1 slide/page
... • constant symbols: Alice, Bob • variables: x, y, z, . . . • predicate symbols of each arity: P , Q, R, . . . ◦ A unary predicate symbol takes one argument: P (Alice), Q(z) ◦ A binary predicate symbol takes two arguments: Loves(Bob,Alice), Taller(Alice,Bob). An atomic expression is a predicate symbo ...
... • constant symbols: Alice, Bob • variables: x, y, z, . . . • predicate symbols of each arity: P , Q, R, . . . ◦ A unary predicate symbol takes one argument: P (Alice), Q(z) ◦ A binary predicate symbol takes two arguments: Loves(Bob,Alice), Taller(Alice,Bob). An atomic expression is a predicate symbo ...
Chapter One {Word doc}
... Note: Equivalent to #14 {A student from this class did not watch any Eagles games last season; OR: It is not true that every student watched an Eagles game last ...
... Note: Equivalent to #14 {A student from this class did not watch any Eagles games last season; OR: It is not true that every student watched an Eagles game last ...
x - Stanford University
... A Correct Proof ∀a. ∀b. ∀c. (aR-1b ∧ bR-1c → aR-1c) Theorem: If R is transitive, then R-1 is transitive. Proof: Consider any a, b, and c such that aR-1b and bR-1c. We will prove aR-1c. Since aR-1b and bR-1c, we have that bRa and cRb. Since cRb and bRa, by transitivity we know cRa. Since cRa, we hav ...
... A Correct Proof ∀a. ∀b. ∀c. (aR-1b ∧ bR-1c → aR-1c) Theorem: If R is transitive, then R-1 is transitive. Proof: Consider any a, b, and c such that aR-1b and bR-1c. We will prove aR-1c. Since aR-1b and bR-1c, we have that bRa and cRb. Since cRb and bRa, by transitivity we know cRa. Since cRa, we hav ...
Why mathematicians do not love logic
... chemistry, say, or to economic history–that much is true. It make sense to house logicians of the university in the department of mathematics–sure. But the differencs between logic and the classical algebra-analysis-geometry kind of “core mathematics” are at least as great as the analogies on which ...
... chemistry, say, or to economic history–that much is true. It make sense to house logicians of the university in the department of mathematics–sure. But the differencs between logic and the classical algebra-analysis-geometry kind of “core mathematics” are at least as great as the analogies on which ...
Proof Theory - Andrew.cmu.edu
... Contemporary logic textbooks often present formal calculi for first-order logic with a long list of axioms and a few simple rules, but these are generally not very convenient for modeling deductive arguments or studying their properties. A system which fares better on both counts in given by Gerhard ...
... Contemporary logic textbooks often present formal calculi for first-order logic with a long list of axioms and a few simple rules, but these are generally not very convenient for modeling deductive arguments or studying their properties. A system which fares better on both counts in given by Gerhard ...
“Sometimes” and “Not Never” Revisited
... logic as either a linear time logic in which the semantics of the time structure is linear, or a system of branching time logic based on the semantics corresponding to a branching time structure. The modalities of a temporal logic system usually reflect the semantics regarding the nature of time. Th ...
... logic as either a linear time logic in which the semantics of the time structure is linear, or a system of branching time logic based on the semantics corresponding to a branching time structure. The modalities of a temporal logic system usually reflect the semantics regarding the nature of time. Th ...
Proof and computation rules
... finite, given arities {ni |i ∈ I}. First order formulas, F(L), over L are defined as usual. The variables in a formula (which range over D) are taken from a fixed set Var = {di |i ∈ N}. Negation ¬ψ can be defined to be ψ ⇒ False. The first order formulas of minimal logic, MF(L), are the formulas in F(L) ...
... finite, given arities {ni |i ∈ I}. First order formulas, F(L), over L are defined as usual. The variables in a formula (which range over D) are taken from a fixed set Var = {di |i ∈ N}. Negation ¬ψ can be defined to be ψ ⇒ False. The first order formulas of minimal logic, MF(L), are the formulas in F(L) ...
Strong Completeness and Limited Canonicity for PDL
... i.e. when | ϕ implies that there is a finite ⊆ with | ϕ, hence | → ϕ. This is, for example, the case in propositional and predicate logic, and in many modal logics such as K and S5. Segerberg’s axiomatization of PDL is only weakly complete, since PDL is not compact: we have that {[a ...
... i.e. when | ϕ implies that there is a finite ⊆ with | ϕ, hence | → ϕ. This is, for example, the case in propositional and predicate logic, and in many modal logics such as K and S5. Segerberg’s axiomatization of PDL is only weakly complete, since PDL is not compact: we have that {[a ...
Critical Terminology for Theory of Knowledge
... Cause and effect: A kind of relation in which one entity or event, called the cause, brings about or helps to bring about another entity or event, called the effect. The proper analysis of cause and effect is a matter of vigorous dispute in philosophy. On most analyses, a cause must exist prior to t ...
... Cause and effect: A kind of relation in which one entity or event, called the cause, brings about or helps to bring about another entity or event, called the effect. The proper analysis of cause and effect is a matter of vigorous dispute in philosophy. On most analyses, a cause must exist prior to t ...
PRESENTATION OF NATURAL DEDUCTION R. P. NEDERPELT
... closely related to the usual way of reasoning and proving in mathematics. In the first instance, the system refers mainly to the nonlogical part of mathematics. However, rules oflogic can be expressed and applied in the system. One may choose natural deduction as a basic for logic, in the manner of ...
... closely related to the usual way of reasoning and proving in mathematics. In the first instance, the system refers mainly to the nonlogical part of mathematics. However, rules oflogic can be expressed and applied in the system. One may choose natural deduction as a basic for logic, in the manner of ...
Definability in Boolean bunched logic
... A property P of BBI-models is said to be definable if there exists a formula A such that for all BBI-models M , A is valid in M ⇐⇒ M ∈ P. We’ll consider properties that feature in various models of separation logic. To show a property is definable, just exhibit the defining ...
... A property P of BBI-models is said to be definable if there exists a formula A such that for all BBI-models M , A is valid in M ⇐⇒ M ∈ P. We’ll consider properties that feature in various models of separation logic. To show a property is definable, just exhibit the defining ...
Day04-InductionProofForVideo - Rose
... Note that what we need to prove for all k is P(k) P(k+1), not P(k+1) P(k). Thus it is incorrect to instead start with P(k+1) and show that the induction assumption leads to P(k) , or that it leads to some true statement. You will lose points on homework assignments if you use the "start with P(k+1 ...
... Note that what we need to prove for all k is P(k) P(k+1), not P(k+1) P(k). Thus it is incorrect to instead start with P(k+1) and show that the induction assumption leads to P(k) , or that it leads to some true statement. You will lose points on homework assignments if you use the "start with P(k+1 ...
PDF
... an initial segment of [0, n], so 0 represents the leftmost tape square. Also assume that n exceeds the integers serving as control states or as tape symbols. Let the domain of the structure A be the interval [0, n] of integers. The vocabulary of A consists of two constant symbols 0 and End, four una ...
... an initial segment of [0, n], so 0 represents the leftmost tape square. Also assume that n exceeds the integers serving as control states or as tape symbols. Let the domain of the structure A be the interval [0, n] of integers. The vocabulary of A consists of two constant symbols 0 and End, four una ...
FIRST DEGREE ENTAILMENT, SYMMETRY AND PARADOX
... induction on the complexity of formulas that this then extends to all of the formulas in the language: for any formula A, if Aρ0 then Aρ′ 0 too, and if Aρ1 then Aρ′ 1 too. The evaluations ρ and ρ′ may still differ, because ρ might leave a gap where ρ′ fills in a value, 0 or 1, or where ρ assigned on ...
... induction on the complexity of formulas that this then extends to all of the formulas in the language: for any formula A, if Aρ0 then Aρ′ 0 too, and if Aρ1 then Aρ′ 1 too. The evaluations ρ and ρ′ may still differ, because ρ might leave a gap where ρ′ fills in a value, 0 or 1, or where ρ assigned on ...
x - Koc Lab
... Proof Strategies for proving p → q Choose a method. 1. First try a direct method of proof. 2. If this does not work, try an indirect method (e.g., try to prove the contrapositive). For whichever method you are trying, choose a strategy. 1. First try forward reasoning. Start with the axioms ...
... Proof Strategies for proving p → q Choose a method. 1. First try a direct method of proof. 2. If this does not work, try an indirect method (e.g., try to prove the contrapositive). For whichever method you are trying, choose a strategy. 1. First try forward reasoning. Start with the axioms ...
Properties of Independently Axiomatizable Bimodal Logics
... Let EL denote the lattice of extensions of a modal logic. We have defined an operation − ⊗ − : (EK)2 → EK2 . ⊗ is a -homomorphism in both arguments. There are certain easy properties of this map which are noteworthy. Fixing the second argument we can study the map − ⊗ M : EK → EK2 . This is a -h ...
... Let EL denote the lattice of extensions of a modal logic. We have defined an operation − ⊗ − : (EK)2 → EK2 . ⊗ is a -homomorphism in both arguments. There are certain easy properties of this map which are noteworthy. Fixing the second argument we can study the map − ⊗ M : EK → EK2 . This is a -h ...
pdf
... if all of its finite subsets are. We gave three proofs for that: one using tableau proofs and König’s lemma, one giving a direct construction of a Hintikka set, and one using Lindenbaum’s construction, extending S to a maximally consistent set, which turned out to be a proof set. In first-order log ...
... if all of its finite subsets are. We gave three proofs for that: one using tableau proofs and König’s lemma, one giving a direct construction of a Hintikka set, and one using Lindenbaum’s construction, extending S to a maximally consistent set, which turned out to be a proof set. In first-order log ...
