A "'non-logical symbol "'only has meaning or semantic content when one is assigned to it by means of an sentence containing a non-logical symbol lacks meaning except under an interpretation, so a sentence is said to be " true or false under an interpretation ".
32.
:* Contains no free variables, and contains logical symbols only ( including identity ), except for one function-symbol "'interpreted ( in advance ) as an Arccosine "'function, and another function symbol "'interpreted ( in advance ) as an Arctan "'function?
33.
I've been asking whether there is a common term for a very specific mapping, i . e . a mapping induced by a first-order formula which contains logical symbols "'only-except for function-symbols for substituting functions which are mapped to one another "'by that formula.
34.
:* Contains no free variables, and contains logical symbols only ( including identity ), except for one function-symbol "'interpreted ( in advance ) as the Sine "'function, and another function symbol "'to be defined by that formula as the Cosine "'function . . .?
35.
In the context of first-order logic, the symbols in a signature are also known as the non-logical symbols, because together with the logical symbols they form the underlying alphabet over which two formal languages are inductively defined : The set of " terms " over the signature and the set of ( well-formed ) " formulas " over the signature.
36.
In the context of first-order logic, the symbols in a signature are also known as the non-logical symbols, because together with the logical symbols they form the underlying alphabet over which two formal languages are inductively defined : The set of " terms " over the signature and the set of ( well-formed ) " formulas " over the signature.
37.
"' Comment " "'If " there is going to be a project for articles related to the golden ratio that is still an open question, as far as I am concerned the most logical symbol would be some version of a lower case ( " not " upper case ) ( Talk ) 13 : 48, 12 March 2008 ( UTC)
38.
While I can imagine that eventually the meaning of mathematical or logical symbols might eventually be established ( for example tranmitting many messages such as " . . + . . = . . . . " would give meaning to + and = ), would it be possible to eventually build up enough meaning from a zero base so that in time they would understand what was meant by the message " Last thursday my Uncle Bill went to the supermarket "?
39.
Walter Isaacson ascribes Lovelace's insight regarding the application of computing to " any " process based on logical symbols to an observation about textiles : " When she saw some mechanical looms that used punchcards to direct the weaving of beautiful patterns, it reminded her of how Babbage's engine used punched cards to make calculations . " This insight is seen as significant by writers such as Betty Toole and Benjamin Woolley, as well as the programmer John Graham-Cumming, whose project Plan 28 has the aim of constructing the first complete Analytical Engine.
40.
A structure over a signature consists of a set " D ", known as the domain of discourse, together with interpretations of the non-logical symbols : Every constant symbol is interpreted by an element of " D ", and the interpretation of an " n "-ary function symbol is an " n "-ary function on " D ", i . e . a function " D n " ?! " D " from the " n "-fold cartesian product of the domain to the domain itself.
