Logic

Logic is the study of the principles and methods of valid reasoning

Arguments and Logic

Reasoning is the process of trying, by rational means, to answer a question, resolve an issue, solve a problem, or make a decision. Logic is the study of the rational means by which thinkers ought to reason.
The focus of Logic is the evaluation of arguments.
- An argument is an instance of reasoning, from premises to a conclusion.
- Arguments are typically set forth to establish their conclusions.

Examples of Arguments

The core of the Declaration of Independence is the following argument:
1. The people have the right to alter or abolish a government if it violates basic human rights such as the rights to life, liberty and the pursuit of happiness.
2. The British government of the colonies has violated those rights.
3. Therefore the American people have a right to alter or abolish the British government of the colonies.
Before he flew a kite in a thunderstorm, Benjamin Franklin presented an argument that lightning was electrical in nature.
1. Lightning and sparks are alike in the following respects:
  - 1. Giving light, 2. Color of the light, 3. Crooked direction, 4. Swift motion, 5. Being conducted by metals, 6. Crack or noise in exploding, 7. Subsisting in water or ice, 8. Rending bodies it passes through, 9. Destroying animals, 10. Melting metals, 11. Firing inflammable substances, 12. Sulphureous smell
2. A spark is electrical in nature.
3. Therefore, lightning is likely electrical in nature.
  - “Let the experiment be made.”
The New York Times presented this argument against capital punishment:
1. Fallible governments should refrain from inflicting irreversible punishments
2. Capital punishment is irreversible.
3. Governments are fallible.
4. Therefore, governments should refrain from inflicting capital punishment.
Hypothesis testing is one kind of statistical inference.
1. In a blind taste test, a wine connoisseur, who claims she can taste the difference between French and California Cabernets, correctly identified 15 out of 20 glasses of the wines.
2. The probability she correctly identified at least 15 out of 20 glasses by chance is 1/50. (In statistical jargon, the p-value = 1/50.)
3. Therefore, she can probably taste the difference between French and California Cabernets.
The Britannica is an authoritative source.
1. In its article on swans the Britannica says there are black swans in Australia.
2. The Britannica is a reliable source of information.
3. Therefore, black swans exist.

Branches of Logic

Formal Logic is the theoretic study of the principles and methods of valid reasoning within a formal (symbolic) language.
Informal Logic is the practical study of the methods of valid reasoning (within a natural language such as English).

Formal (Symbolic, Mathematical) Logic

Formal Logic is the theoretic study of the principles and methods of valid reasoning within a formal (symbolic) language.

Why a Formal Language

The reason for a formal language, long recognized by logicians, is that the anomalies of natural language can impede reasoning. Anomalies such as:
- Multiple senses of words, e.g. the verb to be:
  - Mark Twain wrote the Adventures of Huckleberry Finn (predication).
  - Mark Twain is Samuel Clemens (identity).
  - There is a solution to the problem (existence).
- Ambiguity
  - He’s a poor student.
  - Visiting relatives can be boring.
- Multiple ways of saying that same thing:
  - All whales are mammals. Every whale is a mammal. Each whale is a mammal.
  - The gun won’t fire unless there’s a round in chamber. The gun won’t fire without a round in the chamber. If there’s no round in the chamber the gun won’t fire.
- Figurative and literal uses of language
  - The groom had cold feet.
  - Money talks.
- Obscurity
  - Being, which is indeterminate immediacy, is Nothing, neither more nor less.

What Formal Logic Looks Like

Consider the argument:
- If Newton’s theory of gravitation were true, the gravitational attraction between Sun and Earth would be instantaneous.
- The gravitational attraction between Sun and Earth is not instantaneous. (The Sun’s gravitational pull on Earth takes a little over 8 minutes.)
- Therefore Newton’s theory of gravitation is false.
The argument can be “translated” into the language of Propositional Logic (where capital letters stand for sentences) as follows:
- N → I
- ~I
- Therefore, ~N
In the formalized argument:
- N stands for “Newton’s theory of gravitation is true.”
- I stands for “The gravitational attraction between Sun and Earth is instantaneous.”
- → means “If … then —— “.
- ~ means “It is false that ….”.
It is easily shown in Propositional Logic that the formalized argument is valid, that is, ~N follows necessarily from N → I and ~I. The form of argument even has a name: modus tollens.

How a System of Formal Logic is Defined

A system of formal logic is defined by specifying the syntax and semantics of its underlying language.
The syntax is specified by defining:
- The symbols of the language;
- A sentence of the language;
- The rules of inference;
- The idea of a sentence being derivable from a set of sentences:
  - A sentence is derivable from a set of sentences if it can be derived from the set of sentences using the rules of inference.
The semantics is specified by defining:
- An interpretation of the sentences;
- The idea of a sentence being true under an interpretation;
- The idea of a sentence being a logical consequence of a set of sentences:
  - A sentence is a logical consequence of a set of sentences if there is no interpretation under which the sentence is false and the set of sentences are all true.
It’s then proven that the notions of derivability and logical consequence coincide, i.e.
- a sentence is derivable from a set of sentences if and only if it’s a logical consequence of the sentences.

Pages on Formal Logic and Related Formal Systems

Axiom Systems
Conditional Logic
Decision Theory
Formal Logic
Modal Logic
Paradoxes
Predicate Logic
Propositional Logic
Probability Theory
Syllogisms and Venn Diagrams

Informal (Practical, Applied) Logic

Informal (Practical, Applied) Logic is the practical study of the methods of valid reasoning (within a natural language such as English).

Evaluating Arguments

The focus of Informal Logic is the evaluation of arguments as they naturally occur in writing and conversation.
Evaluating an argument means determining whether the argument establishes its conclusion, other things being equal.
- “Establish” typically means to prove beyond a reasonable beyond.
- The reason for keeping other things equal is that there may be arguments on both sides of an issue and, before making a final judgment, all the arguments need to be considered. The NY Times argument against capital punishment, for example, may be a good argument in its own right. But before making a final determination on whether “governments should refrain from inflicting capital punishment,” arguments for capital punishment need to be looked at, e.g. arguments based on retributive justice, deterrence, and incapacitation.
Evaluating an argument involves three things:
- Reconstructing the argument so its premises, conclusion, and logic (reasoning) are clear.
  - View Argument Reconstruction
- Determining whether the premises are true.
- Determining whether the argument’s logic is any good.

Pages on Informal Logic

Topics

Arguments
- An argument is an instance of reasoning, from premises to a conclusion.
- View Arguments
Argument Reconstruction
- Argument reconstruction is the process of restating a naturally-occurring “real life” argument so its premises, conclusion, and reasoning are clear.
- View Argument Reconstruction
Argumentation (Dialectic)
- Argumentation (Dialectic) is the exchange of arguments, objections, and replies in the hope of shedding light on the matter at issue.
- View Argumentation
Epistemic Probability
- The epistemic probability of a proposition is how much it’s supported by the evidence and arguments.
- Epistemic probabilities are expressed by locutions such as: “it is certain that,” “it is beyond a reasonable doubt that,” “it is likely that,” “it is doubtful that,” and “it is impossible that.”
- View Epistemic Probability
Fact-checking
- Fact-checkers rate claims false, misleading, or unsupported based on the evidence and arguments.
- View Fact-checking
Framework for Determining What’s True
- View Determining What’s True
Framework for Decision-making
- View Decision-making
Kinds of Arguments
- Deductive Arguments
  - A deductive argument is an argument whose conclusion (purportedly) follows necessarily from its premises.
  - View Deductive Arguments
- Evidential Arguments
  - An evidential argument is an argument whose premises are evidence (purportedly) making its conclusion probable.
  - View Evidential Arguments
- Analogical Arguments
  - An analogical argument is an argument that, because things are alike in certain respects, they are therefore (purportedly) alike in a further respect.
  - View Analogical Arguments
- Normative Arguments
  - A normative argument is an argument whose premises are (purportedly) reasons why a particular action should (or shouldn’t) be done.
  - View Normative Arguments
- Defeasible Arguments
  - A defeasible argument is an argument whose premises (purportedly) support its conclusion other things being equal (and which support can therefore be “defeated” by additional information).
  - View Defeasible Arguments
Logical (Metaphysical) Modalities
- The logical modalities are the concepts of necessary truth, logical entailment, logical incompatibility, and logical impossibility that underly deductive logic.
- View Logical (Metaphysical) Modalities
Syllogisms and Venn Diagrams
- View Syllogisms and Venn Diagrams

Pitfalls

Anomalies of Language
- The “mist and veil of words,” using George Berkeley’s phrase, can be a source of confusion that impedes reasoning.
- View Anomalies of Language
Artifices of Deception and Distraction
- The methods of deception and distraction are used to fool people and divert their attention.
- View Artifices of Deception and Distraction
Bias
- Bias is a predisposition that can result in systematic errors in a process of reasoning.
- View Bias
Conspiracy Theories
- A conspiracy theory explains events by invoking a secret plot by a group of conspirators.
- View Conspiracy Theories
Disinformation
- “Those who can make you believe absurdities can make you commit atrocities.” — Voltaire
- View Disinformation
Fallacies
- A fallacy is an error in reasoning having an air of plausibility.
- View Fallacies
Getting Fooled by Statistics
- Statistics is tricky. It’s easy to be fooled.
- View Fooled by Statistics
Why people believe irrational things
- People sometimes believe what they’re predisposed to believe through the process of motivated reasoning.
- View Motivated Reasoning