Introduction to Logic
... • Formal logic replaces the ordinary language of argument with a symbolic language. • This language is meant to be free of all ambiguity and vagueness. • The language is meant to wear its logical structure on its face. • Our formal languages: SL and QL. ...
... • Formal logic replaces the ordinary language of argument with a symbolic language. • This language is meant to be free of all ambiguity and vagueness. • The language is meant to wear its logical structure on its face. • Our formal languages: SL and QL. ...
First-Order Loop Formulas for Normal Logic Programs
... graph of P , written GP , is the infinite graph (V, E), where V is the set of atoms that do not mention any constants other than those in P , and for any A, A0 ∈ V , (A, A0 ) ∈ E if there is a rule (1) in P and a substitution θ such that hθ = A and bθ = A0 for some b ∈ Body. A finite non-empty subse ...
... graph of P , written GP , is the infinite graph (V, E), where V is the set of atoms that do not mention any constants other than those in P , and for any A, A0 ∈ V , (A, A0 ) ∈ E if there is a rule (1) in P and a substitution θ such that hθ = A and bθ = A0 for some b ∈ Body. A finite non-empty subse ...
Quantification - Rutgers Philosophy
... substitutions for x on the right of the disjunction sign, and hence we can without any alteration of meaning use different variables on the two sides of the disjunction. So, for example, the formula (∀x)R(x)→(∀y)B(y)) means the same thing as the formula (∀x)R(x)→(∀x)B(x)). We translate these equival ...
... substitutions for x on the right of the disjunction sign, and hence we can without any alteration of meaning use different variables on the two sides of the disjunction. So, for example, the formula (∀x)R(x)→(∀y)B(y)) means the same thing as the formula (∀x)R(x)→(∀x)B(x)). We translate these equival ...
On the Finite Model Property in Order-Sorted Logic
... computing a refutationally-complete upper bound on the size of a single sort (as a function of the user-provided bounds on the other sorts). The conditions defining this fragment are not directly comparable to ours, but in some respects constrain the sentences rather severely. For example existentia ...
... computing a refutationally-complete upper bound on the size of a single sort (as a function of the user-provided bounds on the other sorts). The conditions defining this fragment are not directly comparable to ours, but in some respects constrain the sentences rather severely. For example existentia ...
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... 3. If Γ is not consistent, Γ `⊥. If Γ ⊆ ∆, then ∆ `⊥ as well, so ∆ is not consistent. 4. Since ∆ is consistent, ⊥∈ / Ded(∆). Now, if Ded(∆) `⊥, but by the remark below, ⊥∈ Ded(∆), a contradiction. 5. Suppose ∆ is consistent and A any wff. If neither ∆ ∪ {A} and ∆ ∪ {¬A} are consistent, then ∆, A `⊥ ...
... 3. If Γ is not consistent, Γ `⊥. If Γ ⊆ ∆, then ∆ `⊥ as well, so ∆ is not consistent. 4. Since ∆ is consistent, ⊥∈ / Ded(∆). Now, if Ded(∆) `⊥, but by the remark below, ⊥∈ Ded(∆), a contradiction. 5. Suppose ∆ is consistent and A any wff. If neither ∆ ∪ {A} and ∆ ∪ {¬A} are consistent, then ∆, A `⊥ ...
Definition: A proof is a system of reasoning or argument to convince
... **There are many kinds of direct proofs, but we will only study 4 types. Types of Deductive Arguments that we will study: 1. Law of Deduction 2. Modus ponens 3. Modus tollens 4. Transitivity Definitions: The Law of Deduction is a method of deductive proof with the following symbolic form: p q1, q2, ...
... **There are many kinds of direct proofs, but we will only study 4 types. Types of Deductive Arguments that we will study: 1. Law of Deduction 2. Modus ponens 3. Modus tollens 4. Transitivity Definitions: The Law of Deduction is a method of deductive proof with the following symbolic form: p q1, q2, ...
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... In statement (13) we are including the information that “P” is a statement in symbolic logic which corresponds to the statement in English, “It is raining today”. 1.3 Connectives In this section we introduce rather complicated statements from simple statements using connective words or expressions ( ...
... In statement (13) we are including the information that “P” is a statement in symbolic logic which corresponds to the statement in English, “It is raining today”. 1.3 Connectives In this section we introduce rather complicated statements from simple statements using connective words or expressions ( ...
An Independence Result For Intuitionistic Bounded Arithmetic
... We first briefly describe the first-order theories of bounded arithmetic introduced by Samuel Buss [B1]. The language of these theories extends the usual language of first-order arithmetic by adding function symbols x x2 y (= x2 rounded down to the nearest integer), |x| (=the number of digits in the ...
... We first briefly describe the first-order theories of bounded arithmetic introduced by Samuel Buss [B1]. The language of these theories extends the usual language of first-order arithmetic by adding function symbols x x2 y (= x2 rounded down to the nearest integer), |x| (=the number of digits in the ...
The Expressive Power of Modal Dependence Logic
... Väänänen [17] introduced modal dependence logic MDL. In the context of modal logic a team is just a set of states in a Kripke model. Modal dependence logic extends standard modal logic with team semantics by modal dependence atoms, =(p1 , . . . , pn , q). The intuitive meaning of the formula =(p1 , ...
... Väänänen [17] introduced modal dependence logic MDL. In the context of modal logic a team is just a set of states in a Kripke model. Modal dependence logic extends standard modal logic with team semantics by modal dependence atoms, =(p1 , . . . , pn , q). The intuitive meaning of the formula =(p1 , ...
On not strengthening intuitionistic logic
... Lemma 1: Any weakly C-valid rule which is closed under substitution is C-valid. Proof: Suppose R is closed under substitution and is not C-valid. Then there is bound to be an instance of R, say < Σ , T > , which is not Cvalid, and hence there is bound to be an assignment of truth-values to the propo ...
... Lemma 1: Any weakly C-valid rule which is closed under substitution is C-valid. Proof: Suppose R is closed under substitution and is not C-valid. Then there is bound to be an instance of R, say < Σ , T > , which is not Cvalid, and hence there is bound to be an assignment of truth-values to the propo ...
Logic - Humble ISD
... Get Ready To Be Logical! • 1. Books and notebooks out. • 2. Supplies ready to go. • 3. Answer the following: The sum of 2 positive integers is ___________ True or False ...
... Get Ready To Be Logical! • 1. Books and notebooks out. • 2. Supplies ready to go. • 3. Answer the following: The sum of 2 positive integers is ___________ True or False ...
Plural Quantifiers
... such that none of the C s admires herself or anyone outside of the C s . To get clearer about it, try drawing some models in which it is true, and some in which it is false. Your models can consist of circles (critics), squares (non-critics), and arrows indicating who admires whom. (See for example ...
... such that none of the C s admires herself or anyone outside of the C s . To get clearer about it, try drawing some models in which it is true, and some in which it is false. Your models can consist of circles (critics), squares (non-critics), and arrows indicating who admires whom. (See for example ...
