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predicate_logic [2015/10/13 20:54] nikolaj |
predicate_logic [2016/05/01 15:38] nikolaj |
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| This is then always true for logical equality. Conversely, the axioms for equality in mathematical theory (Peano axioms for natural numbers or set theory axioms like in the theory of ZFC) are rules that tell you when you can follow that $a=b$. For example, the almost always adopted extensionality axioms for sets says that if two sets have the same members, then they are equal. | This is then always true for logical equality. Conversely, the axioms for equality in mathematical theory (Peano axioms for natural numbers or set theory axioms like in the theory of ZFC) are rules that tell you when you can follow that $a=b$. For example, the almost always adopted extensionality axioms for sets says that if two sets have the same members, then they are equal. | ||
| - | In the following, we introduce some predicates which are used to formulate logical sentences more concisely. To make reading easier, I've split this predicate defintion in three conceptually parts. The first couple of definitions are really just common abbreviations in logic. After that follow some set theoretical notions, which are used in formulations of the axioms, set specifications, theorems and so on. | + | In the following, we introduce some predicates which are used to formulate logical sentences more concisely. To make reading easier, I've split this predicate defintion in three conceptual parts. The first couple of definitions are really just common abbreviations in logic. After that follow some set theoretical notions, which are used in formulations of the axioms, set specifications, theorems and so on. |
| === Abbreviations === | === Abbreviations === | ||
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| the first time, which we use to introduce new notation. | the first time, which we use to introduce new notation. | ||
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| + | I came up with | ||
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| + | $P(!a) \equiv \left(\forall x.\,P(x)\implies x=a\right)$ | ||
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| + | for a predicate $P$, expressing that $P$ only holds for the term $a$. | ||
| We also use the abbreviation | We also use the abbreviation | ||
