Difference between revisions of "First order language"
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* {{link|Formula|FOL}} - the set of all formulas is denoted {{M|\mathscr{L}_F}} | * {{link|Formula|FOL}} - the set of all formulas is denoted {{M|\mathscr{L}_F}} | ||
** {{link|Sentence|FOL}} - a special kind of formula | ** {{link|Sentence|FOL}} - a special kind of formula | ||
+ | * {{link|Structure|FOL}} | ||
+ | ** {{link|Domain|FOL}} | ||
+ | ** {{link|Interpretation|FOL}} | ||
+ | * {{link|Model|FOL}} | ||
+ | ** {{link|Assignment|FOL}} | ||
{{Todo|Add more}} | {{Todo|Add more}} | ||
+ | |||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Formal logic navbox|plain}} | {{Formal logic navbox|plain}} | ||
{{Definition|Formal Logic}} | {{Definition|Formal Logic}} |
Latest revision as of 10:45, 8 September 2016
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Needed for set theory
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This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Tasks:
- Expand on the logical connective symbol caution.
- There are notes on this topic
Definition
A first order language, [ilmath]\mathcal{L} [/ilmath] consists[1] of two types of symbols, non-logical and logical, these are described in the tree below:
- Non-logical symbols - these are the same for all first order languages
- [ilmath]V[/ilmath] - The set of (at most countably many, possibly empty) variable symbols: [ilmath]x_1,x_2,\ldots,x_n,\ldots[/ilmath]
- [ilmath]C[/ilmath] - The set of logical connective symbols. Caution:Not all of these are needed, you can write some in terms of others
- [ilmath]\neg[/ilmath] - logical not
- [ilmath]\wedge[/ilmath] - logical and
- [ilmath]\vee[/ilmath] - logical or
- [ilmath]\rightarrow[/ilmath] - logical implication, "if ... then ..."
- [ilmath]\leftrightarrow[/ilmath] - logical equivalence (AKA: if and only if)
- [ilmath]Q[/ilmath] - The set of quantifier symbols. Caution:Given [ilmath]\neg[/ilmath] you can define [ilmath]\forall x(A)[/ilmath] as [ilmath]\neg(\exists x(\neg(A)))[/ilmath] or define [ilmath]\exists x(A)[/ilmath] as [ilmath]\neg(\forall x(\neg(A)))[/ilmath]
- [ilmath]E[/ilmath] - The set containing the equality symbol. We will use [ilmath]\doteq[/ilmath] for this (to separate it from equality in the meta-language)
- [ilmath]B[/ilmath] - The set of brackets, that is "(" and ")".
- Non-logical symbols - these vary from language to language
- [ilmath]\mathscr{L}_c[/ilmath] - the set of (possibly zero, at most countably many) constant symbols, [ilmath]c_1,c_2,\ldots,c_n,\ldots[/ilmath].
- [ilmath]\mathscr{L}_f[/ilmath] - the set of (possibly zero, at most countably many) function symbols.
- Each function has an arity, and we write [ilmath]ft_1t_2\cdots t_m[/ilmath] for an [ilmath]m[/ilmath]-ary function (here the [ilmath]t_i[/ilmath] are terms)
- Here [ilmath]m\ge 1[/ilmath][1] Caution:This is disputed, Kunen's Set Theory gives an example of [ilmath]0[/ilmath]-ary predicates and functions!
- Each function has an arity, and we write [ilmath]ft_1t_2\cdots t_m[/ilmath] for an [ilmath]m[/ilmath]-ary function (here the [ilmath]t_i[/ilmath] are terms)
- [ilmath]\mathscr{L}_P[/ilmath] - the set of (possibly zero, at most countably many) predicate symbols, [ilmath]P_1,P_2,\ldots,P_n,\ldots[/ilmath].
See next
- Term - the set of all terms is denoted [ilmath]\mathscr{L}_T[/ilmath]
- Formula - the set of all formulas is denoted [ilmath]\mathscr{L}_F[/ilmath]
- Sentence - a special kind of formula
- Structure
- Model
TODO: Add more