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Beau Wright

# Learn Logic with Language Proof and Logic: A Fun and Interactive Course

## What is Language Proof and Logic?

Language Proof and Logic (LPL) is a textbook and software package, intended for use in undergraduate level logic courses. The text covers topics such as the boolean connectives, formal proof techniques, quantifiers, basic set theory, and induction. The last few chapters include material on soundness, completeness, and Godel's incompleteness theorems. The book is appropriate for a wide range of courses, from first logic courses for undergraduates (philosophy, mathematics, and computer science) to a first graduate logic course.

### Why is it important to learn?

Logic is the study of reasoning, or the principles and methods of valid inference. Logic can help us to evaluate arguments, distinguish correct reasoning from incorrect reasoning, and construct valid arguments of our own. Logic can also help us to understand various disciplines and domains that rely on formal systems, such as mathematics, computer science, artificial intelligence, linguistics, philosophy, and more. Learning logic can improve our critical thinking skills, our creativity, and our communication abilities.

### What are the main topics covered?

The main topics covered in LPL are:

• The boolean connectives: These are symbols that are used to form complex sentences from simpler ones. For example, the symbol "&" means "and", the symbol "v" means "or", the symbol "" means "not", etc. The boolean connectives allow us to express various logical relations between sentences, such as conjunctions, disjunctions, negations, conditionals, biconditionals, etc.

• Formal proof techniques: These are methods that are used to show that one sentence necessarily follows from another or from a set of sentences. For example, modus ponens is a proof technique that allows us to infer "Q" from "P" and "P -> Q". Formal proof techniques are based on rules of inference that are valid for any interpretation of the sentences involved.

• Quantifiers: These are symbols that are used to express general statements about individuals or sets of individuals. For example, the symbol "" means "for all", the symbol "" means "there exists", etc. The quantifiers allow us to express various logical relations between predicates (properties or relations) and individuals or sets of individuals.

• Basic set theory: This is a branch of mathematics that studies sets, which are collections of objects. Basic set theory introduces concepts such as sets, subsets, elements, membership, union, intersection, complement, etc. Basic set theory also introduces the notion of functions, which are mappings from one set to another.

• Induction: This is a method of reasoning that allows us to infer general statements from specific cases. For example, if we observe that 1+1=2, 2+2=4, 3+3=6, etc., we can use induction to infer that for any natural number n, n+n=2n. Induction is based on the principle of mathematical induction, which states that if a statement is true for some base case and if it is true for any case given that it is true for the previous case, then it is true for all cases.

## How to use the LPL software package?

LPL contains three logic programs (Boole, Fitch and Tarski's World), and an Internet-based grading service (which is free to students who purchase the package). These programs are designed to help students learn and practice the concepts and techniques of logic.

### Tarski's World

Tarski's World is a program that teaches the basic first-order language and its semantics. The program allows students to create and explore worlds made of blocks of different shapes, sizes, and colors. Students can then write sentences in the first-order language to describe the properties and relations of the blocks in the world. The program can check whether a sentence is true or false in a given world, and whether a sentence logically follows from another sentence or from a set of sentences.

### Fitch

Fitch is a natural deduction proof environment for giving and checking first-order proofs. The program allows students to construct proofs using various proof techniques, such as modus ponens, modus tollens, conjunction introduction and elimination, disjunction introduction and elimination, negation introduction and elimination, conditional proof, reductio ad absurdum, universal introduction and elimination, existential introduction and elimination, etc. The program can check whether a proof is correct or not, and provide feedback and hints.

### Boole

Boole is a program that facilitates the construction and checking of truth tables and related notions (tautology, tautological consequence, etc.). The program allows students to enter sentences in the propositional language and display their truth values under different assignments of truth values to the atomic sentences. The program can also check whether a sentence is a tautology or not, whether a sentence is a tautological consequence of another sentence or of a set of sentences or not, etc.

Submit is a program that allows students to submit exercises done with the above programs to the Grade Grinder, the online grading service. The Grade Grinder can grade exercises automatically and provide feedback and scores. The Grade Grinder can also keep track of students' progress and performance.

## How to write proofs in LPL?

Writing proofs in LPL is a skill that requires practice and understanding of the logic rules and concepts. Here are some tips on how to write proofs in LPL:

### Basic proof techniques

The basic proof techniques are the rules of inference that allow us to derive new sentences from existing ones. For example, modus ponens is a rule that allows us to derive "Q" from "P" and "P -> Q". To use a rule of inference in Fitch, we need to cite the rule name and the line numbers of the sentences involved. For example:

1. P -> Q 2. P 3. Q MP 1 2

The above proof shows that Q follows from P -> Q and P using modus ponens (MP) on lines 1 and 2.

The basic proof techniques include:

• Modus ponens (MP): From P -> Q and P infer Q.

• Modus tollens (MT): From P -> Q and Q infer P.

• Conjunction introduction (I): From P and Q infer P Q.

• Conjunction elimination (E): From P Q infer P or Q.

• Disjunction introduction (I): From P infer P Q or Q P.

• Disjunction elimination (E): From P Q, if P then R, and if Q then R infer R.

• Negation introduction (I): From P -> Q and Q infer P.

Negation elimination ( 71b2f0854b