Predicate Logic I: Syntax



|Predicate Logic: Syntax |

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|Predicate Logic (PL) is an enrichment (or elaboration) of sentential logic, in which the “atoms” of sentential logic (the atomic |

|sentences) are “split” into terms and predicates, and the additional logical device of quantification over terms is introduced. As|

|a result, we can extend our understanding of entailment to capture logical relationships such as those exemplified by the |

|categorical syllogisms of Aristotelian logic, which crucially involve expressions like “all”, “only,” “some,””one,” and “none.” |

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|Here are some examples: |

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|Aristotle is a philosopher. |

|All philosophers are logicians. |

|/Aristotle is a logician. |

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|and: |

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|Lewis Carroll is a Poet |

|No philosopher is a poet. |

|Only philosophers are logicians. |

|/Lewis Carroll is not a logician. |

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|I.  We begin by characterizing the language of predicate logic (PL):  |

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|In addition to the truth functors and other elements of SL, PL includes the following: |

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|1.  Predicates, represented by capital letters:  Predicates are distinguished in part by the number of places they have (or “slots”|

|into which terms may be put).  As necessary, a superscript may also be added to specify how many places that predicate takes (that |

|is, Px represents an x-place predicate).  The sentences of sentential logic may simply be regarded as 0-place predicates. |

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|2.   Terms, represented by lower-case letters:  In turn, these may be divided into individual constants (or names) and individual |

|variables.  Typically, we will reserve the letters x, y, z, etc. for the variables. |

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|3.  The universal and existential Quantifier symbols: ( and (. |

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|II.  The syntax of predicate logic: |

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|1.  Atomic formulas:  An atomic formula in PL will consist of an n-place predicate followed by n terms. |

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|2.  Truth-functional compounds:  …are constructed in exactly the same way as they were for sentential logic. |

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|3.  Quantified formulas: A quantifier in PL is formed by appending a variable to a quantifier symbol. That variable is to be |

|understood as that quantifier’s variable of quantification. If φ is a formula of predicate logic, then another formula can be |

|formed by prefixing φ with a quantifier. |

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|Here some more terminology that applies specifically to quantifier formulas:  |

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|The formula φ to which a quantifier is prefixed is said to be the scope of that quantifier.  |

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|The variable attached to that quantifier specifies the type of variable that that quantifier may bind.  A quantifier will bind any |

|unattached variable of that type falling within its scope, provided that that variable cannot be bound by another variable with a |

|smaller scope. |

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|A vacuous quantifier is one that doesn’t bind any variables.  |

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|A free variable is one that isn’t bound by any quantifier.  |

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|A closed formula (or sentence) is one that contains no free variables.  An open formula does. Note that open formulas in PL do |

|not, strictly speaking, count as sentences. |

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|Finally, an instantiation of some quantified formula is one in which the quantifier has been removed, and all of the variables that|

|it bound have been uniformly replaced with a single specific individual constant (name).  We can represent an instantiation of some|

|quantifier formula, Qσ Ф, as Ф (ν /σ).  |

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|[In the above schema, Q stands for some quantifier symbol, σ for some variable, ν for some name, and Ф for the scope of some |

|quantified formula].  The latter formula, Ф (ν /σ), is |

|simply the scope of the original quantifier formula where all occurrences of the bound variable σ has been substituted with the |

|name ν. |

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|Got it? |

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|This notion of instantiation will figure prominently in our presentation of the tableau rules for PL. It’s to that that we shall |

|now turn…. |

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