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# SYMBOLIC LOGIC II PHI 520

UK

GPA 3.56

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This 8 page Class Notes was uploaded by Emerald Altenwerth on Friday October 23, 2015. The Class Notes belongs to PHI 520 at University of Kentucky taught by Brandon Look in Fall. Since its upload, it has received 22 views. For similar materials see /class/228287/phi-520-university-of-kentucky in PHIL-Philosophy at University of Kentucky.

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Date Created: 10/23/15

Beyond Standard Predicate Logic Identity What we have done so far has gone a long way toward allowing us to symbolize expressions of natural language but we aren39t done yet We need another logical symbol which will express identity in order to take care of many more expressions Brandon c Look Symbolic Logic ii Lecture 9 Consider Tarski39s world We want to say Only a is a cube We can39tjust write Ca because thatjust says a is a cube Nor can we write CaA N HCX for that is a contradiction What we need to write is something that gets at the only something along the lines of a is a cube and there exists nothing else which is a cube and which is not identical to 3 Thus CaA N 3XCXA N X 3 Or alternatively Ca AVXCX a X 3 Brandon c Look Symbolic Logic ii Lecture 9 Grammar and Semantics for the Identity Sign We will simply add the following clause to our defintion of a wff gt If 04 and B are terms then 04 B is a wff And to our definition of truth in a model we will add the following clause gt Vffg04 1 lffi 04ng BLlg So for example Hesperus Phosphorus iff Hesperus and Phosphorus refer to the same object Brandon c Look Symbohc Loglc ll Lecture 9 A translation guide Only a is F The only F that is G is a No F except a is C All F except a are C There is at most one F There are at least two Fs There are exactly two Fs Brandon c Look Symbohc Logic TL Lecture 9 Fa AVXFX a X a Fa A Ga AVXFX A Gx ax a Fa A Ga AVXFX A Gx ax a FaA N Ga AVXFX AX 7 a a GX VXVyFXA Fy ax y 3X3yFX A Fy AX 7 y 3X3yFXFyX7 yVZFZgt zXVzy Beyond Standard Predicate Logic Function Symbols A singular term such as Socrates Fido and Lexington refer to individual objects in a world But there is another class of terms that refer to individual objects but which are not among the class of names from PL like a7 b7 c functions For example Socrates39 wife refers to Xanthippe And we will add a function symbol to our vocabulary to allow for such references Thus fa is the function that maps a to some referent and where f is the function for is the wife of and s is Socrates we can write fs and know that the referent will be Xanthippe Moreover we can iterate these functions if f is the father function then ffa is a39s paternal grandfather and fma could be a39s maternal grandfather for example Brandon c Look Symbolic Logic ll Lecture 9 Grammar and Semantics for Functions We will then add to our vocabulary the following gt for each n gt 0 n place function symbols g with or without subscripts And to our definition of terms gt if f is an n place function symbol and a1an are terms then fa1an is a term Brandon c Look Symbohc Loglc ll Lecture 9 And to the definition of a model gt If f is an n place function symbol then f is an n place total function defined on 9 And to the definition of denotation gt avg jf041g OWLLg if 04 is a complex term fa1 04 Brandon c Look Symbolic Logic ii Lecture 9 Beyond Standard Predicate Logic Definite Descriptions What about expressions that deal with definite descriptions The so and so As Sider shows one response is to introduce a new symbol 7 to stand for the 7XBX A CX Thus the black cat becomes Brandon c Look Symbohc Logic ii Lecture 9

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