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LIS590IML Information Modeling — Class 01b Unit 1: First Order Logic

Basic Logical Concepts:Logic, Argument, Assertion, Validity

and Introduction to Unit 1: First Order Logic

Slides for August 26th lecture

[Last revision: August 26th]

LIS590IM: Information ModelingAllen Renear, renear@uiuc.edu

Graduate School of Library and Information ScienceUniversity of Illinois, Urbana-Champaign

Fall 2008

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Information modeling

• What is information modeling?.

• Oddly (or not?) this question can be best answered at the end of the course. but for now;

• information modeling is the representation of information in precisely defined well-understood formal languages that can be processed by computer systems, or can be easily converted into languages that can be processed by computer systems.

• Why study information modeling? Same comment as above, but for now let me say:

• science, commerce, culture, social justice, political equity -- everything -- depends on making sound decisions about information modeling, therefore … ec.

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Introduction to Unit 1: First Order Logic

• Most of this unit consists of a presentation of First Order Logic (FOL), which is in a sense the foundational information modeling language as well as the principal tool for the presentation, analysis, and comparison of all other information modeling methods and query languages.

• In fact, FOL itself is arguably the science of information modeling. • Teller will present logic to you as “the science of arguments” and not the

“science of modeling”. But to see the connection think of the premises of an argument as modeling the world, and the conclusion as representing a yes/no query against that model

• In addition the use of FOL as a modeling language in it own right, and not just a tool for constructing, analyzing, and exploiting other modeling languages, has recently acquired new importance:

the W3C Semantic Web languages are all based directly and immediately on First Order Logic.

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Unit 1: First Order Logic

• Class 01a: Basic Concepts

• Class 01b: Sentence Logic: Syntax and Semantics

• Class 02: Sentence Logic: Inference (Natural Deduction)

• Class 03: Predicate Logic: Syntax and Semantics

• Class 04: Predicate Logic: Inference (Truth Trees)

• Class 05: Metatheory

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Basic Concepts

• What is logic?

• What is an argument?

• What is an assertion?• including the concept of truth value

• What is [deductive] validity?

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What is formal logic?

• Broadly…• Formal logic is a system for the clear and exact representation of information and for reasoning with

or about that information• Logic as an academic discipline is

(i) the development of logical systems and (ii) an analysis of the nature of these systems

• Narrowly…• The “science of deduction”, … of a particular kind of reasoning• Systems and techniques for identifying valid arguments

I.e. determining whether or not conclusions follow from premises—Logic takes arguments and separates them into two mutually exclusive and jointly exhaustive

piles: valid, invalid.• and generalizations about those logical systems

— Such generalizations about logical systems are at the heart of this course.• how does it do this?

—in part, by taking arguments expressed in natural language and representing (modeling) them in a rigorously defined formal language that has procedures for determining validity.

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What is an argument?

We can explain that by giving examples…

1) All politicians are liars

2) That man is a politician

3) Therefore that man is a liar

What is the stuff arguments are made of ?• Assertions

— The argument consists of three assertions

What is the gross anatomy of an argument?• Premises (one or more) and a conclusion (one)

What is the gross physiology of an argument:

one assertion (the conclusion) is claimed to follow from the others (the premises)

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What are these assertions?What are exactly the things the arguments are made of?

• Sentences?• Utterances (instances of sentence use)• The meanings of sentences? … or of those utterances?• Possible states of affairs? (situations, putative facts, ways the world might be)?• The proper bearers of truth of falsity (things that can be true or false).• The objects of thought (things that can be believed or doubted?)

There are many interesting issues here as Teller hints (p. 4)

Contra Teller: can’t two people who don’t share a language nevertheless make “the same argument”? And if so they will use different sentences to “say the same thing”, or “express the same proposition”. In which case arguments consist not of sentences, but of the propositions expressed by sentences.

Doesn’t this seem right? And doesn’t it let us avoid those problems with “ambiguous sentences” and faux declarative sentences without truth value that Teller has to fuss so much about? So why bother talking about sentences? Linguistics is about sentences; logic is about propositions!

[As Teller notes, those of us (and your instructor is one) who prefer this view of what arguments are and what logic is about also prefer the name “Propositional Logic” to “Sentence Logic”. ]

However we will now leave these abstruse and controversial topics to the philosophers.

In this course we use Teller’s terminology. And to be fair, the sentence approach is:

(i) more widely accepted; (ii) easier for beginners; and(iii) more naturally suited to understanding logic as a information modeling language.]

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Arguments and sentences (from Teller)

• “An argument is a collection of declarative sentences … “• “… declarative sentences …. which are unambiguous … and [each] either

true or false … ”

• “… one of which is called the conclusion and the rest of which are called premises”

• This is a very general definition of argument. It makes any collection of sentences with a distinguished as the conclusion an argument. That works fine of our purposes even though as Teller notes..

• “Ordinarily the premises of an argument are supposed to support, or give us reasons, for believing the conclusion….”

Note: This is a fairly informal definition, we will develop a more rigorous concept of argument later. In this course, we proceed relatively informally at first, and then as we acquire the tools we need we sometimes use those tools to redefine previously defined concepts more formaly and rigorously. It is hard to proceed any other way -- there is a sort of a bootstrapping problem if you try to. In addition, this approach is also better (far better) for ease of learning.

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Validity

If there is a single big idea at the center of logic, this is it.

Here goes…

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Examples of valid and invalid arguments

1) All politicians are liars

2) Renear is a politician

3) Therefore Renear is a liar

1) All politicians are liars

2) Renear is a liar

3) Therefore Renear is a politician

Everybody recognizes that the first has a property, a good property, that the second does not: in the first case the conclusion follows from the premises (or is a valid deduction from the premises), but not in the second case.

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What is [deductive] validity?• Validity has many senses, we are trying to focus on one: “deductive” or “logical”

validity

Teller’s examples…

1) Adam just got an ‘A’ on his logic exam.

2) Anyone who gets a ‘A’ on an exam is happy.

3) Therefore, Adam is happy

1) Adam has smiled a lot today

2) Adam has not been frowning today

3) Adam has said lots of nice things and no unfriendly things today

4) Therefore Adam is happy.

The first case is an example of deductive validity, the second of a “good inductive argument”. If we were to say the second is valid we would not be using “valid” in the sense of “deductively valid”.

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More on empirical science and deductive validity

• Also compare these two arguments about biology each using the word “valid”.• The evolution of the species by differential reproduction, mutation, and

sexual assortment of genetic material is a valid conclusion from the evidence of the fossil record, animal and plant husbandry, and molecular biology.

• That only the fittest survive is a valid conclusion from your definition of "fittest" as those who in fact survive.

• Again, we are interested in only the second sense of validity

• Both inductive and deductive reasoning are important in the sciences.• In fact the scientific method has been called “hypothetico-deductive”,

• But empirical science cannot proceed on deduction alone, inductive methods are also required.

• We are concerned however only with deductive reasoning. • Not only here in the first unit, but throughout the course.

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The emblematic valid Argument

All humans are mortalSocrates is a humanTherefore: Socrates is mortal

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Examples of valid arguments … ?No librarian is an athleteTherefore: No athlete is a librarian

All librarians are athletesTherefore: All athletes are librarians

Some librarian is an athleteTherefore it is not the case that no librarian is an athlete.

Some librarian is an athleteTherefore it is not the case that no athlete is a librarian.

Some librarian is not an athleteTherefore it is not the case that all librarian is an athletes.

All librarians are humanSue Searing is a humanTherefore: Sue Searing is librarian

No human is perfectAll librarians are humanTherefore: No librarian is perfect

No human is perfectAll librarians are perfectTherefore: All librarians are human

Some librarians are friendly.Some librarians are athletic.Therefore: Some athletic people are friendly.

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Examples of valid arguments

1) If John is home then Jill is home;2) John is home3) Therefore Jill is home

1) If John is home then Jill is home; 2) Jill is not home3) Therefore John is not home

1) If John is home then Jill is home; 2) Jill is home3) Therefore John is home

1) If John is home then Jill is home; 2) John is not home3) Therefore Jill is not home

1) John is home or Jill is home; 2) John is not home3) Therefore Jill is home

1) John is home and Jill is home2)Therefore John is home

1) If John is home then Jill is home; 2) If Jill is home then Jane is home; 3) John is home4) Therefore Jane is home

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NB!!!

• A invalid argument may have

1. all true premises, and a true conclusion

2. all true premises, and a false conclusion

3, one or more false premises, and a true conclusion

4. one or more false, and a false conclusion

• An valid argument may have• …?

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Defining deductive validity: getting started

Two possible characterizations…

• The conclusion follows from the premises

• …without fail, if the premises are true, then the conclusion will also be true. (Teller)

These may seem to say the same thing, and perhaps they do. But some people believe the first says something slightly different than the second, and they may be right.

So for now we will focus on the second definition as an account of our intuitive notion of validity.

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Defining deductive validity: variations on a theme

Focusing now just on the first definition…

Teller: …without fail, if the premises are true, then the conclusion will also be true.

Alternatively:• In every possible situation where the premises are true the

conclusion is true also—There is no possible situation in which the premises are true and the

conclusion false.

• It is impossible for the premises to be true and the conclusion false• In every possible world in which the premises are true, the

conclusion is true.

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Deductive validity in general: A provisional definition

It is impossible for the premises to be true and the conclusion false• In every possible situation where the premises are true,

the conclusion is true also.• It is not possible for the premises to be true and the conclusion false.• If the premises are true then the conclusion must be true (*)• There is no possible case (no counterexample),

in which the premises are true and the conclusion false. • In every possible world in which the premises are true,

the conclusion is true.

In other words, I prefer a version that uses the word “impossible” (and its companion terms, possible, necessary, etc.) in place of Teller’s “without fail”.

I also prefer parentheses to make it clear what is being said about what:

It is impossible for (the premises to be true and the conclusion false)

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Again

It is impossible for …

(the premises to be true and the conclusion false)

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Terminology

• Assertions (sentences/propositions) are true or false• not valid or invalid

—we may revise this terminological resolution later.

• Arguments are valid or invalid• not true or false

—never true or false

• An argument that is valid and has true premises is sound• (what is the truth value of its conclusion?)

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Looking forward…

• Kinds of deductive validity• In the next slide set, on Sentence Logic, we will focus on one

kind of deductive validity: truth-functional deductive validity.

• Problems with the concept of validity• At the end of our unit on logic we will return to the notion of

deductive validity and note some problems and puzzles.