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## Propositional Logic

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**Propositional Logic**Propositional Language Translations Truth Tables Propositional Proofs Appendix: Model Theory**Introduction**• Propositional logic studies arguments whose validity depends on “if-then,” “and,” “or,” “not,” and similar notions. • In the following, firstly we will introduce a formal language that sets up the framework of propositional logic, and then explain the relationship between this language and natural languages. • Secondly, we will introduce a simple model of it, i.e. the one made by truth-tables. • Finally, we will introduce inference rules and show how to construct a proof.**1. Propositional Language**• vocabulary • formation rules PL**Vocabulary**• Logical constants: logical connectives • “,” “,” “,” “,” and “.” • Logical variables: propositional variables • P, Q, R…, and so on. (if necessary with subscripts appended—like ”P1”) • Auxiliary Signs: brackets • “ (,”and “).”**The Function of Brackets**• Consider the case in mathematics: • 2+35=? • Either (2+3)5=25 or 2+(35)=17. • Similarly consider the following case: • What does “PR” mean? • It means either that (P)R or that (PR). • The function of brackets is to disambiguate the meaning of wffs (well-formed formula).**Formation Rules**• Any capital letter is a well-formed formula. • The result of prefixing any wff with “” is a wff. • The result of joining any two wffs by “,” “,” “,” or “”and enclosing the result in parentheses is a wff. • Only that which can be generated by the rules (i)-(iii) in a finite number of steps is a wff in PL.**Some examples**• PP • P(RP) • R • ((PR))Q • ((PR)(Q))P • PP • P(RP) • R • ((PR))Q • ((PR)(Q))P**Construction Tree (Top-Down)**• For any wff in PL, we can construct a tree for it. • Ex. PP PP (iii, ) P (i)P (ii, ) P (i)**Construction Tree (Bottom-Up)**• Ex. P(RP) P (i) R (i) P (i) RP (iii, ) P(RP) (iii, )**Main Connectives**• The main connective of a formula is at the top of the tree (Top-down)—at the bottom if bottom-up. • In another words, if the whole sentence was constituted at last by one connective, then we call this one as main connective of this sentence. • Note: the leaf of tree must be atomic.**Some examples**• (PR) • (P(RP)) • (P)(R) • ((PR))Q • ((PR))P • PP • P(RP) • R • ((PR))Q • ((PR)(Q))P**2. Translations**• As mentioned earlier, propositional logic studies arguments whose validity depends on “if-then,” “and,” “or,” “not,” and similar notions. • It follows that PL must contain these notions in order to represent natural languages to certain degree. • In a way, we can translate arguments in natural languages into wffs in PL.**Connectives**• “” stands for “not.” • “” stands for “and.” (or “but,” “though”) • “” stands for “or.” (inclusive-or) • “” stands for “if-then.” • “” stands for “if and only if.”**Some examples**• It will rain tomorrow. • It will not rain tomorrow. • It will rain tomorrow and I will bring my umbrella. • If it rains tomorrow, I will bring my umbrella. • Either it rains tomorrow, or it will not rain. • It will rain heavily if and only if the sky will be covered with dark clouds.**More…**• Not both Alan and Bill like to play baseball. • If Alan took this course and Bill dropped this course, I would take this course. • Only if Cindy took this course, I would take this course. • Not either Alan drops this course or Bill drops this course.**3. Truth Tables**• Logical connectives are represented by some truth functions—by which we can calculate the truth table of each compound wff. • Before we introduce logical connectives, let’s see what a function is and what kind of function is called “truth functional.”**Set**• A set is a collection of entities (or objects). • The principle of extensionality: • A set is defined by the members it contains. • Ex. Suppose our domain is {Alan, Bill, Cindy} • M: {Alan, Bill} (M: {x|x is a man}) • W: {Cindy} (W: {x|x is a woman})**Membership**• Suppose we use “a” for Alan, “b” for Bill, and “c” for Cindy: • a M and b M, but c M. • c W, but a W and b W. • Suppose that B: {x|x is Cindy’s brother}. Alan is Cindy’s brother, and Bill is not. • a B and b B.**Subset**• If A is a subset of B, then if x A, then x B. • A B • If A is a proper subset of B, then if x A then x B and, for some y, such that y B and y A. • A B • Ex. • Given that M: {a, b} and B: {a}, we call B is a proper subset of M—symbolized as B M.**Intersection and Union**• AB : {xxA and xB} • AB : {xxA or xB} • Ex. • {1, 2, 3}{2, 4, 6}={1, 2, 3, 4, 6} • {1, 2, 3}{2, 4, 6}={2}**Function: Many-one Relation between Domain and Range**brotherhood Alan Cindy Bill**Suppose both Alan and Bill are Cindy’s brothers.**Alan Cindy Bill**This is not a function**Alan Cindy Bill**Function**1 1 0 0**Compound Function**1 1 1 0 0 0**1 1**1 0 0 0**1 1**1 0 0 0**1 1**1 0 0 0**1**1 0 0**1 1**1 0 0 0**1 1**1 0 0 0**1 1**1 0 0 0**1 1**1 0 0 0**Truth Function**• Connectives which give rise to sentences whose truth value depends only on the truth values of the connected sentences are said to be truth-functional. • Therefore, “not” (), “or” (), “and” (), “if-then” (), and “if and only if” () are truth-functional.**Some examples**• U F • U T • U F • F U • F U • (T) U**Complex Truth Tables**• (P (Q R)) • ((P Q) R) • ((P Q) Q) • ((P Q) R) • (P (Q R))**Functional Completeness**• A system of connectives can express all truth functions is said to be functionally complete. • Ex. Define “” by “” and “.” • Actually, we can use “” and “” to represent others, and we call these two functions functionally complete.**Can you find another system of connectives which is also**functionally complete?**The Truth-table Test**• Recall how valid and invalid are defined for arguments: • VALID = no possible case has premises all true and conclusion false. • INVALID = some possible case has premises all true and conclusion false. • Now, we can use the truth-table test on a propositional argument.**More example**• It is in your left hand or your right hand. • It is not in your left hand. • It is in your right hand. (L: it is in your left hand; R: it is in your right hand) • L R • L • R