Meanings, Equivalence, & Independence Flashcards
How can “use” and “reference” be applied to formal systems?
a) We have assign specific meaning to primitives by means of interpretation, i.e. by specifying particalur referents in the structure
b) Since many different interpretations possible, say the meanings of the primitives are definied implicitly by the relations in which they are assumed to stand according to axoims
Interpretation: Both syntax and semantics
What is the meaning of ‘dog’?
In terms of reference and use
Meaning in reference: the term ‘dog’ refers to the concept of DOG;
They say ‘dogs are nice’ means that the concept of DOG is subsumed under the concept of NICE; i.e. everything falls under the concept of DOG falls under the concept NICE
Meaning in use: the meaning of dog consists of how we use the term in a sentence; all true statements one can make about ‘dogs’ constitutes it meaning
Ex: ‘Dogs are furry’
Meaning: Complicated
Definitions: Assigned meaning
What is explicit definitions (within a system)?
Introduce of a new terms, expressed by old ones that are already accepted in the vocabulary
- Abbreviation; e.g. 2 <=>df …
-‘C–’ iff
Eliminable in principle (‘circle’ > points at given distance from a given point)
Not all terms can be defined explicitly
What are implicit definitions (in a system)?
new terms defined through a context of use (“contextual definitions)
- “Between any two points there is another point”
Not all terms can be defined explicitly
What is syntacticaly equivalence?
To show that P is equivalent to Q (relative to A1, …, An):
- Assume A1, …, An and P. Prove Q.
- Assume A1, …, An and Q. Prove P.
Whatever you prove from A1, …, An and P, you can also prove from A1, …, An and Q.
What are Euclid’s postulates in The Elements:
Postulate 01: To draw a straight line from any point to any point
Postulate 02: To produce a finite straight line continuously in a straight line
Postulate 03: To describe a circle with any center and radius
Postulate 04: The all right angles equal one another
Postulate 05: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if prodced indefinitely, meet on that side on which are the angles less than the right angles.
What postulates are equivalent to Euclid’s fifth postulate?
(relative to his other postulates)
(1) Proclus’ Axiom: If a line intersectts one of two parallels, it must intersect the other also.
(2) Equidistance Postulates: Parallel lines are everywhere equidistant
(3) Playfair’s Postulate: Through a point not on a given line, there can be drawn one and only one parallel to the given line
(4) The Triangle Postulate: The sum of the angles of a triangle is two right angles
How to show syntactical equivalence between Euclid’s fifth postulate and Playfair’s postulate:
- Assume Euclid’s postulates 1-4 (A1, …, A4) and his postulate 5 (P): Prove Playfair’s postulate (Q)
- Assume Euclid’s postulates 1-4 (A1, …, A4) and Playfair’s postulate (Q): Prove Euclid’s postulate (P)
How do you show independence?
A statement P is independent of a set of statements A1, …, An, if
a) P cannot be proven from A1, …, An, and,
b) ¬P cannot be proven from A1, …, An, either
How to show that something cannot be proven?
How can we show tha something cannot be proven?
Give a model that satisfies A1, …, An but makes P false
What is truth preservation?
Want logical inferences to preserve truth:
If the premises are true, then the conclusion must also be true
If we have a model that makes all premises A1, …, An true, then every logical conclusion must also be true in the model
To show, that a conclusion P does not follow from the premises A1, …, An we have to give a model that satisfies A1, …, An, but in which P is false
In the case of Euclid, need to interpret the terms ‘points’ and ‘lines’ in such a way that postulates 1-4 come true, but the Parallel Postulate is false
How to slow that the Parallel Postulate does not follow from the premises:
In the case of Euclid, need to interpret the terms ‘points’ and ‘lines’ in such a way that postulates 1-4 come true, but the Parallel Postulate is false
Implicitly define the postulates
How to slow that the Parallel Postulate is independent from the other axoims:
Playfair’s Postulate: Through a point not on a given line, there can be drawn one and only one parallel to the give line
What is its negation?
Two Options:
1. There is no parallel to the given line through the given point
2. There are many parallel to the given line through the given point
No Parallels: Model for Elliptic Geometry
Many Parallels: Model for hyperbolic geometry
Model for Elliptic Geometry (no parallels)
To prove that parallel postulate is indepenent
Model for Hyperbolic Geometry (Many Parallels):
