OLI - 2. Syntax and symbolization
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Basic Symbols of Sentential Logic
All and only the following are _____:
1. The capital letters we use to symbolize atomic sentences, called ***atomic formulae*** or occasionally ***sentential letters***: A, B, C, and so on (possibly with numeric subscripts).
2. The symbols for the logical connectives: &, ∨, →, and ¬.
3. The parentheses used to disambiguate the scope of the connectives: ( and ).
1. The capital letters we use to symbolize atomic sentences, called ***atomic formulae*** or occasionally ***sentential letters***: A, B, C, and so on (possibly with numeric subscripts).
2. The symbols for the logical connectives: &, ∨, →, and ¬.
3. The parentheses used to disambiguate the scope of the connectives: ( and ).
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Expressions of Sentential Logic
Any finite sequence or string of basic symbols.
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Formulae of Sentential Logic
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1. Every atomic formula φ is a formula of sentential logic.
2. If φ is a formula of sentential logic, then so is ¬φ.
3. If φ and ψ are formulae of sentential logic, then so are each of the following:
1. (φ&ψ)
2. (φ∨ψ)
3. (φ→ψ)
4. An expression φ of sentential logic is a formula only if it can be constructed by finitely many applications of the first three rules.
1. Every atomic formula φ is a formula of sentential logic.
2. If φ is a formula of sentential logic, then so is ¬φ.
3. If φ and ψ are formulae of sentential logic, then so are each of the following:
1. (φ&ψ)
2. (φ∨ψ)
3. (φ→ψ)
4. An expression φ of sentential logic is a formula only if it can be constructed by finitely many applications of the first three rules.
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Parse Tree Construction Rules
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Subformula
A formula ψ is a subformula of a formula φ if and only if ψ appears (as a node) in the parse tree of φ.
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Procedure for Reinserting Omitted Parentheses
1. First, insert parentheses around every occurrence of & and its two conjuncts, starting with the rightmost & and ending with the leftmost,
2. Next, insert parentheses in the same fashion for each ∨ and its two disjuncts, from rightmost occurrence first, to the leftmost occurrence last,
3. Finally, insert parentheses for each →, and its antecedent and consequent, from rightmost occurrence first, to the leftmost occurrence last,
As you follow this procedure, keep in mind that parentheses are never inserted around negations, and that a single parenthesis should never be inserted within another set, i.e., do not break up any existing pairs of parentheses with ones you insert.
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Atmoic formulae
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Logical connectives
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Conjunction
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Disjunction
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Conditional
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Negation
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Logical operators
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Idealizations
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Compound
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Truth-functional
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Semantics
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Meaning
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Truth-values
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Compound sentence
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Compound formula
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Symbolize
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Translate
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Conjuncts
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Disjuncts
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Inclusive
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Exclusive
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Conditional
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Antecedent
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Consequent
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Unary
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Binary
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Main connective
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Scope
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Ambiguous
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Basic symbols
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Formula
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Expression
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Well-formed formula
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Variables
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Inductive
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Semantics
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Parse tree
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