Logic

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

  1. Logic
  2. Formal Logic
  3. Informal Logic

Logic

  • Logic is the systematic study of the principles and methods of valid reasoning, invented by Aristotle in the 4th century BCE.
  • Logic has always focused on evaluating arguments, an argument being a set of propositions, one of which (the conclusion) is purportedly supported by the others (the premises).
  • There are two approaches to the subject:
    • Formal Logic is the study of valid reasoning within a formal (symbolic) language.
    • Informal Logic is the practical study of valid reasoning in the context of reasoning in real life.

Formal (Symbolic, Mathematical) Logic

  • Formal Logic is the study of the principles and methods of valid reasoning within a formal (symbolic) language.
The Need for 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 (is of predication).
      • Mark Twain is Samuel Clemens (is of identity).
      • There is a solution to the problem (is of existence).
    • Ambiguity
      • He’s a poor student.
      • Visiting relatives can be boring.
    • Multiple ways of saying the same thing:
      • The car won’t go unless there’s gas in the tank.
      • The car won’t go without gas in the tank.
      • The car won’t go if there’s no gas in the tank.
    • Figurative and literal uses of language
      • The groom has cold feet.
      • Money talks.
    • Obscurity
      • “Being, the indeterminate immediate, is in fact nothing, and neither more nor less than nothing.”
        • Georg Wilhelm Friedrich Hegel.
What a Simple 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.
  • Translated into the language of Propositional Logic, the argument becomes:
    • N → I
    • ~I
    • Therefore, ~N.
  • Where
    • N = Newton’s theory of gravitation is true.
    • I = the gravitational attraction between Sun and Earth is instantaneous.
    • → means “if … then — “.
    • ~ means “It is false that ….”.
  • It is readily shown in Propositional Logic that, ~N follows 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.
  • 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:
  • It’s then shown 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.

Informal (Practical, Applied) Logic

  • Informal Logic is the practical study of the principles and methods of valid reasoning.
Evaluating Arguments
  • The focus of Informal Logic is the evaluation of arguments as they occur in real life.
  • Evaluating an argument means determining the extent its premises support its conclusion.
  • Which involves three things:
    • Reconstructing the argument so its premises, conclusion, and reasoning are clear.
    • Determining whether the premises are true, typically beyond a reasonable doubt.
    • Assessing the reasoning by which the premises are supposed to support the conclusion.
Pages on Informal Logic
Topics
  • Arguments
    • An argument is a set of propositions, one of which (the conclusion) is purportedly supported by the others (the premises)
    • 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
    • Epistemic probability is the kind of probability relative to evidence and expressed by locutions such as:
      • it is certain that, it is beyond a reasonable doubt that, it is likely that, it is doubtful that, 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
  • Framework for Decision-making
  • Kinds of Arguments
    • Deductive Arguments
      • A deductive argument is an argument whose conclusion follows necessarily from its premises.
      • View Deductive Arguments
    • Inductive Arguments
      • An inductive argument is an argument whose conclusion is probable based on its premises.
      • View Inductive Arguments
    • Analogical Arguments
      • An analogical argument is an argument that things are alike in a certain respect because they are alike in others respects.
      • View Analogical Arguments
    • Normative Arguments
      • A normative argument is an argument that an action should be done based on reasons for doing it.
      • View Normative Arguments
  • Metaphysical Modalities
    • The metaphysical modalities are the concepts of necessity, possibility, contingency, entailment, incompatibility, and equivalence that underly deductive logic.
    • View Metaphysical Modalities
  • 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
  • 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
  • Why people believe irrational things
    • People sometimes believe what they’re predisposed to believe through the process of motivated reasoning.
    • View Motivated Reasoning