Introduction to Discrete Structures Introduction
... of B is |B|=25 because there are 25 primes less than or equal to 100. – The cardinality of the empty set is ||=0 – The sets N, Z, Q, R are all infinite ...
... of B is |B|=25 because there are 25 primes less than or equal to 100. – The cardinality of the empty set is ||=0 – The sets N, Z, Q, R are all infinite ...
Chapter 4. Logical Notions This chapter introduces various logical
... at least two cats, so there is at least one cat is formally valid the logician may paraphrase there are at least two cats as there is an x such that there is a y such that x is a cat and y is cat and it is not the case that x is identical to y. The numerical sentence and its "identity" paraphrase a ...
... at least two cats, so there is at least one cat is formally valid the logician may paraphrase there are at least two cats as there is an x such that there is a y such that x is a cat and y is cat and it is not the case that x is identical to y. The numerical sentence and its "identity" paraphrase a ...
A. Formal systems, Proof calculi
... The reason why proof calculi have been developed can be traced back to the end of 19 th century. At that time formalization methods had been developed and various paradoxes arose. All those paradoxes arose from the assumption on the existence of actual infinities. To avoid paradoxes, D. Hilbert (a s ...
... The reason why proof calculi have been developed can be traced back to the end of 19 th century. At that time formalization methods had been developed and various paradoxes arose. All those paradoxes arose from the assumption on the existence of actual infinities. To avoid paradoxes, D. Hilbert (a s ...
XR3a
... Prove: The sum of an irrational number and a rational number is irrational. Proof: Let q be an irrational number and r be a rational number. Assume that their sum is rational, i.e., q+r=s where s is a rational number. Then q = s-r. But by our previous proof the sum of two rational numbers must be ra ...
... Prove: The sum of an irrational number and a rational number is irrational. Proof: Let q be an irrational number and r be a rational number. Assume that their sum is rational, i.e., q+r=s where s is a rational number. Then q = s-r. But by our previous proof the sum of two rational numbers must be ra ...
Quadripartitaratio - Revistas Científicas de la Universidad de
... membership relation and that the common noun following is really a proper name of a class. See my 2013 “Errors in Tarski’s 1983 truth-definition paper”. The is of identity can make a predicate out of a proper name as in ‘two plus one is three’, where ‘two plus one’ is the subject and ‘is three’ the ...
... membership relation and that the common noun following is really a proper name of a class. See my 2013 “Errors in Tarski’s 1983 truth-definition paper”. The is of identity can make a predicate out of a proper name as in ‘two plus one is three’, where ‘two plus one’ is the subject and ‘is three’ the ...
BASIC COUNTING - Mathematical sciences
... • Propositional logic is the study of propositions (true or false statements) and ways of combining them (logical operators) to get new propositions. It is effectively an algebra of propositions. In this algebra, the variables stand for unknown propositions (instead of unknown real numbers) and the ...
... • Propositional logic is the study of propositions (true or false statements) and ways of combining them (logical operators) to get new propositions. It is effectively an algebra of propositions. In this algebra, the variables stand for unknown propositions (instead of unknown real numbers) and the ...
BEYOND FIRST ORDER LOGIC: FROM NUMBER OF
... distinguish it from first order model theory. We give more detailed examples accessible to model theorists of all sorts. We conclude with questions about countable models which require only a basic background in logic. For the past 50 years most research in model theory has focused on first order lo ...
... distinguish it from first order model theory. We give more detailed examples accessible to model theorists of all sorts. We conclude with questions about countable models which require only a basic background in logic. For the past 50 years most research in model theory has focused on first order lo ...
The disjunction introduction rule: Syntactic and semantics
... Obviously, this fact could be interpreted as evidence that the mental models theory holds, since it appears to show that people only reason considering semantic models, and not formal or syntactic rules. However, this problem does not really affect theories such as the mental logic theory. As indica ...
... Obviously, this fact could be interpreted as evidence that the mental models theory holds, since it appears to show that people only reason considering semantic models, and not formal or syntactic rules. However, this problem does not really affect theories such as the mental logic theory. As indica ...
The Herbrand Manifesto
... weaker. In fact, it is stronger. There are more things that are true. We cannot prove them all, but we can prove everything we could prove before. Some may be disturbed by the fact that Herbrand entailment is not semi-decidable. But a similar argument could be leveled against Tarskian semantics. Sem ...
... weaker. In fact, it is stronger. There are more things that are true. We cannot prove them all, but we can prove everything we could prove before. Some may be disturbed by the fact that Herbrand entailment is not semi-decidable. But a similar argument could be leveled against Tarskian semantics. Sem ...
Part3
... Example: Prove that there is no largest prime number. Solution: Assume that there is a largest prime number. Call it pn. Hence, we can list all the primes 2,3,.., pn. Form None of the prime numbers on the list divides r. Therefore, by a theorem in Chapter 4, either r is prime or there is a smaller p ...
... Example: Prove that there is no largest prime number. Solution: Assume that there is a largest prime number. Call it pn. Hence, we can list all the primes 2,3,.., pn. Form None of the prime numbers on the list divides r. Therefore, by a theorem in Chapter 4, either r is prime or there is a smaller p ...
The Complete Proof Theory of Hybrid Systems
... true. Soundness should be sine qua non for formal verification, but is so complex for hybrid systems [PC07] that it is often inadvertently forsaken. In logic, we can simply ensure soundness locally per proof rule. More intriguingly, however, our logical setting also enables us to ask the converse: i ...
... true. Soundness should be sine qua non for formal verification, but is so complex for hybrid systems [PC07] that it is often inadvertently forsaken. In logic, we can simply ensure soundness locally per proof rule. More intriguingly, however, our logical setting also enables us to ask the converse: i ...
Insights into Modal Slash Logic and Modal Decidability
... well make use of the previous choices of either player, his or her own ones included. Hodges (2007) points out that in the literature some authors discussing IF first-order logic have opted for formulating the semantics as Hintikka does while others have utilized strategy functions in the standard g ...
... well make use of the previous choices of either player, his or her own ones included. Hodges (2007) points out that in the literature some authors discussing IF first-order logic have opted for formulating the semantics as Hintikka does while others have utilized strategy functions in the standard g ...
Separation Logic with One Quantified Variable
... first-order quantifiers can be found in [11, 4]. However, these known results crucially rely on the memory model addressing cells with two record fields (undecidability of 2SL in [6] is by reduction to the first-order theory of a finite binary relation). In order to study decidability or complexity ...
... first-order quantifiers can be found in [11, 4]. However, these known results crucially rely on the memory model addressing cells with two record fields (undecidability of 2SL in [6] is by reduction to the first-order theory of a finite binary relation). In order to study decidability or complexity ...
ordinal logics and the characterization of informal concepts of proof
... is also a proof predicate, but Gonx8 is provable in (S) itself. To give a precise treatment of this idea of recognizing a proof predicate as such we shall consider formal systems whose constants are not only numerical terms and function symbols, but also proof predicates. This is independently justi ...
... is also a proof predicate, but Gonx8 is provable in (S) itself. To give a precise treatment of this idea of recognizing a proof predicate as such we shall consider formal systems whose constants are not only numerical terms and function symbols, but also proof predicates. This is independently justi ...
Annals of Pure and Applied Logic Automata and logics
... canonical if n is even, for each even i the interval Ii is singular (i.e. of the form [t , t ]), and for no even i such that 0 < i < n is ai−1 = ai = ai+1 . The canonical interval representation for the signal of Fig. 1 is ([0, 0], a)((0, 0.5), a) ([0.5, 0.5], b)((0.5, 2), c )([2, 2], c )((2, 4), a) ...
... canonical if n is even, for each even i the interval Ii is singular (i.e. of the form [t , t ]), and for no even i such that 0 < i < n is ai−1 = ai = ai+1 . The canonical interval representation for the signal of Fig. 1 is ([0, 0], a)((0, 0.5), a) ([0.5, 0.5], b)((0.5, 2), c )([2, 2], c )((2, 4), a) ...
Simple multiplicative proof nets with units
... conclusion of a ⊥-link) works. Worse, this new jump is by no means natural (if A is B ⊗ C, the new jump can either be B or C), which is quite unpleasant. As far as we know, the only solution consists in declaring that jumps are not part of the proof-net, but rather some control structure. It is then ...
... conclusion of a ⊥-link) works. Worse, this new jump is by no means natural (if A is B ⊗ C, the new jump can either be B or C), which is quite unpleasant. As far as we know, the only solution consists in declaring that jumps are not part of the proof-net, but rather some control structure. It is then ...
Formal Reasoning - Institute for Computing and Information Sciences
... language, specifically speaking, when such a statement of logic would be true. For the atomic propositions, we can think of any number of simple statements, such as ‘2=3,’ or ‘Jolly Jumper is a horse,’ or ‘it rains.’ In classical logic, which is our focus in this course, we simply assume that these ...
... language, specifically speaking, when such a statement of logic would be true. For the atomic propositions, we can think of any number of simple statements, such as ‘2=3,’ or ‘Jolly Jumper is a horse,’ or ‘it rains.’ In classical logic, which is our focus in this course, we simply assume that these ...
P - Bakers Math Class
... Example: Prove that there is no largest prime number. Solution: Assume that there is a largest prime number. Call it pn. Hence, we can list all the primes 2,3,.., pn. Form None of the prime numbers on the list divides r. Therefore, by a theorem in Chapter 4, either r is prime or there is a smaller p ...
... Example: Prove that there is no largest prime number. Solution: Assume that there is a largest prime number. Call it pn. Hence, we can list all the primes 2,3,.., pn. Form None of the prime numbers on the list divides r. Therefore, by a theorem in Chapter 4, either r is prime or there is a smaller p ...
Chapter 1: The Foundations: Logic and Proofs
... Logical Equivalences Example 5: Use De Morgan’s laws to express the negations of “Miguel has a cellphone and he has a laptop computer”. Example 5: Use De Morgan’s laws to express the negations of “Heather will go to the concert or Steve will go to the concert”. ...
... Logical Equivalences Example 5: Use De Morgan’s laws to express the negations of “Miguel has a cellphone and he has a laptop computer”. Example 5: Use De Morgan’s laws to express the negations of “Heather will go to the concert or Steve will go to the concert”. ...
Identity and Harmony revisited ∗ Stephen Read University of St Andrews
... Is identity a logical operator? The rules for identity in a natural deduction setting are usually given in the form of Reflexivity and Congruence (see, e.g., [9] p. 77): a=b p Congr Refl a=a p(b/a) Here, p(b/a) denotes the result of replacing one or more occurrences of the term a in p by b. Refl wou ...
... Is identity a logical operator? The rules for identity in a natural deduction setting are usually given in the form of Reflexivity and Congruence (see, e.g., [9] p. 77): a=b p Congr Refl a=a p(b/a) Here, p(b/a) denotes the result of replacing one or more occurrences of the term a in p by b. Refl wou ...
Bounded Functional Interpretation
... the (intuitionistically acceptable) FAN theorem is interpreted by the b.f.i.: ∀g ≤1 f ∃nA(g, n) → ∃k∀g ≤1 f ∃n ≤ kA(g, n), where A is any formula, provided that we read the relation ≤1 intensionally (more on this below). This is a blatantly false principle in classical mathematics. Intuitionistic ma ...
... the (intuitionistically acceptable) FAN theorem is interpreted by the b.f.i.: ∀g ≤1 f ∃nA(g, n) → ∃k∀g ≤1 f ∃n ≤ kA(g, n), where A is any formula, provided that we read the relation ≤1 intensionally (more on this below). This is a blatantly false principle in classical mathematics. Intuitionistic ma ...
A Crevice on the Crane Beach: Finite-Degree
... • The graph of any nondecreasing unbounded function is exactly one power of two strictly greater than x, using the f : N → N defines a finite-degree predicate, since f −1 (n) monadic predicate true on powers of two. Moreover, we can is a finite set for all n; define formulas trans(i) (x, y), 1 ≤ i ≤ ...
... • The graph of any nondecreasing unbounded function is exactly one power of two strictly greater than x, using the f : N → N defines a finite-degree predicate, since f −1 (n) monadic predicate true on powers of two. Moreover, we can is a finite set for all n; define formulas trans(i) (x, y), 1 ≤ i ≤ ...