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45 Cards in this Set
- Front
- Back
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Logically Valid Argument- (Intuitive Logical Notions)
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An argument is logically valid just in case it is not logically possible for the premises to be true and the conclusion false.
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Logically Sound Argument- (Intuitive Logical Notions)
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An argument is logically sound just in case it is logically valid and all its premises are true
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Logical Consistency- (Intuitive Logical Notions)
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A set of sentences is logically consistent just in case it is logically possible for the sentences in that set to jointly be true
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Logical Truth- (Intuitive Logical Notions)
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A sentence is logically true just in case it is not logically possible for the sentence to be false.
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Logical Equivalence- (Intuitive Logical Notions)
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Two sentences are logically equivalent just in case it is not logically possible for the sentences to differ in their truth values.
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Truth-Value Assignment- (Truth-Functional)
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A truth value assignment is a function that maps each sentence letter to a truth value {either true or false)
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Satisfiable Set of Schemas- (Truth-Functional)
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A set of schemas is satisfiable just in case there is an interpretation under which the schemas are jointly true.
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Disjunctive Normal Form- (Truth-Functional)
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A disjunction of conjunctions of sentence letters and negations of sentence letters
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Conjunctive Normal Form- (Truth-Functional)
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A conjunction of disjunctions of sentence letters and negations of sentence letters
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Expressive Adequacy- (Truth-Functional)
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A set of connectives is expressively adequate just in case every truth-functional compound can be expressed using the connectives in the set
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Substitution- (Truth-Functional)
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A schema is obtained by another by substitution if it is the result of uniformly substituting some schema letters with some schema
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Contraposition- (Truth Functional)
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General property that, if A ⊃ B holds then –B ⊃ -A also holds.
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Theorem- (Proof Theory)
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A schema is a theorem just in case there is a deduction with a final line containing the schema and no premise numbers.
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Effective- (Proof Theory)
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A system of deduction is effective if there is a mechanical procedure for checking that it’s a derivation
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System of Natural Deduction- (Proof Theory)
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A deduction is a finite set of pairs, each of which contains a schema and a finite set of schemas.
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Interderivability- (Proof Theory)
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A pair of schemas are interderivable just in case each schema can be derived from the other.
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Inconsistency- (Proof Theory)
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A set of schemas is inconsistent if you can derive a contradiction from it.
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Decidability- (Proof Theory)
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There is a mechanical procedure that is guaranteed to tell you the right answer in a finite amount of time
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Soundness- (Proof Theory)
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If A |- B then A |= B. That is, if B is derivable from A then A implies B.
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Completeness- (Proof Theory)
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If A |= B then A |- B. That is, if A implies B then B is derivable from A.
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Monadic Predicate- (Quantificational)
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Stands for a property that can be held by something. (✪ is male)
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Extension- (Quantificational)
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The class of all things that satisfy a predicate
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Coextensive- (Quantificational)
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Predicates that are coextensive have the very same extension
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Extensional- (Quantificational)
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A sentence is extensional just in case its truth value depends only on the extension of the predicates and the things named by its names
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Atomic Sentences- (Quantificational)
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Sentences are considered atomic if they have no connectives or qualifiers
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Binding- (Quantificational)
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Sentences are “closed” if each of the variables has a quantifier and “open” if not
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Scope- (Quantificational)
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How many variables are under a certain quantifier
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Universe of Discourse- (Quantificational)
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The class of relevant objects; must be non-empty.
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Validity- (Quantificational)
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There is no interpretation (includes UD, translation of predicate letters, constants, and variables) in which it is false. That is, its negation is unsatisfiable.
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Satisfiability- (Quantificational)
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There is an interpretation under which all schemas are true. That is, given an interpretation, an open schema is satisfied by an object in the UD just in case the schema is true in the interpretation where x is mapped to the object.
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Unsatisfiability- (Quantificational)
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There is no interpretation under which all schemas are true
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Equivalence- (Quantificational)
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Two schemas are equivalent if they have the same truth value under all interpretations. That is, the negation of one is not satisfiable with the other.
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Inequivalence- (Quantificational)
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Two schemas are inequivalent if there exists an interpretation on which their truth values are different.
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Implication- (Quantificational)
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Under any interpretation on which the set is true the sentence is also true. That is, the set and the negation of the sentence are not satisfiable.
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Abstract Interpretation- (Quantificational)
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Avoids having to use real-world empirical knowledge, extensions are just integers
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Simple Schema- (Quantificational)
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Entire schema is in the scope of one quantifier
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Interpretation- (Quantificational)
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Includes Universe of Discourse, translation of monadic predicate, constants, and free variables mapped to objects
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Law of Interchange- (Quantificational)
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-(∀x)(Fx) and (∃x)-(Fx) are interchangeable
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Polyadic Predicates- (Quantificational)
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Predicates with multiple variables that rely on the order of the variables represented
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Nested Quantifiers- (Quantificational)
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Schemas with more than one quantifier
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Reflexivity- (Quantificational)
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Everything is related to itself, represented (∀x)(Rxx)
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Everything is R related- (Quantificational)
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Any object is related to any object, represented (∀x)( ∀y)(Rxy)
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Symmetry- (Quantificational)
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All R relations are mutual, represented, (∀x)(∀y)(Rxy⊃Ryx)
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Transitivity- (Quantificational)
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If object 1 is related to object 2 and object 2 is related to object 3, object 1 is related to object 3, represented (∀x)( ∀y)( ∀z)(Rxy•Ryz ⊃ Rxz)
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Seriality- (Quantificational)
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Every object is related to something, represented (∀x)(∃y)(Rxy)
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