PRESENTATION OF NATURAL DEDUCTION R. P. NEDERPELT
... Basic units in the system!> we shall call sentences, written in a sequential (not a tree-like) order, one sentence below the other. A sentence can express something like an axiom, a theorem, a definition, an assumption or a derived statement. If desired, one may add comments, e.g. containing justifi ...
... Basic units in the system!> we shall call sentences, written in a sequential (not a tree-like) order, one sentence below the other. A sentence can express something like an axiom, a theorem, a definition, an assumption or a derived statement. If desired, one may add comments, e.g. containing justifi ...
Modus ponens
... While modus ponens is one of the most commonly used concepts in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution". Modus ponens allows one to el ...
... While modus ponens is one of the most commonly used concepts in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution". Modus ponens allows one to el ...
6/FORMAL PROOF OF VALIDITY
... This proves the argument valid using MP and CJ as rules of inference to deduce the conclusion of the argument. The formal proof for this argument is written as 1. O>-M ...
... This proves the argument valid using MP and CJ as rules of inference to deduce the conclusion of the argument. The formal proof for this argument is written as 1. O>-M ...
Notes Predicate Logic II
... We mentioned in “Semantics of Predicate Logic” that equality is usually interpreted to mean identity, which means that in a model a =M b holds if and only if a and b are the same elements of the model’s universe. It is safe to assume t = t for any term t, because both sides of the equation will eval ...
... We mentioned in “Semantics of Predicate Logic” that equality is usually interpreted to mean identity, which means that in a model a =M b holds if and only if a and b are the same elements of the model’s universe. It is safe to assume t = t for any term t, because both sides of the equation will eval ...
Overview of proposition and predicate logic Introduction
... The syntax of a language is concerned with formulating expressions in the language correctly, semantics deals with the meaning of the expressions. Since the formal syntactical definition considers expression as abstract objects, which have no meaning by themselves, semantics can only be given to exp ...
... The syntax of a language is concerned with formulating expressions in the language correctly, semantics deals with the meaning of the expressions. Since the formal syntactical definition considers expression as abstract objects, which have no meaning by themselves, semantics can only be given to exp ...
Methods of Proof for Boolean Logic
... Why truth tables are not sufficient: • Exponential sizes • Inapplicability beyond Boolean connectives ...
... Why truth tables are not sufficient: • Exponential sizes • Inapplicability beyond Boolean connectives ...
Exercise
... P(x) it is not enough to show that P(a) is true for one or some a’s. 2. To show that a statement of the form x P(x) is FALSE, it is enough to show that P(a) is false for one a ...
... P(x) it is not enough to show that P(a) is true for one or some a’s. 2. To show that a statement of the form x P(x) is FALSE, it is enough to show that P(a) is false for one a ...
POSSIBLE WORLDS AND MANY TRUTH VALUES
... constructed, via ¬, ∨, , from formulas τai pj , and again α00 ⇔ α000 is valid on every frame. Finally, let β be obtained from α000 by replacing each τai pj by a new variable qij , let γ be a formula which “says” that, necessarily, for each j exactly one qij holds, and let α∗ be γ ⇒ β. Then α∗ is a ...
... constructed, via ¬, ∨, , from formulas τai pj , and again α00 ⇔ α000 is valid on every frame. Finally, let β be obtained from α000 by replacing each τai pj by a new variable qij , let γ be a formula which “says” that, necessarily, for each j exactly one qij holds, and let α∗ be γ ⇒ β. Then α∗ is a ...
ND for predicate logic ∀-elimination, first attempt Variable capture
... While the proof of Model Existence Lemma is still based on (an updated version of) maximally consistent sets, it is much harder than in the propositional case. ...
... While the proof of Model Existence Lemma is still based on (an updated version of) maximally consistent sets, it is much harder than in the propositional case. ...
Bound and Free Variables Theorems and Proofs
... domain D, an interpretation I, and a valuation V , written (I, D, V ) |= A The definition is by induction: (I, D, V ) |= P (x) if I(P )(V (x)) = true (I, D, V ) |= P (c) if I(P )(I(c))) = true (I, D, V ) |= ∀xA if (I, D, V 0) |= A for all valuations V 0 that agree with V except possibly on x • V 0(y ...
... domain D, an interpretation I, and a valuation V , written (I, D, V ) |= A The definition is by induction: (I, D, V ) |= P (x) if I(P )(V (x)) = true (I, D, V ) |= P (c) if I(P )(I(c))) = true (I, D, V ) |= ∀xA if (I, D, V 0) |= A for all valuations V 0 that agree with V except possibly on x • V 0(y ...
PHILOSOPHY 326 / MATHEMATICS 307 SYMBOLIC LOGIC This
... proofs in a system of natural deduction for the logic of propositions. This course does not have a fixed and predetermined syllabus. There is core content which will be covered in class, and a number of options for additional content. The specific content of the course, the type and frequency of exa ...
... proofs in a system of natural deduction for the logic of propositions. This course does not have a fixed and predetermined syllabus. There is core content which will be covered in class, and a number of options for additional content. The specific content of the course, the type and frequency of exa ...
Lecture Notes on Sequent Calculus
... The proof of contraction actually exposes an imprecision in our presentation of the sequent calculus. When there are two occurrences of a proposition A among the antecedents, we have no way to distinguish which one is being used, either as the principal formula of a left rule or in an initial sequen ...
... The proof of contraction actually exposes an imprecision in our presentation of the sequent calculus. When there are two occurrences of a proposition A among the antecedents, we have no way to distinguish which one is being used, either as the principal formula of a left rule or in an initial sequen ...
Propositional Logic Predicate Logic
... Name of Symbols ∀ (universal quantifier), and ∃ (existential quantifier). Definition. A formula A is valid if A is true no matter how we replace the individual constants in A with concrete individuals and the predicate variables in A with concrete predicates. Note. The set of individuals must be in ...
... Name of Symbols ∀ (universal quantifier), and ∃ (existential quantifier). Definition. A formula A is valid if A is true no matter how we replace the individual constants in A with concrete individuals and the predicate variables in A with concrete predicates. Note. The set of individuals must be in ...