Difference between revisions of "Formal logic language"
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Formal logic plays an important role, especially so in set theory, but overall an important role. It is important to have a concrete understanding of this. | Formal logic plays an important role, especially so in set theory, but overall an important role. It is important to have a concrete understanding of this. | ||
| − | ===Example 1 | + | ==Definitions== |
| + | ===Functions=== | ||
| + | These take some arguments (the number of which is called "Arity") and return something. | ||
| + | ====Examples==== | ||
| + | *<math>\text{Plus}(\text{Natural }a,\text{Natural }b)\rightarrow\text{Natural}</math> | ||
| + | ===Predicate=== | ||
| + | Predicates are basically [[Relation|relations]] - they are either true or false, they are similar to functions in the sense they are either true or false, but they are not [[Function|functions]] as such (arguably they are functions, <math>P:X\times Y\rightarrow\{\text{true},\text{false}\}</math>) | ||
| + | ====Examples==== | ||
| + | * <math>\text{Mortal}(\text{Person }x)</math> | ||
| + | ===Quantifiers=== | ||
| + | We cannot express things like "all men have a mother" in this language, we require quantifiers: | ||
| + | * <math>\forall</math> - forall | ||
| + | * <math>\exists</math> - exists | ||
| + | * <math>\exists_1</math> - exists only one thing, a unique thing | ||
| + | ====Examples==== | ||
| + | * <math>\forall x:\text{Man}\exists_1 y:\text{Woman}\ \text{MotherOf}(x,y)</math> | ||
| + | |||
| + | ==Example 1== | ||
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Revision as of 15:03, 9 March 2015
Contents
Introduction
Formal logic plays an important role, especially so in set theory, but overall an important role. It is important to have a concrete understanding of this.
Definitions
Functions
These take some arguments (the number of which is called "Arity") and return something.
Examples
- [math]\text{Plus}(\text{Natural }a,\text{Natural }b)\rightarrow\text{Natural}[/math]
Predicate
Predicates are basically relations - they are either true or false, they are similar to functions in the sense they are either true or false, but they are not functions as such (arguably they are functions, [math]P:X\times Y\rightarrow\{\text{true},\text{false}\}[/math])
Examples
- [math]\text{Mortal}(\text{Person }x)[/math]
Quantifiers
We cannot express things like "all men have a mother" in this language, we require quantifiers:
- [math]\forall[/math] - forall
- [math]\exists[/math] - exists
- [math]\exists_1[/math] - exists only one thing, a unique thing
Examples
- [math]\forall x:\text{Man}\exists_1 y:\text{Woman}\ \text{MotherOf}(x,y)[/math]
Example 1
| [math]\forall x\forall y\forall z((P(x,y)\wedge P(y,z))\rightarrow P(x,z))[/math] | [math]\forall x\forall y\forall z(P(x,y)\wedge P(y,z)\rightarrow P(x,z))[/math] | ||||||||||||||||||||||||
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| Whenever we have ([ilmath]P(x,y)[/ilmath] and [ilmath]P(y,z)[/ilmath]) we also have [ilmath]P(x,z)[/ilmath] | [ilmath]P(x,y)[/ilmath] is true and whenever [ilmath]P(y,z)[/ilmath] is true then [ilmath]P(x,z)[/ilmath] is true | ||||||||||||||||||||||||
| It is always true that if ([ilmath]P(x,y)[/ilmath] and [ilmath]P(y,z)[/ilmath]) then [ilmath]P(x,z)[/ilmath] | It is always true that [ilmath]P(x,y))[/ilmath] and (if [ilmath]P(y,z)[/ilmath] then [ilmath]P(x,z)[/ilmath]) | ||||||||||||||||||||||||
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