problem of multiple generality was recognized in medieval times. Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions.

Today, some academics claim that Aristotle's system is generally seen as having little more than historical value (though there is some current interest in extending term logics), regarded as made obsolete by the advent of Sentential Logic and the predicate calculus.

### Propositional Logic

Logic as it is studied today is a very different subject from Classical Logic; the principal difference is the innovation of Propositional Logic and Predicate Logic. Propositional Logic allows for more complex argument forms than classical syllogisms. In propositional logic, propositions are represented by symbols and connectors, so that the statement's logical form can be assessed for cases of truth and falsity, which in turn allows us to assess the entire argument's form for validity or invalidity. In symbolic, or propositional logic, a simple statement, containing one proposition, is is referred to as an atomic statement, and is symbolized by one letter, such as p. A compound statement, with more than one proposition holding some relationship to another proposition, is referred to as a molecular statement, which may be symbolized as p v q. The v symbol just used is a connective: Atomic propositions become molecular propositions when they are joined by connectives.

### Predicate Logic

However, just as limits in Classical Logical led to Propositional Logic, limits in Propositional Logic pointed to the need for a new logic. Propositional Logic is not adequate for formalizing valid arguments that rely on the internal structure of the propositions involved. For example, a translation of the valid argument:

```   * All men are mortal
* Socrates is a man
* Therefore, Socrates is mortal
```

into propositional logic yields

```   * A
* B
* ∴ C (∴ means "therefore"
```

which is invalid, because there are no connectors between the premises and between the premises and the conclusion.

Therefore, the need for a First Order, or Predicate Logic became apparent. The new ingredient of first-order logic not found in propositional logic is quantification: where φ is any (well-formed) formula, the new constructions ∀x φ and ∃x φ — read "for all x, φ" and "for some x, φ" — are introduced, where x is an individual variable whose range is the set of individuals of some given universe of discourse (or domain). For example, if the universe consists solely of people, then x ranges over people. For convenience, we write φ as φ(x) to show that it contains only the variable x free and, for b a member of the universe, we let φ[b] express that b satisfies (i.e. has the property expressed by) φ. Then ∀x φ(x) states that φ[b] is true for every b in the universe, and ∃x φ(x) means that there is a b (in the universe) such that φ[b] holds.

The argument about Socrates can be formalized in first-order logic as follows. Let the universe of discourse be the set of all people, living and deceased, and let Man(x) be a predicate (which, informally, means that the person represented by variable x is a man) and Mortal(x) be a second predicate. Then the argument above becomes

```   * ∀ x (Man(x) → Mortal(x))
* Man(Socrates)
* ∴ Mortal(Socrates)
```

A literal translation of the first line would be "For all x, if x is described by 'Man', x must also be described by 'Mortal'." The second line states that the predicate "Man" applies to Socrates, and the third line translates to "Therefore, the description 'Mortal' applies to Socrates."

### Modal logic

In language, modality deals with the phenomenon that subparts of a sentence may have their semantics modified by special verbs or modal particles. For example, "We go to the games" can be modified to give "We should go to the games", and "We can go to the games"" and perhaps "We will go to the games". More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied.

The logical study of modality dates back to Aristotle, who was concerned with the alethic modalities of necessity and possibility, which he observed to be dual in the sense of De Morgan duality. While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1918, who formulated a family of rival axiomatisations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include deontic logic and epistemic logic. The seminal work of Arthur Prior applied the same formal language to treat temporal logic and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of frame semantics which revolutionized the formal technology available to modal logicians and gave a new graph-theoretic way of looking at modality that has driven many applications in computational linguistics and computer science, such as dynamic logic.

Students following the Course in Logic 101 should proceed to the section entitled: The Laws of Classical Logic

## References

• Birkhoff, G., and von Neumann, J. (1936), "The Logic of Quantum Mechanics", Annals of Mathematics 37, 823–843.
• Cohen, R.S, and Wartofsky, M.W. (1974), Logical and Epistemological Studies in Contemporary Physics, Boston Studies in the Philosophy of Science, D. Reidel Publishing Company, Dordrecht, Netherlands. ISBN 90-277-0377-9.
• Finkelstein, D. (1969), "Matter, Space, and Logic", in R.S. Cohen and M.W. Wartofsky (eds. 1974).
• Gabbay, D.M., and Guenthner, F. (eds., 2001-2005), Handbook of Philosophical Logic, 13 vols., 2nd edition, Kluwer Publishers, Dordrecht.
• Hodges, W. (2001), Logic. An introduction to Elementary Logic, Penguin Books.
• Hughes, R.I.G. (ed., 1993), A Philosophical Companion to First-Order Logic, Hackett Publishing.
• Priest, Graham (2004), "Dialetheism", Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), Eprint.
• Smith, B. (1989), "Logic and the Sachverhalt", The Monist 72(1), 52–69.