Questions regarding the first half of the course are here.
Barbara. (Lecture).
It is not expressively complete. There are many valid inferences that cannot be expressed using syllogisms. (Lecture).
The arity of a predicate is the number of terms that it applies to, e.g., mother(x,y) has arity two. (Lecture).
A sentence can be true or false, a term refers to an object. (Lecture).
No. ((1+1)+1) or (1+(1+1)) would be terms, since they are built up from the recursive rules for terms. (Lecture).
1. Variables: X, Y, Z, ...
2. If A and B are propositional formulas, so are not A, A v B, A & B,
A -> B.
3. Nothing else is.
This is a recursive definition.
(Lecture).
B & B C
----- & Elim
B B --> A
---------------------- ->ELim (Modus Ponens)
A
A B | A v B -> A
----+-----------
T T | T T
T F | T T
F T | T F
F F | F T
Since the colum under the arrow is not completely T, the formula is
not a tautology.
(Lecture).
It means "Gamma entails S", i.e., when all the formulas in Gamma are true, also S is true. (Lecture).
Predicate logic is propositional logic plus predicates and quantifiers, (Lecture).
It means that there is a number x, such that all numbers (including itself) is less or equal than it. This is true, since 0 is such a number. (Lecture).
Intuitionistic, constructive, modal, fuzzy, linear... (Lecture).
1. Base clause(s).
2. Recursive clause(s).
3. Final clause.
(Lecture).
Georg Cantor (1845-1918)
The modern axioms of set theory, due to Zermelo and Fraenkel.
No two different elements of the set A are mapped to the same element in B.
"forall x exists y ( x < y )" is true if for all numbers there exists a greater number. This can be true only with infinite sets and it certainly the case for the natural numbers.
"exists x forall y ( x < y )" is true if there is one number such that all numbers are greater than it. This sentence is false, since no number is greater than itself. If would be true if the relation were "greater or equal", because in this case there is a number such that all numbers are greater or equal, namely 1.