Formal Languages For First-order Logic
Given a signature σ, the corresponding formal language is known as the set of σ-formulas. Each σ-formula is built up out of atomic formulas by means of logical connectives; atomic formulas are built from terms using predicate symbols. The formal definition of the set of σ-formulas proceeds in the other direction: first, terms are assembled from the constant and function symbols together with the variables. Then, terms can be combined into an atomic formula using a predicate symbol (relation symbol) from the signature or the special predicate symbol "=" for equality (see the section "Interpreting equality" below). Finally, the formulas of the language are assembled from atomic formulas using the logical connectives and quantifiers.
Read more about this topic: Interpretation (logic), First-order Logic
Famous quotes containing the words formal, languages and/or logic:
“The conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.”
—David Elkind (20th century)
“Science and technology multiply around us. To an increasing extent they dictate the languages in which we speak and think. Either we use those languages, or we remain mute.”
—J.G. (James Graham)
“Neither Aristotelian nor Russellian rules give the exact logic of any expression of ordinary language; for ordinary language has no exact logic.”
—Sir Peter Frederick Strawson (b. 1919)