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Introductory Lecture.docx

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Department
Philosophy
Course
Philosophy 2250
Professor
Lorne Falkenstein
Semester
Fall

Description
Introductory Lecture Kayla Ann Marie This is a course in basic formal logic Formal logic is the study of an argument’s form.  When we reason we argue from a collection of known things: the premise of our argument.  To conclude to another thing that is not so well known. The forms arguments take can be good or bad.  EX: good o I have five toes on my left foot. The left and right sides of animals are generally approximate reflections of one another. I have five toes on my right foot.  EX: bad o I have five toes on my left foot. Stephen Harper is the Prime Minister of Canada. Different arguments can follow a common form or pattern:  All A’s are B’s. All B’s are C’s. All A’s are C’s.  Some A’s are B’s. Some B’s are C’s. Some A’s are C’s.  a, b, and c are A’s. d is like a, b, and b. d is an A  If P then Q. But not-P, Not-q. If P then Q. P, Q.  a is a B. Most B’s are C’s. a is a C. What separates good and bad arguments?  Good argument forms are of two kinds: o Deductive o Inductive  We are concerned with deductively good arguments in this course. o Inductively good argument forms establish the likelihood of the truth of the conclusion.  When an argument form is deductively good, the form itself makes it impossible to have all true premises and a false conclusion. So, when an argument form is deductively good, it is impossible to think of a substitution instance (an argument with that form) that does have either: at least one false premises OR a true conclusion.  EX: o All A’s are B’s. All B’s are C’s. All A’s are C’s. o All lawyers are greedy. All greedy people are untrustworthy. All lawyers are untrustworthy o All cats are felines. All felines are mammals. All cats are mammals.  Method for proving an argument form is bad: a counter example, think up a single argument that shares that form that has all true premises and a false conclusion.  EX: o Some cats weigh over 12 pounds. Some animals weighing over 12 pounds are dogs. Therefore, some cats are dogs. o How do you prove that a deductive argument form is a good one?  First step: logicians construct a formal language that represents just those features of natural language that are relevant to the forms of arguments.  Most basic and general formal elements have to do with the way entire sentences are compounded in arguments using logically relevant “connective” expressions such as “and,” “or,” and “not.”  This course begins with the study of that logic, sentence logic. Why the study of formal deductive logic is important:  Constructing formal language that represents logical features of natural language tells us something about how natural languages work. It also tells us how to construct machines that will compute the right results.  Tells us how to understand how the mind processes information. nd  It was once thought that mathematics could be reduced to 2 order predicate logic, the study of formal logic contributes to understanding the foundations of mathematics.  Formal logic gives us an example of an especially rigorous and precise way of approaching any topic. The way you learn to pove results in formal logic is highly instructive for effectively proving results in any other field.  F.L has been the model for discourse in contemporary analytic philosophy and a basic understanding of it is required to understand papers on a wide range of philosophical topics, from philosophy of language to mathetics. When we reason we make claims that are generally well known to be true, and then draw conclusions from those claims that are less well known to be true.  To make a claim that is generally regarded as trust is to say or write a certain kind of sentences: a “declarative” sentence. Sentences of this sort are the most basic thing that logic is concerned with. Exercise 1.3#1: Declarative Setences When we reason we make claims that are generally well known to be true, and then draw conclusions from those claims that are less well known to be true.  Arguments are therefore used to persuade and to make discoveries.  They are to be distinguished from collections of sentences that merely serve to report a number of facts without attempting to justify any of them.  Also to be distinguished from collections of sentences that make claims that are generally not well known to be true and appeal to those claims to explain why something that is much better known came to be. In this courses, arguments will be put in “standard form.” This means numbering and listing premises first and placing the conclusion last under a bar. Deductive Validity:  An argument form is deductively valid if and only if each substitution instance of that form either has at least one false premise or has a true conclusion.  An arugment form is deductively invalid as long as there is even on substitution instance of that form that has all true premises and a false conclusion.  An argument is deductively valid if and nly if it has a deductively valid form.  An argument is deductively invalid if and only if it does not have a deductively valid form.  Deductively Valid Arguments can have: o True premises and true conclusions. o False Premises and true conclusions. o False premises and false conclusions. o False premises and true conclusion.  Deductively Invalid Arguments can have: o True premises and true conclusions. o True premises and false conclusions. o False premises and true conclusions. o False premises and false conclusions. Deductive Soundness:  An argument is deductively sound if and only if it both has a deductively valid form and it has all true premises.  An argument is not deductively sound if and only if it either has at least one false premise or has deductively invalid form.  Deductively sound arguments must have true conclusions.  This is not necessarily the case for deductively valid arguments Objects of Classical Formal Logic:  Sentences  Sets of sentences  Arguments  Sentence of Classical Logic: a set of sentences of classical logic must have one or the other, but not both, of exactly two truth values, true (T) and false (F). Other sentences do not exist as far as classical logic is concerned.  Set of sentences of classical logic: a set of sentences of classical logic is a list of from none to infinitely many sentences of classical logic placed between braces ({,}). A set with no members , is the empty or null set. A set with exactly one member is a unit set.  Argument: an argument is a collection of two or more sentences, one of which is identified or intended as a conclusion. The Properties and Relations of these Objects:  Logical truth, falsity, and indeterminacy  Logical equivalence  Logical entailment  Deductive Validity and invalidity  Logical truth, falsity, and indeterminacy are properties of sentences. o It is improper to describe arguments or sets as true, false, or indeterminate.  Logical equivalence is a relation between certain pairs of sentences.  Logical consistency and inconsistency are properties of sets of sentences. o It is improper to describe sentences or arguments as consistent or inconsistent.  Logical entailment is a relation that holds between certain sets and certain sentences.  Deductive validity and invalidity are properties of arguments. o It is improper to describe sentences or sets as valid or invalid. These properties and relations can be defined in terms of:  Form (just done for deductive validity)  Non-contradiction (handout definitions)  Possibility (textbook definitions)  Contradiction: a contradiction is either a sentence of the form “P but not-P” or two sentences, one of the form “P,” and the other of the form, “not-P.” o You are caught in a contradiction when you end up saying something that reduces to a contradiction.  E.g., denying that all fish are fish reduces to saying that there something that is a fish but that thing is not a fish. Definitions in terms of contradiction:  A sentence of classical logic is logically true if and only if you would get caught in a contradiction were you to deny it. E.g., “All fish are fish.”  A sentence of classical logic is logically false if and only if you would get caught in a contradiction were you to affirm it. E.g., “Some fish are not fish.”  A sentence of classical logic is logically indeterminate if and only if there is no contradiction in either affirming or denying it. E.g., “There are more than 80 people in this room.”  A pair of sentences of classical logic is logically equivalent if and only if there would be a contradiction in supposing that they have different truth values. E.g., “Some hawks are osprey.” “Some osprey are hawks.”  A pair of sentences of classical logic is not logically equivalent if and only if there would be a contradiction in supposing that they have different truth values. E.g., “All whales are mammals.” “All mammals are whales.”  A set of sentences of classical logic is logically consistent if and only if there is no contradiction in suppos
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