Symbolic Logic - Chapter 3 - The Predicate Calculus 1
|The propositional calculus reveals validity when this depends on propositional structure alone. (T/F)||True.|
|What do lower case letters represent in the predicate calculus?||They represent proper names.|
|What do upper-case letters represent in the predicate calculus?||They represent property-expressions or predicates.|
|What are upper-case letters called in the predicate calculus?||They are called predicate letters.|
|What are lower-case letters called in the predicate calculus? (93)||They are called proper names.|
|How does one write that m does not have G? (93)||-Gm|
|How do we say that m has property F? (93)|| We juxtapose the symbols 'F' and 'm' in that order, we write:|
|What do we obtain from "all robins are migrants" or "every robin is a migrant"? (94)|| We obtain everything with the property of being a robin has the property of being a migrant.|
It can be written as:
Everything with F has G.
|To what do we naturally abbreviate 'x has F'? (95)||Fx|
|How do we write 'for any x'? (95)||We enclose 'x' in brackets.|
|How do we write 'For any x, if x has F then x has G'? (95)||(x)(Fx -> Gx)|
|How do we write 'nothing with F has G'? (95)|| (x)(Fx -> -Gx)|
For any x, if x has the property of F, then it does not have the property of G.
|How do we write 'nothing with F has both G and H'? (95)||(x)(Fx -> -(Gx & Hx))|
|The device of universal quantifier enables us to render into logical notation many sentences that contain what such words? (95)||'All', 'every', 'any', 'everything', 'no', 'none', and/or 'nothing'.|
|What is (3x)? (96)||It is the existential quantifier.|
|What does (3x) represent? (97)|| There is an x such that..|
An object x can be found which...
|How do we write 'some felons are German'?||(3x)(Fx & Gx)|
|If F means felon and G means German, how do we read: (3x)(Fx & Gx)? (97)||There exists some object that is both a felon and a German.|
|How can one write 'something with F has not G', such as 'some Frenchmen are not generous'? (97)||(3x)(Fx & -Gx)|
|How might we summarize the task of translation into the quantifier-notation? (97)|| 1) Render into a sentence about properties, and employ predicate-letters for these properties.|
2) Introduce variables
3) Introduce propositional calculus connectives and quantifiers.
|A predicate-letter followed by one name expresses a ___________. A predicate-letter followed by two names expresses a ___________. (98)||Property, Relation|
|If we give 'Pmn' the interpretation 'm is a parent of n', how can we express 'Prince Charles has a parent', where n is Prince Charles? (98)|| (3x)Pxn|
This translates to: "There exists an 'x', that is a parent of Prince Charles."
(I'm a bit confused as to why the book does not use 'm', but rather 'x'.)
|If we give 'Pmn' the interpretation of 'm is a parent of n', and Prince Philip is represented by m, how can we write 'Prince Philip has a child'? (99)|| (3x)(Pmx)|
This translates to: "There exists an 'x' of which Prince Philip is a parent."
|If Pmn reads as 'm is a parent of n', what is the difference in reading (3x)Pnx in contrast with (3x)Pxn if 'n' represents Prince Charles? (99)|| The first reads: there exists an 'x' of which Prince Charles is a parent.|
The second reads: there exists an 'x' of which Prince Charles is a son, or there exists an 'x' of which 'x' is a parent of Prince Charles.