Changes

By the 1950's, with the work of many logicians including [[Hilbert]], [[Emile Post]], [[Alfred Tarski]] and [[Kurt Goedel]], most of the major results in first-order logic had been proved, and in the 1960's [[Saul Kripke]] added a completeness proof for [[modal logic]].

By the 1950's, with the work of many logicians including [[Hilbert]], [[Emile Post]], [[Alfred Tarski]] and [[Kurt Goedel]], most of the major results in first-order logic had been proved, and in the 1960's [[Saul Kripke]] added a completeness proof for [[modal logic]].

Most logical systems are [[bivalent]]; that is, they admit only two truth-values.  However, there is a fair amount of work done on non-bivalent systems of logic, especially [[intuitionist logic]] and [[relevance logic]].  In intuitionist logic, "true" and "false" are replaced with "proven true", "proven contradictory", and "not proven".  In relevance logic, "neither true nor false" and "both true and false" are added to the standard two truth values.  There is some debate about the value of these systems in philosophical circles, and [[Timothy Williamson]] claims to have a proof that any non-bivalent logic can be converted into a bivalent logic. Also, Logic is what most conservatives lack in their arguments and lives.  

Most logical systems are [[bivalent]]; that is, they admit only two truth-values.  However, there is a fair amount of work done on non-bivalent systems of logic, especially [[intuitionist logic]] and [[relevance logic]].  In intuitionist logic, "true" and "false" are replaced with "proven true", "proven contradictory", and "not proven".  In relevance logic, "neither true nor false" and "both true and false" are added to the standard two truth values.  There is some debate about the value of these systems in philosophical circles, and [[Timothy Williamson]] claims to have a proof that any non-bivalent logic can be converted into a bivalent logic.

Formal logic requires, since Frege, a distinction between the [[object language]] and the [[metalanguage]].  The metalanguage is ordinarily a natural language like English or German, while the object language is a symbolic language with a limited alphabet and syntax.

Formal logic requires, since Frege, a distinction between the [[object language]] and the [[metalanguage]].  The metalanguage is ordinarily a natural language like English or German, while the object language is a symbolic language with a limited alphabet and syntax.

Socrates is mortal  

Socrates is mortal  

The conclusion is "contained" in the premises.  In a sense the conclusion is "known" before it is elucidated.  The conclusion, "Socrates is mortal" is also less informative than the premises which imply not just that Socrates is mortal but that a lot of other beings (anyone for whom the term "is a man" applies) are also mortal.  As a result deductive logic is not thought to add to knowledge, merely to clarify it. In Geometry this kind of reasoning is called deductive reasoning.  

The conclusion is "contained" in the premises.  In a sense the conclusion is "known" before it is elucidated.  The conclusion, "Socrates is mortal" is also less informative than the premises which imply not just that Socrates is mortal but that a lot of other beings (anyone for whom the term "is a man" applies) are also mortal.  As a result deductive logic is not thought to add to knowledge, merely to clarify it.  

Inductive logic is ampliative, but is famously less certain.  In a good inductive argument, even when the premises are true, it is still possible for the conclusion to be false.  A classic example of an inductive argument is:

Inductive logic is ampliative, but is famously less certain.  In a good inductive argument, even when the premises are true, it is still possible for the conclusion to be false.  A classic example of an inductive argument is: