Lecture Two: Chapter One

Philosophy 202

1. Administration

 
1. Has everyone looked at the syllabus?
2. The books are here, at Book People.  You need to pick up a copy ASAP.  
3. Remember the e-mail message. I will reply to all the e-mails I receive.
4. Concerning the Submit program: you need only send me the final grade report that you generate for your homeworks; the intermediate reports are for your own use. If you have questions about how to interpret those reports, please let me know.
5. Problem Set #1: This is due a week from Thursday.   
6. Questions?
   
   
2. Brief Review

 
1. The Methodological Argument

 
1. Logic is the study of Reason
2. We can study Reason by studying language.
3. We can study language by studying artificial language.
4. Therefore, we can study logic by studying artificial language.
2. Language Study

 
1. Our primary focus will be on syntax, or the structure of the language, and semantics, or the meaning of the language.
2. Relationship between syntax (i.e., structure) and semantics (i.e., meaning).

 
1. In making a relevant contribution to a discourse, you must select words appropriate to your goals and put them together in the correct way; the first of these tasks is semantic, the second syntactic.
2. Structure supports semantics, both at the sentence level and at the argument level.

 
1. Sentence Level: Something qualifies as a sentence because of its structure, not its meaning. (E.g., "Colorless green ideas sleep furiously.")
2. Argument Level: Structural relationships between sentences determine what can be said about the distribution of truth to those sentences, where truth is a semantic notion. In particular, logical consequence—a relationship between sentences in virtue of their structure—has important implications for truth.
   
   
3. Chapter One

 
1. Today we will learn the most basic syntactical elements of FOL—names (or "individual constants"), predicates, and atomic sentences.
2. First of all, FOL is not a single language—you can think of it as a language schema, one that sets constraints that can be satisfied in a number of ways (28-31).

 
1. As B&E put it, FOL is really a "family of languages," related by the fact that they all represent realizations of certain general types—individual constants, predicates, etc. In the text, these are called "General First-Order Languages".
2. You will learn about the general characteristics of all these languages—what types of terms they have, acceptable ways of combining those terms, etc.—but until they are made specific, you don't have an actual language.
3. E.g.:

 
1. Logic Class: students, teacher, text, classroom, etc.
2. Set Theory: you fix the predicates (=, element of), and names, and then you can talk about sets of objects (37-38).
3. We begin this study by specifying the types of expressions.

 
1. Terms: symbols that refer to individuals.

 
1. Individual Constants (19-20): these are the names of FOL. These names stand in a certain relation to objects, a relation called "reference". We can say the following things about this relation in FOL:

 
1. It is unanalyzed—we assume the connection as a primitive. It is SEMANTIC, though.
2. A particular name will refer to one and only one object—it must name an object (i.e., no empty names) and it cannot name more than one.
3. An object can have no name or many names.
2. Functions (31-36): A new term type.

 
1. Function symbols combine with a term or terms to create another term.
2. Function symbols have arity, like predicates, but there is a difference: when you combine functions sybols with terms, you get new terms; when you combine predicates with terms, you get sentences.
3. The result of a composition of a function symbol with a number of terms equal to its arity yields a new term. The result is a term and it functions like one when in a sentence. You can compose functional terms inside other function symbols to yield new terms. Examples: Leftfoot(x), Father(x), sum(x,y), etc.
2. Predicates (20-23): symbols that refer to a property or relation.

 
1. Predicates like 'is red', 'is tall', 'saw a great concert over the break' refer to properties that single individuals have.
2. Predicates like 'loves', 'hates', (...) refer to relations between objects. These can be between any number of objects.
3. Interesting metaphysical question re: the relation between properties and relations—Why such a difference between unary predicates and those of higher arity?
4. Arity: a predicates arity is determined by the number of objects it applies to. Predicates that refer to properties are unary; predicates that refer to relations have their arity determined by the number of objects they relate.
5. Predicates always refer to a determinate property or relation.
3. Atomic Sentences (23-27): A sentence is a piece of language formed by the concatenation of a predicate with a number of terms equal to its arity.

 
1. The order of these terms is crucial—e.g., `owes money'.
2. Sentences make claims, which can be evaluated for truth. (Frege had them refer to a truth value.)
4. Differences with Natural Language

 
1. Names (Individual Constants): they refer to one and only one object.
2. Predicates:

 
1. The distinction between properties and relations is not made precise in natural language.
2. Predicates are not vague.
3. In FOL, there will not be any predicates that have multiple arity.
5. Example I: Set Theory (37-38)

 
1. Individual Constants: 'a', 'b', 'c', ...
2. Predicates: '=', 'E'
3. Atomic Sentences: 'a=b', 'a E c', ...
4. Example: (Talk about numbers) Let 'a' stand for 6, and 'b' stand for {3, 5, 6}.
6. Example II: Arithmetic (38-39)

 
1. Individual Constants: ‘0’, ‘1’
2. Predicates: ‘=’, ‘<’
3. Function Symbols: ‘+’, ‘x’
4. Combinations:

 
1. New Terms: ‘0=1’, ‘0x1’, ‘(0+(1+1))’
2. Atomic Sentences: ‘1=1’, ‘0=(1x0)’, etc.
5. Note: An inductive definition is a way of characterizing all things in a very large set that involves giving the ingredients and the recipe for creating the members of the set and ruling everything else out.

4. More on General First-order Languages

 
1. A first-order language is specified by fixing the names, predicates, and function symbols.
2. Necessary Elements: Predicates (at least one), Connectives (Chs. 3-8), and Quantifiers (Chs. 9-14).

 
1. You can get by without function symbols or even individual constants.
2. Quantifiers allow you to make claims about objects insofar as they satisfy certain conditions: no one, everyone, someone, etc. These differ from individual constants and function symbols in that terms of this sort refer to individual things, whereas quantifiers do not.
   
   
5. Problem Set #1 -- Go through it.
   
   
6. Translations

 
1. Michael slept through logic class on Thursday.
2. Paul sent Clay to Jan for money.
3. a is a dodecahedron.
4. b is between a and c.