The ( well-formed ) formulas of the "'language of first-order arithmetic "'are built up from these symbols together with the logical symbols in the usual manner of first-order logic.
22.
If it is treated as a logical symbol, then any interpretation will be required to interpret the equality sign using true equality; if interpreted as a non-logical symbol, it may be interpreted by an arbitrary equivalence relation.
23.
If it is treated as a logical symbol, then any interpretation will be required to interpret the equality sign using true equality; if interpreted as a non-logical symbol, it may be interpreted by an arbitrary equivalence relation.
24.
It is common to divide the symbols of the alphabet into "'logical symbols "', which always have the same meaning, and "'non-logical symbols "', whose meaning varies by interpretation.
25.
It is common to divide the symbols of the alphabet into "'logical symbols "', which always have the same meaning, and "'non-logical symbols "', whose meaning varies by interpretation.
26.
Here's what I think is a pretty concrete, intuitive example, that shows up some of the issues : Suppose you're just looking at linear orders; your language has just one non-logical symbol, namely <.
27.
IConji debuted with 1183 unique characters, known as the " lexiConji " ( vocabulary ), culled from base words used in common daily communications, word frequency lists, often-used mathematical and logical symbols, punctuation symbols, and the flags of all nations.
28.
So my question is : "'do we actually allow other non-logical symbols in first-order Peano arithmetic in certain conditions ? or do the arithmetical hierarchy formulas actually not belong to first-order Peano arithmetic ? or am I missing something ? "'
29.
:: : : While it is clear to me such a formula can be constructed by using new non-logical symbols, it is unclear how to do that "'without "'such symbols . talk ) 09 : 32, 16 June 2016 ( UTC)
30.
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 ".
