Philosophy 120

Symbolic Logic I

H. Hamner Hill

CSTL-CLA.SEMO.EDU/HILL/PL120

Logic is the science of arguments

• Separate good arguments from bad ones

• Identify the characteristics of good arguments (validity and soundness)

• Produce good arguments of our own

Student Objectives• learn the vocabulary of logic• master methods and principles• explain important concepts in logic• improve communication skills• symbolize arguments using logical notation• test arguments for validity• evaluate reasoning using the tools of logic

Requirements

• 3 in class examinations• 10 routine graded homework assignments• a comprehensive final examination

Cell Phones

• Turn it off. We are in class, your call can wait. Do not text message during class.

• Cell phones and logic do not mix.• Hang up and derive!• Read this column from the New York

Times.

Textbook and Associated Computer Program

• The Power of Logic, 5th edition, available at the Textbook Services.

• SUBJECT TO CHANGE: ALL homework will be completed using the program The Logic Tutor. The Logic Tutor is provided by the textbook publisher and is available online.

Logic is the science of arguments

• All rational inquiry turns on the ideal of a logical consequence, the idea that some claim must necessarily follow from others.

• Arguments are designed to show that one claim logically follows from others.

• Logic allows to determine whether the arguments succeed.

What is an argument?• An argument is not a disagreement or a form

of verbal battle.• An argument is a set of statements, one of

which (the conclusion) is supposed to follow from the others (the premises).

Statement

• A sentence that has a truth value, i.e., a sentence that is either true or false (but never both).

• Statements are true when what they assert about the world is the case.

• Can you think of a sentence that is not a a statement?

Can you think of a sentence that is not a a statement?

• OK, this is the sort of question logicians love to ask, because the question itself is a legitimate answer! The sentence “Can you think of a sentence that is not a statement?” is itself a sentence that is not a statement. Questions are neither true nor false. Commands, exclamations, and exhortations (Let’s . . .) are other sentences that do not express statements.

Types of statements

• Simple--A simple statement asserts exactly one fact about the world

• Compound--A compound statement is one or more simple statements plus logical connectives.

• 5 logical connectives: not, and, or, if-then, if and only if

NOTE: TRUTH is a property of statements. VALIDITY is a

property of arguments

Conclusion• A statement one is urged to accept on

the basis of reasons given.

Premise

• A statement given as a reason for believing some other statement.

Identifying premises and the conclusion

• Correctly identifying the premises and conclusion of an argument are essential if we are to evaluate it.

• English uses many discrete premise and conclusion indicators (review your handout) that serve as guideposts in arguments.

Deductive Validity

• A characteristic of arguments in which the truth of the premises guarantees the truth of the conclusion. It is impossible for both the premises of a valid argument to be true and the conclusion to be false.

• Any argument that is not valid is invalid or non-valid

Validity does NOT guarantee the truth of the conclusion

• It is possible for the conclusion of a valid argument to be false. If this is the case, then at least one premise must be false.

The following argument is VALID:

• All trout are mammalsAll mammals have wingsSO, all trout have wings

• This argument is valid because IF the premises are true THEN the conclusion MUST be true. This holds even though the premises are in fact false.

Soundness

• A characteristic of valid arguments whose premises are in fact true. It is impossible for the conclusion of a sound argument to be false.

• It is irrational to reject the conclusion of an argument one admits to be sound.

Logical Form and Grammatical Form

• Logic is not a matter of grammar. “Following logically’ is not a matter of grammatical placement.

Logic is a matter of form

• Logic is a formal discipline. It is concerned with the formal or structural properties (patterns) and relations in statements and arguments.

Argument Forms

• An argument form is a pattern of argument, the logical structure of an argument. Argument forms are either valid or non-valid. Valid arguments have valid argument forms.

Consistency• Consistency is a property of sets of

statements• A set of statements is consistent if, but only

if, it is possible for all of the statements in the set to be true.

• A set of statements is inconsistent if, but only if, it is impossible for all of the statements in the set to be true.

Consistency and Validity

• We can use the concept of consistency to test an argument for validity.

• How? Suppose I gave you a consistency checking machine (a machine that tests a set of statements for consistency). How could you use that machine to determine whether an argument is valid?

Hamner’s Helpful Home Consistency Checker

• Input Output• (set of statements) (verdict)• Consistent

Not Consistent

Using the Consistency Checker

• Negate the conclusion of the argument and then ask whether the set of statements consisting of the premises and the negation of the conclusion is consistent. If yes, then the argument is NON-VALID. If no, if that set is inconsistent, then the argument is VALID.

Historical Significance

• Indirect Proof (Reductio ad Absurdum)• Euclidean and Non-Euclidean Geometry• Lobachevsky Reimann

Indirect Proof

• Both Lobachevsky and Reimann tried to establish the truth of all 5 of the core postulates of Euclidian geometry using indirect proof. They succeeded in proving 4 out of 5, but efforts to prove the parallel postulate by indirect proof never led to a contradiction.

• In fact, the failure to prove the parallel postulate led to the development of Non-Euclidian geometry.

Logic and Psychology

• Contexts of DISCOVERY and contexts of JUSTIFICATION are different.

• LOGIC is concerned with the context of justification, the business of defending beliefs.

• The "logic" of discovery is a matter for the discipline of psychology.

Justification and Discovery

• Ramanujan and the difference between justification and discovery.

Justification and Discovery• Ramanujan was one of the greatest

mathematicians of the 20th Century. Today’s mathematicians are still trying to prove some of his theorems.

• He insisted that his ideas came to him in dreams, presented by the Goddess Namakaal. Even if this is true, it doesn’t concern the logician.

• Logicians are interested in the justification of the theorems (How they are proved), not how the are discovered.

Arguments are often confused with explanations

• Sometimes the language of arguments is used when one is not arguing for a conclusion but rather trying to explain a phenomenon.

Arguments:

• Answer the question "Why should I believe this?“

• Give reasons for believing that something is the case.

Explanations:

• Answer the question "Why is this the case?“• Give an account of something already

believed to be the case (the facts are not in dispute).

Key Ideas• Definition of “argument”• Validity is a matter of form• Validity does not guarantee the truth of

the conclusion• Consistency as a test for validity• Contexts of discovery and justification• Arguments and explanations