PDF
... The first approach is axiomatic and abstract. We state logical properties of the numbers using first-order logic. It might be the case that these first-order properties describe numbers so well that they capture our intuition completely. The classical first-order theory of numbers is called Peano Ar ...
... The first approach is axiomatic and abstract. We state logical properties of the numbers using first-order logic. It might be the case that these first-order properties describe numbers so well that they capture our intuition completely. The classical first-order theory of numbers is called Peano Ar ...
axioms
... terms. These are terms that we make no attempt to define, rather we accept their existence without necessarily placing a meaning upon them. • Similarly, we must have some initial statements which are accepted without justification. These initial statements are called axioms. ...
... terms. These are terms that we make no attempt to define, rather we accept their existence without necessarily placing a meaning upon them. • Similarly, we must have some initial statements which are accepted without justification. These initial statements are called axioms. ...
Russell`s logicism
... talking about the number 3, and number in general, as properties or characteristics. Now he is moving from this to talking about the number 3, and number in general, as sets. The next question is, what sets are they? Russell says: “Reurning now to the definition of number, it is clear that number i ...
... talking about the number 3, and number in general, as properties or characteristics. Now he is moving from this to talking about the number 3, and number in general, as sets. The next question is, what sets are they? Russell says: “Reurning now to the definition of number, it is clear that number i ...
A(x)
... in every model of the set of the premises. But the set of models can be infinite! And, of course, we cannot examine an infinite number of models; but we can verify the ‘logical form’ of the argument, and check whether the models of premises do satisfy the conclusion. ...
... in every model of the set of the premises. But the set of models can be infinite! And, of course, we cannot examine an infinite number of models; but we can verify the ‘logical form’ of the argument, and check whether the models of premises do satisfy the conclusion. ...
pdf
... Church and Turing in 1936 laid the foundations for computer science by defining equivalent notions of computability – Church for software, Turing for hardware. Their ideas were used to make precise the insights of Brouwer from 1900 that mathematics is based on fundamental human intuitions about numb ...
... Church and Turing in 1936 laid the foundations for computer science by defining equivalent notions of computability – Church for software, Turing for hardware. Their ideas were used to make precise the insights of Brouwer from 1900 that mathematics is based on fundamental human intuitions about numb ...
IntroToLogic - Department of Computer Science
... The validity of first order logic is not decidable. (It is semi-decidable.) If a theorem is logically entailed by an axiom, you can prove that it is. But if it is not, you can’t necessarily prove that it is not. (You may go on infinitely with your ...
... The validity of first order logic is not decidable. (It is semi-decidable.) If a theorem is logically entailed by an axiom, you can prove that it is. But if it is not, you can’t necessarily prove that it is not. (You may go on infinitely with your ...
HISTORY OF LOGIC
... – Russell’s Paradox: If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself. ...
... – Russell’s Paradox: If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself. ...
Relating Infinite Set Theory to Other Branches of Mathematics
... to Other Branches of Mathematics Roads to Infinity: The Mathematics of Truth and Proof. By John Stillwell, AK Peters, Natick, Massachusetts, 2010, 250 pages, $39.00. The infinite, wrote Jorge Luis Borges, is a concept that “corrupts and confuses the others.” Certainly, the theory of large infinite s ...
... to Other Branches of Mathematics Roads to Infinity: The Mathematics of Truth and Proof. By John Stillwell, AK Peters, Natick, Massachusetts, 2010, 250 pages, $39.00. The infinite, wrote Jorge Luis Borges, is a concept that “corrupts and confuses the others.” Certainly, the theory of large infinite s ...
ppt
... statements are true, what other statements can you also deduce are true? • If I tell you that all men are mortal, and Socrates is a man, what can you deduce? ...
... statements are true, what other statements can you also deduce are true? • If I tell you that all men are mortal, and Socrates is a man, what can you deduce? ...
Lecture 34 Notes
... Mike then goes on to note that the function rule essentially leads to Russell’s paradox. He gives the “Russell” version (p.17). Next Mike shows that Musser’s attempted fix also fails. That was for the programming language Euclid. He comments that in our book, A Programming Logic, 1978, we use a tota ...
... Mike then goes on to note that the function rule essentially leads to Russell’s paradox. He gives the “Russell” version (p.17). Next Mike shows that Musser’s attempted fix also fails. That was for the programming language Euclid. He comments that in our book, A Programming Logic, 1978, we use a tota ...
slides - Department of Computer Science
... Definable Functions of TC What kind of functions our theory TC can (essentially) prove to For simplicity: only string exist? inputs to function When do we say that a theory can prove the existence of a function f(X) (aka, a provably total function in the theory) ? ...
... Definable Functions of TC What kind of functions our theory TC can (essentially) prove to For simplicity: only string exist? inputs to function When do we say that a theory can prove the existence of a function f(X) (aka, a provably total function in the theory) ? ...
Sub-Birkhoff
... that s is equal to t in equational logic iff s is convertible to t in the rewrite system induced by the specification. This extends to subequational logics (cf. [3, Lem. 2.6] for rewrite logic). Lemma 8 For a subequational logic L = hS,Ii with (axiom),(congruence) ∈ I we have ` s L t iff s →IS t. He ...
... that s is equal to t in equational logic iff s is convertible to t in the rewrite system induced by the specification. This extends to subequational logics (cf. [3, Lem. 2.6] for rewrite logic). Lemma 8 For a subequational logic L = hS,Ii with (axiom),(congruence) ∈ I we have ` s L t iff s →IS t. He ...
PDF
... 3. We will in due course explore the notion of constructive “truth” or evidence. We will see that we can’t decide whether there is evidence for a given proposition, i.e. whether a programming task is solvable. 4. The notion of evidence/truth depends on a theory of types and programs which we are gra ...
... 3. We will in due course explore the notion of constructive “truth” or evidence. We will see that we can’t decide whether there is evidence for a given proposition, i.e. whether a programming task is solvable. 4. The notion of evidence/truth depends on a theory of types and programs which we are gra ...
Löwenheim-Skolem theorems and Choice principles
... called Skolem functions, which can be found in the modern standard proofs of the theorem. A careful examination of the proof shows that the axiom of dependent choice is sufficient already (see [3]) and in fact it happens to be equivalent to it, as shown in the following1 : Theorem 2. LS(ℵ0 ) is equi ...
... called Skolem functions, which can be found in the modern standard proofs of the theorem. A careful examination of the proof shows that the axiom of dependent choice is sufficient already (see [3]) and in fact it happens to be equivalent to it, as shown in the following1 : Theorem 2. LS(ℵ0 ) is equi ...
logical axiom
... ponens”, which states that from formulas A and A → B, one my deduce B. It is easy to see that this rule preserves logical validity. The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In addition, it is also complete. Note that in the above se ...
... ponens”, which states that from formulas A and A → B, one my deduce B. It is easy to see that this rule preserves logical validity. The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In addition, it is also complete. Note that in the above se ...
pdf
... + is represented by the formula Multiplication ∗ is represented by the formula The constant function ck can be represented by the formula n The projection function π i can be represented by the formula The composition h = g◦f1 , .., fk can be represented by the formula (∃z1 ,..zk )( Rf1 (x1 ,..xn ,z ...
... + is represented by the formula Multiplication ∗ is represented by the formula The constant function ck can be represented by the formula n The projection function π i can be represented by the formula The composition h = g◦f1 , .., fk can be represented by the formula (∃z1 ,..zk )( Rf1 (x1 ,..xn ,z ...
PDF
... Since the language only provides two function symbols (all others would be an abbreviation for combinations of these) there are only four substitution axioms. This means that the theory Q is finitely axiomatizable. ...
... Since the language only provides two function symbols (all others would be an abbreviation for combinations of these) there are only four substitution axioms. This means that the theory Q is finitely axiomatizable. ...
hilbert systems - CSA
... Derivation 1: Z1, Z2, ... Zn is a derivation of Y from S U {X}, Zn = Y Derivation 2: Prefix X >. X > Z1, X > Z2, .... X > Y If Zi is an axiom or a member of S, then insert Zi and Zi > (X > Zi) If Zi is the formula X, insert steps of derivation of X > X If Zi comes from MP, then there exists Zj and Z ...
... Derivation 1: Z1, Z2, ... Zn is a derivation of Y from S U {X}, Zn = Y Derivation 2: Prefix X >. X > Z1, X > Z2, .... X > Y If Zi is an axiom or a member of S, then insert Zi and Zi > (X > Zi) If Zi is the formula X, insert steps of derivation of X > X If Zi comes from MP, then there exists Zj and Z ...
Syntax of first order logic.
... symbols together with a signature σ : I ∪ J → N. In addition to the symbols from L, we shall be using the logical symbols ∀, ∃, ∧, ∨, →, ¬, ↔, equality =, and a set of variables Var. Definition of an L-term. Every variable is an L-term. If σ(f˙i ) = n, and t1 , ..., tn are L-terms, then f˙i (t1 , .. ...
... symbols together with a signature σ : I ∪ J → N. In addition to the symbols from L, we shall be using the logical symbols ∀, ∃, ∧, ∨, →, ¬, ↔, equality =, and a set of variables Var. Definition of an L-term. Every variable is an L-term. If σ(f˙i ) = n, and t1 , ..., tn are L-terms, then f˙i (t1 , .. ...
Document
... numbers.) From this it follows that no function's range will ever be able to include any object defined in terms of the function itself. As a result, propositional functions (along with their corresponding propositions) will end up being arranged in a hierarchy of exactly the kind Russell proposes. ...
... numbers.) From this it follows that no function's range will ever be able to include any object defined in terms of the function itself. As a result, propositional functions (along with their corresponding propositions) will end up being arranged in a hierarchy of exactly the kind Russell proposes. ...
ppt
... Mathematicians will never be completely replaced by computers – There are mathematical truths that cannot be determined mechanically – We can build a computer that will prove only true theorems about number theory, but if it cannot prove something we do not know that that is not a true theorem. ...
... Mathematicians will never be completely replaced by computers – There are mathematical truths that cannot be determined mechanically – We can build a computer that will prove only true theorems about number theory, but if it cannot prove something we do not know that that is not a true theorem. ...
The Origin of Proof Theory and its Evolution
... Sequent Calculus: a technical device for proving consistency of predicate logic in natural deduction; Cut Elimination: states that every sequent calculus derivation can be transformed into another derivation with the same end sequent and in which the cut rule does not occur. ...
... Sequent Calculus: a technical device for proving consistency of predicate logic in natural deduction; Cut Elimination: states that every sequent calculus derivation can be transformed into another derivation with the same end sequent and in which the cut rule does not occur. ...
HW 12
... b. Construct a formal proof that shows that for any sets A, B, and C: A (B \ C) = (A B) \ C 5. Any set without any elements is an empty set a. Provide a definitional axiom that defines a 1-place predicate Empty(x) expressing that x is an empty set b. Construct a formal proof that shows that ther ...
... b. Construct a formal proof that shows that for any sets A, B, and C: A (B \ C) = (A B) \ C 5. Any set without any elements is an empty set a. Provide a definitional axiom that defines a 1-place predicate Empty(x) expressing that x is an empty set b. Construct a formal proof that shows that ther ...