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Proposition a broad use in contemporary philosophy the primary bearers of truth-value the objects of belief and other propositional attitudes (i.e., what is believed, doubted) the referents of that-clauses the meanings of sentences Propositions are the sharable objects of the attitudes the primary bearers of truth and falsity Does not certain candidates for propositions, thought- and utterance-tokens, which presumably are not sharable concrete events or facts, which presumably cannot be false. 3
Predicate (grammar) The part of the sentence (or clause) which states something about the subject or the object of the sentence. In "The dog barked very loudly", the subject is "the dog" and the predicate is "barked very loudly". (logic) A term of a statement, where the statement may be true or false depending on whether the thing referred to by the values of the statement's variables has the property signified by that (predicative) term. From Middle French predicate (French prédicat), from post-classical Late Latin praedicatum ( thing said of a subject ), a noun use of the neuter past participle of praedicare ( proclaim ). 4
Predicate and Proposition A nullary predicate is a proposition. Also, an instance of a predicate whose terms are all constant e.g., P(2,3) acts as a proposition. A predicate can be thought of as either a relation (between elements of the domain of discourse) or as a truth-valued function (of said elements). A predicate is either valid, (all interpretations make the predicate true) satisfiable, or (an interpretation makes the predicate true) unsatisfiable. (no interpretations make the predicate true) There are two ways of binding a predicate's variables: one is to assign constant values to those variables, the other is to quantify over those variables (using universal or existential quantifiers). If all of a predicate's variables are bound, the resulting formula is a proposition. 5
Satisfiability and Validity In mathematical logic, satisfiability and validity are elementary concepts of semantics. A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true. some S are P A formula is valid if all interpretations make the formula true. every S is a P A formula is unsatisfiable if none of the interpretations make the formula true. no S are P A formula is invalid if some such interpretation makes the formula false. no S are P a theory is satisfiable if one of the interpretations makes each of the axioms of the theory true. a theory is valid if all of the interpretations make each of the axioms of the theory true. a theory is unsatisfiable if all of the interpretations make each of the axioms of the theory false. a theory is invalid if one of the interpretations makes each of the axioms of the theory false. These four concepts are related to each other in a manner exactly analogous to Aristotle's square of opposition. 6
Reduction of Validity to Unsatisfiability For classical logics, can reexpress the validity of a formula to satisfiability, because of the relationships between the concepts expressed in the square of opposition. In particular φ is valid if and only if φ is unsatisfiable, which is to say it is not true that φ is satisfiable. Put another way, φ is satisfiable if and only if φ is invalid. 7
Square of Opposition A categorical proposition is a simple proposition containing two terms, subject and predicate, in which the predicate is either asserted or denied of the subject. four logical forms. The 'A' proposition, the universal affirmative (universalis affirmativa), 'omne S est P', usually translated as 'every S is a P'. The 'E' proposition, the universal negative (universalis negativa), 'nullum S est P', usually translated as 'no S are P'. The 'I' proposition, the particular affirmative (particularis affirmativa), 'quoddam S est P', usually translated as 'some S are P'. The 'O' proposition, the particular negative (particularis negativa), Latin 'quoddam S non est P', usually translated as 'some S are not P'. 8
Material Conditional (1) a logical connective (or a binary operator) that is often symbolized by a forward arrow " " is used to form statements of the form "p q" (termed a conditional statement) which is read as "if p then q" and conventionally compared to the English construction "If...then...". But unlike as the English construction may, the conditional statement "p q" does not specify a causal relationship between p and q is to be understood to mean "if p is true, then q is also true" such that the statement "p q" is false only when p is true and q is false. antecedent p q consequent The material conditional is also to be distinguished from logical consequence. In classical logic p q is logically equivalent to (p q) and by De Morgan's Law to p q 9
Material Conditional (2) is to be understood to mean "if p is true, then q is also true" such that the statement "p q" is false only when p is true and q is false. the negative compound: not both p and not q. p q is false if and only if both p is true and q is false. p q is true if and only if either p is false or q is true (or both). The compound p q is logically equivalent also to p q (either not p, or q (or both)), and to q p (if not q then not p). But it is not equivalent to p q, which is equivalent to q p. In classical logic p q is logically equivalent to (p q) and by De Morgan's Law to p q p q p q T T T T F F F T T F F T q p p q p q T T T T F F F T T F F T 10
Comparison implication if P then Q first statement implies truth of second inverse if ~P then ~Q negation of both statements converse if Q then P reversal of both statements contrapositive if ~Q then ~P reversal and negation of both statements negation P and ~Q contradicts the implication p q converse q p inverse contrapositive inverse p q q p converse 11
Contraposition In logic, contraposition is a law, which says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of P Q is thus Q P. For instance, the proposition "All bats are mammals" can be restated as the conditional "If something is a bat, then it is a mammal". Now, the law says that statement is identical to the contrapositive "If something is not a mammal, then it is not a bat." Note that if P Q is true and we are given that Q is false, Q, it can logically be concluded that P must be false, P. This is often called the law of contrapositive, or the modus tollens rule of inference. 12
Conversion and Negation The contrapositive can be compared with three other relationships between conditional statements: Inversion (the inverse): P Q "If something is not a bat, then it is not a mammal." Unlike the contrapositive, the inverse's truth value is not at all dependent on whether or not the original proposition was true, as evidenced here. The inverse here is clearly not true. Conversion (the converse): Q P. "If something is a mammal, then it is a bat." The converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition. Negation: (P Q) = P and Q "There exists a bat that is not a mammal. " If the negation is true, the original proposition (and by extension the contrapositive) is untrue. Here, of course, the negation is untrue. ~(~PVQ) = P and ~Q 13
Contradiction (falsum): represents an arbitrary contradiction Upside-down T (tee): denotes an arbitrary tautology (turnstile): "yields", "proves" T ϕ a proposition is a contradiction iff In the classical logic, especially for a contradictory proposition ϕ propositional and it is true that first for all ( ) order logic ϕ ψ ψ ψ "from falsity, whatever you like" one may prove any proposition from a set of axioms which contains contradictions. principle of explosion (ex falso quodlibet) ϕ a proposition ϕ is a contradiction iff it is unsatisfiable In complete logic 14
Soundness An argument is sound if and only if The argument is valid. All of its premises are true. For instance, All men are mortal. Socrates is a man. Therefore, Socrates is mortal. The argument is valid (because the conclusion is true based on the premises, that is, that the conclusion follows the premises) and since the premises are in fact true, the argument is sound. The following argument is valid but not sound: All organisms with wings can fly. Penguins have wings. Therefore, penguins can fly. Since the first premise is actually false, the argument, though valid, is not sound. 15
Formal System A formal system is broadly defined as any well-defined system of abstract thought based on the model of mathematics. In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. a tautology (from the Greek word ταυτολογία) is a formula which is true in every possible interpretation. 16
Tautology a tautology (from the Greek word ταυτολογία) is a formula which is true in every possible interpretation. 17
Complete Logic In logic, semantic completeness is the converse of soundness for formal systems. A formal system is "semantically complete" when all its tautologies are theorems A formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas: formulas that are true under every interpretation of the language of the system that is consistent with the rules of the system). A formal system is consistent if for all formulas φ of the system, the formulas φ and φ (the negation of φ) are not both theorems of the system (that is, they cannot be both proved with the rules of the system). 18
Completeness and Soundness 19
Well Formed Formula 20
References [1] en.wikipedia.org [2] en.wiktionary.org [3] U. Endriss, Lecture Notes : Introduction to Prolog Programming [4] http://www.learnprolognow.org/ Learn Prolog Now! [5] http://www.csupomona.edu/~jrfisher/www/prolog_tutorial [6] www.cse.unsw.edu.au/~billw/cs9414/notes/prolog/intro.html 21
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