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# Logic Unit 2.pdf

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School
University of Toronto St. George
Department
Philosophy
Course
PHL245H1
Professor
Niko Scharer
Semester
Winter

Description
UNIT 2 SENTENTIAL LOGIC: SYMBOLIZATION 2.1 WHAT IS SENTENTIAL LOGIC? Sentential Logic (SL): A branch of logic in which sentences or propositions are used as the basic units. It is also called Propositional Logic or Propositional Calculus. We will use a symbolic language that will let us move from English sentences to symbolic sentences and back again. Each truth-valuable English sentence (statements that can be true or false, rather than questions, orders, exclamations, etc.) will be assigned a symbol and then we can use symbols for logical operators to combine those sentences together. Sentential logic allows us to focus on the logical relations between sentences. By symbolizing truth- functional sentences of natural languages (English, French, Mandarin…), we can focus on the logical structures without being distracted by what the sentences mean. Of course, the disadvantage of this is that it ignores the logical structures within a sentence. Some of those logical relations within sentences will be addressed in the second part of the course (Predicate Logic). The logical connectives that join the simple sentences are ‗truth-functional‘ – they operate on the truth- values of sentences rather than their meanings. For that to work, the atomic sentences need to be simple and (for classical logic) bivalent. The logical operators work on the simple sentences in a systematic way allowing us to calculate the truth-value of complex sentences from the truth-values of simple sentences. Then, we can use the techniques of sentential logic to determine whether sets of complex sentences are consistent and whether arguments are valid or invalid, etc. The truth-functional nature of the logical operators of sentential logic makes it relatively easy to interpret the logical operations electronically or mechanically. ‗Logic gates‘ (AND, OR, NOT…) control the flow of information (truth-values) and are used in logical circuits, calculators and computers. They work by taking the truth-values of one or two sentences as input and outputting truth- values according to the logical function of the ‗gate‘. By assembling and arranging such logic gates, so that the output of one gate is the input for another, one can build more and more complex computing devices. Indeed, you can build simple computing machines out of wooden ®evers and ba®ls, dominoes, Popsicle sticks, or out of Lego or Meccano . Logic gates made out of Lego . Babbage difference engine made of Meccano ® Constructed by Tim Robinson Logic Unit 2: Symbolization 1 ©2011 Niko Scharer 2.2 THE SYMBOLS FOR SL In sentential logic we need three types of symbols 1. Symbols for sentences or propositions: capital letters. 2. Symbols for the logical relationships between those sentences: ~     3. Symbols to keep things clear and organized: brackets and parentheses. It is just a matter of convention which symbols get used for sentences and logical operators – and there are many different conventions. But, the different conventions share many features, and once you get used to one set of conventions, it is easy to understand symbolic sentences or arguments symbolized according to different conventions. Sentence Letters: Capital letters P through Z (with or without numerical subscripts) P, Q, R … Z or we can use them with numerical subscripts so that we have an infinite supply: P 1 Q1, …1 P ,2Q ,2R ,2… Each letter symbolizes a complete sentence or proposition. These are ‗atomic sentences‘, the basic building blocks of the symbolic language we are using. ‗P‘ could symbolize ‗Plato is a Greek philosopher.‘, ‗Frank is the fastest runner in town.‘ or ‗Toronto is in Ontario.‘ or any other sentence. Likewise, ‗Q‘ could symbolize ‗Anteaters eat termites.‘ or ‗Suzy loves Adam.‘ or ‗Plato is a Greek philosopher.‘ or any other sentence. Most symbolic languages use letters – some use small letters, others use capitals. We will use the capital letters in the latter part of the alphabet for atomic sentences. Sentential Connectives or Logical Operators:  Conditional sign (if ... then)  Negation sign (not)  Conjunction sign (and)  Disjunction sign (or)  Biconditional (if and only if) These operate syntactically on sentences creating compound symbolic sentences (well-formed formulas). Most of them are binary operators, used to combine two sentences together. the negation sign is unary operator, operating on a single sentence. Logic Unit 2: Symbolization 2 ©2011 Niko Scharer In other textbooks or in philosophical articles you might see other symbols. These are other symbols that might be used for the logical operators we are using. Negation ~ ~ ¬ Conditional     Disjunction   Conjunction   &  Biconditional     Parentheses or Brackets (round brackets/parentheses) [square brackets] These keep things clear in complex symbolic sentences by indicating the order of operation for the logical operators. Whatever is inside a set of brackets or parentheses must be done first. This is just like in math: 3  (2 + 4) is equal to 3  6. Connecting Sentences in Ordinary Language We connect sentences or clauses together informally with words such as ‘and’, ‘or’ and ‘not’. For instance, consider the following two sentences: 1.1 It’s raining. 1.2 It’s windy. We use logical operators to connect them, to form compound sentences: 2.1 It’s raining and it’s windy. 2.2 It’s raining or it’s windy 2.3 It’s not windy. The meaning of these new sentences depends on the meaning of the two simple sentences 1.1 and 1.2 and on the logical operation performed by the word connecting them. The difference between 2.1 and 2.2 is due just to the difference in the logical operators. The logical connective (and, or) determines the inferential relations between sentences. For instance, from 2.1 we can infer either 1.1 or 1.2. (If we are told that it is raining and it’s windy, we can infer that it is raining.) From 2.2 we cannot! (If we are told that it is raining or it is windy, we cannot conclude that it is raining and we cannot conclude that it is windy.) However, from 2.2 and 2.3 together, we can infer 1.1 Logic Unit 2: Symbolization 3 ©2011 Niko Scharer The Building Blocks … Atomic Sentences These are sentences that have truth-value (they can be true or false). Such sentences must be statements or propositions, as opposed to questions, commands, exclamations, promises, etc. We can symbolize them with capital letters (P-Z). We use an ‗abbreviation scheme‘ or ‗symbolization scheme‘ to show what letter stands for what English sentence. P: Plato was a Classical Greek philosopher.. S: Socrates was a Classical Greek philosopher. T: Thales was a Classical Greek philosopher. Any letter can be used to stand for any atomic sentence. When making an abbreviation scheme, you can use any letter that hasn‘t been already assigned, but it‘s more convenient to start at P and put them in alphabetical order, or to use a letter related to the sentence if you can. Although we will rarely do it in this course, you can also use numeric subscripts: P2or T 3. Since subscripts are available, you can have an abbreviation scheme for an unlimited number of sentences! Remember, each letter symbolizes the whole sentence. Each sentence letter is an atomic sentence. The Mortar … Sentential Connectives or Logical Operators We can use logical operators to build new, well-formed sentences out of simpler ones: (P  S) Plato was a Classical Greek philosopher and Socrates was a Classical Greek philosopher. ~ T It is not the case that Thales was a Classical Greek philosopher. (S  ~T) If Socrates was a Classical Greek Philosopher then Thales was not a Classical Greek philosopher. ((S  P)  ~T) If Socrates and Plato were Classical Greek Philosophers then Thales was not a Classical Greek philosopher. ―and‖ ―not‖ and ―if ... then‖ are operating on atomic sentences, connecting them together in increasingly complex ways. Official Notation In official notation, a set of parentheses is used every time a binary connective is used to connect two sentences. In informal notation, the outmost set of these parentheses are often left off, and sometimes inner parentheses, provided it does not lead to ambiguity or lack of clarity. Sentential Connectives are logical operators that can be used to join or operate on atomic sentences to form well-formed formulas. Compound sentences are the sentences formed when you use sentential connectives to operate on one or more atomic sentences (single sentence letters without any logical operators). Official notation is the formal, grammatically correct way of symbolizing a sentence. Logic Unit 2: Symbolization 4 ©2011 Niko Scharer 2.3 NEGATION ~ P ~P This is the characteristic truth table for negation. It defines the use of the negation sign: ~ T F The truth table shows the truth-value of the complex sentence given the F T truth value of the atomic sentences. Here it shows that when the truth- value of P is T (true) then the truth-value of ~P is F (false); and when the truth-value of P is F (false) then the truth-value of ~P is T (true). Note: the ‗squiggle‘ ~ is called a ‗tilde‘. Negation is a unary connective  it acts on a single sentence. Putting ‗~‘ in front of a sentence negates that sentence. Thus, the negation of ‗ P ‘ is ‗ ~P ‘ – it is not the case that P. The negation of any sentence is the sentence with a negation sign in front of it. The negation of ‗ ~P ‘ is ‗ ~ ~P ‘. ~ ~ P is logically equivalent to P. Consider the following scheme of abbreviation: P: Mental states are identical to brain states. This abbreviation scheme shows that the atomic sentence, ―Mental states are identical to brain states.‖ is symbolized with the sentence letter ‗P‘. Now we can symbolize ―Mental states are not identical to brain states.‖ using the negation sign. First, paraphrase the sentence, ―Mental states are not identical to brain states.‖ using ―it is not the case that‖ for the negation. It‘s not the case that mental states are identical to brain states. Second, replace the sentence with the sentence letter: It‘s not the case that P Third, replace the logical operator with the negation sign: ~P Logic Unit 2: Symbolization 5 ©2011 Niko Scharer Stylistic Variances: Many different English sentences express the same idea. W: Mary will win the election. All of the following sentences can be symbolized ~W Mary won‘t win the election. The election won‘t be won by Mary. Mary will fail to win the election. Mary won‘t be elected. Mary will lose the election.* *Note that some words can sometimes be negated with a different word altogether: win/lose pass/fail succeed/fail stop/continue You need to be careful using such words when negating sentences since in many contexts they aren‘t true negations. For example, if you don‘t win a game you don‘t necessarily lose (you may tie or draw). But, if you don‘t win an election, you lose it. Any properly formed symbolic sentence can be negated by putting a negation sign at the front. Thus, we can negate the same sentence twice: Mary won‘t lose the election. We can paraphrase that: It‘s not the case that Mary will lose the election. Above we symbolized ‗Mary will lose the election‘ with ~ W. It‘s not the case that ~ W. Now we can use the negation sign to replace ‗it‘s not the case that‘. ~ ~ W ~~W is logically equivalent to W (they have exactly the same truth-table). Logic Unit 2: Symbolization 6 ©2011 Niko Scharer 2.4 CONDITIONAL  This is the characteristic truth table for the conditional (if ... P Q P  Q then). (Also called the material conditional) It defines the use of the conditional sign:  T T T T F F The truth table shows the truth-value of the complex sentence given the truth value of the atomic sentences. For instance, F T T when both P and Q are assigned the truth-value T, then the F F T sentence P  Q also has the truth-value T (true). You can see that there is only one truth-value assignment on which the sentence P  Q is false: line 2, when P has the truth-value T (true) and Q has the truth-value F (false). P is the antecedent of the conditional (it comes before ) and Q is the consequent of the conditional (it comes after ). antecedent  consequent Unlike negation, which is a unary connective, the conditional is a binary connective, joining two sentences together. Consider the following scheme of abbreviation: T: Tom is taking logic at U of T. S: Tom is a student. We can symbolize the complex sentence, ―If Tom is taking logic at U of T then he‘s a student.‖ using the conditional symbol. If Tom is taking logic at U of T then Tom is a student. We replace the atomic sentences according to the scheme of abbreviation: If T then S Then, we replace the logical operator with the conditional sign and put it in parentheses: (T  S) The material conditional is similar to the conditional we use in everyday speech; however, in everyday speech the antecedent and the consequent are usually semantically related (their meanings are related) so that the conditional makes sense (as it does in this example.) But, in logic, any two sentences can be connected with the material conditional – they don‘t have to have anything to do with one another. The resulting material conditional statement may sound odd, but it will be a well-formed sentence. For example: If Tom is a student then Harper will win the next election. Like any material conditional sentence, it is a true statement unless the antecedent is true and the consequent is false. Logic Unit 2: Symbolization 7 ©2011 Niko Scharer Stylistic variances for “if … then …” There are many stylistic variances for ―if ... then‖ in English. When using the conditional, think about which clause is the antecedent and which is the consequent. Think about when the sentence is false. A material conditional is false only when the antecedent is true and the consequent is false. So if you know what will make it false, you can figure out which clause is the antecedent. Sarah will quit her job if she wins the lottery. This is false only if Sarah wins the lottery but she doesn‘t quit her job. Thus, ―Sarah wins the lottery‖ is the antecedent and ―Sarah quits her job‖ is the consequent. W  Q. Only The word ‗only‘ often causes some confusion! Consider the difference between: You will pass the course if you complete the problem sets. You will pass the course ONLY if you complete the problem sets. In the first case: You will pass the course (P) if you complete the problem sets (S). If your professor reassures you of this, you know that if you just complete the problem sets, you will get that credit. S  P In the second case: You will pass the course (P) ONLY if you complete the problem sets (S). If your professor warns you of this, you know that if you end up passing the course, then you will have completed the problem sets. You won‘t pass if you don‘t complete them, but completing them won‘t guarantee a pass. P  S The expression ‗only if‘ introduces the consequent not the antecedent! The word ‗only‘ seems to reverse the antecedent and the consequent in conditional sentences. You will pass only provided you study. P  S Only on the assumption that you study will you pass. P  S You pass only when you study. P  S Logic Unit 2: Symbolization 8 ©2011 Niko Scharer Consider the following stylistic variances for the conditional. This list is by no means complete! See if you think of more expressions or create new variants of these by using commas or changing the order of the clauses. P: I am alert Q: I have had coffee. If P then Q P  Q If I am alert then I have had coffee. P only if Q P  Q I‘m alert only if I have had coffee. Whenever P, Q P  Q Whenever I‘m alert, I‘ve had coffee. Q if P P  Q I have had coffee if I am alert. P provided that Q Q  P I am alert provided that I have had coffee. P on the condition that Q. Q  P I am alert on the condition that I have had coffee. P only on the condition that Q. I am alert only on the condition that I‘ve had coffee. P  Q Q is necessary for P P  Q Having coffee is necessary for my being alert. Q is sufficient for P Q  P Having coffee is sufficient for my being alert. Note, in the table above, there are no parentheses around the conditional sentences. In Official Notation, a pair of parentheses is used for every binary connective. In Informal Notation the parentheses can be left off provided there is no ambiguity. "How did people ever do philosophy before coffee?" 1 © 1996 Gerald Grow 1This illustration is the property of Gerald Grow, Professor of Journalism, Florida A&M University. http://www.longleaf.net/ggrow/CartoonPhil.html Logic Unit 2: Symbolization 9 ©2011 Niko Scharer 2.4 E1: UNDERSTANDING THE MATERIAL CONDITIONAL… A LITTLE LOGIC PUZZLE Every card has a number on one side and a letter on the other. Suppose there is a rule: If one side of a card has an odd number on it, then the other side has a vowel on it. Which of the following cards must you turn over in order to test whether the rule was broken? 11 20 B E Now consider this rule: A person can drink alcohol only if he/she is 19 years or older. Each of the following people is drinking a beverage. Which of the following people must you learn more about in order to determine whether the rule was broken? Carol: Darren: Adam: Betty: drinking drinking age 23 age 17 soda beer Both the rules above are conditionals. What is the antecedent? What is the consequent? If the rule is broken, then that conditional is false. Remember, a material conditional is false only if the antecedent is true and the consequent is false! Logic Unit 2: Symbolization 10 ©2011 Niko Scharer 2.5 SYMBOLIZATION WITH NEGATION AND CONDITIONAL Parts of the Sentence: An atomic sentence is a sentence letter: P-Z. A molecular sentence (or compound sentence) is one that is composed of an atomic sentence or sentences that have been made into more complex sentences through operations of the logical connectives: ~ and  A Syntax for Sentential Logic with Negation and Conditional: Now we can provide a syntax or grammar for our symbolic language. The Greek letters  (phi) and  (psi) can represent any sentence, whether atomic or molecular. Only symbolic sentences formed through the following steps are well-formed formulas in SL: 1. Sentence letters (P-Z with or without numerical subscripts) are symbolic sentences. 2. If  is a sentence then ~  is a symbolic sentence. 3. If  and  are sentences then (  ) is a symbolic sentence. 4. Anything that can be constructed through recursive application of steps 1-3 is a symbolic sentence. Every sentence we form through steps 1-3 is a well-formed symbolic sentence. Since steps 2 and 3 work on any symbolic sentence, we can use steps 1-3 recursively to make even more complex sentences. Thus, these steps provide a technique for generating all possible truth-valuable sentences in SL. Official notation: brackets or parentheses must be used whenever two atomic sentences are joined with a binary connective, such as . Informal notation: brackets or parentheses must be used to avoid ambiguity (it is customary to leave the outermost brackets off). When a negation sign, ~, is directly in front of a sentence letter, then it operates on that sentence letter. ~ P When a negation sign, ~, is directly in front of a leftmost parenthesis then it is operating on the molecular sentence inside the parentheses: ~ (P  Q) In informal notation, the outermost parentheses or brackets are left off. Thus ((P  Q)  ~R) can be written (P  Q)  ~R Logic Unit 2: Symbolization 11 ©2011 Niko Scharer We can symbolize complex sentences using negation and conditional together. When symbolizing English sentences, there are three basic steps: Step 1: Paraphrase the sentence using standard expressions: ―it is not the case that‖ for negation; ―if … then …‖ for the conditional; and the atomic sentences. Underline the standard logical expressions. Use parentheses to keep the order of operations clear. Step 2: Replace the atomic sentences with their sentence letters. Step 3: Replace the logical terms with their symbols. P: I am alert Q: I have had coffee. Example: If I am not alert then I have not had coffee. Step 1: Paraphrase using standard expressions If it is not the case that I am alert then it is not the case that I have had coffee. Step 2: Replace the atomic sentences with their sentence letters. If it is not the case that P then it is not the case that Q. Step 3: Replace the logical terms with their symbols. If ~ P then ~ Q ~P  ~Q Example: Having coffee is not sufficient for my being alert. Step 1: Paraphrase using standard expressions. Use parentheses to keep the order of operations clear. It is not the case that (having coffee is sufficient for my being alert). It is not the case that (if I have had coffee then I am alert). Step 2: Replace the atomic sentences with their sentence letters. It is not the case that (if Q then P). Step 3: Replace the logical terms with their symbols. ~ (Q  P) * NOTE: The parentheses here show that the negation acts on the sentence: Q  P. If there were no parentheses, ~Q  P , then it would mean: If I haven’t had coffee then I am alert. Logic Unit 2: Symbolization 12 ©2011 Niko Scharer Now, let’s expand the abbreviation scheme and try some more: P: I am alert S: I have had a good night’s sleep. Q: I have had coffee. T: I am in a hurry. R: I bump into things. Example: Assuming I haven’t had a good night’s sleep, I bump into things if I haven’t had coffee. Step 1: Paraphrase using standard expressions and parentheses. If it is not the case that I have had a good night’s sleep then (if it is not the case that I have had coffee then I bump into things.) Step 2: Replace the atomic sentences with their sentence letters. If it is not the case that S then (if it is not the case that Q then R.) Step 3: Replace the logical terms with their symbols. ~ S  (~ Q  R.) P: I am alert S: I have had a good night’s sleep. Q: I have had coffee. T: I am in a hurry. R: I bump into things. 2.5 E1 Using the abbreviation scheme above, symbolize the following sentences: (a) I don’t bump into things. (b) It’s not the case that I don’t bump into things. (c) If I’m alert then I’ve had coffee. (d) If I haven’t had coffee, I’m not alert. (e) I bump into things if I haven’t had a good night’s sleep. (f) It’s not the case that whenever I’m in a hurry I bump into things. (g) Assuming I’m in a hurry, I bump into things if I am not alert. (h) Coffee is sufficient for my being alert only if I have had a good night’s sleep. (i) If I haven’t had a good night’s sleep then coffee is necessary for my being alert. (j) Assuming I’m not in a hurry, only if I haven’t had coffee do I bump into things. (k) It is not the case that whenever I bump into things I haven’t had coffee. (l) It’s not the case that if I’m alert I don’t bump into things if I have had coffee. (m) Provided that I have had a good night’s sleep, it’s not the case that only if I have had coffee am I alert. (n) If it is necessary that I have coffee in order to be alert then having a good night’s sleep is sufficient for my not bumping into things. NOW TRY LOGIC 2010 SYMBOLIZATION EXERCISES FOR CHAPTER 1. Logic Unit 2: Symbolization 13 ©2011 Niko Scharer 2.6: DO WE NEED MORE LOGICAL OPERATORS? We could symbolize everything we want to symbolize in SL just using negation and conditional. Indeed, any possible finite truth-value assignment can be expressed using only these sentential connectives. However, we will see that things get a little complex (and less like natural language) if we don‘t introduce some new symbols. Disjunction - Or Suppose I want to symbolize: Simon will study or he will fail the course. S: Simon will study. T: Simon will fail the course. The sentence is logically equivalent to (means the same as): If Simon won‘t study then Simon will fail the course. If it is not the case that Simon will study then Simon will fail the course. If it is not the case that S then T. ~S  T It‘s easier to use a symbol for „or‟:  Simon will study or he will fail the course. S  T Conjunction – And Suppose I want to symbolize: Simon will study and he will fail the course. This sentence is logically equivalent to: It is not the case that if Simon studies then he won‘t fail the course. It is not the case that (if Simon will study then it is not the case that Simon will fail the course.) It is not the case that (if S then it is not the case that T). ~ ( S  ~ T) It‘s easier to use a symbol for „and‟ :  Simon will study and Simon will fail the course. S  T Logic Unit 2: Symbolization 14 ©2011 Niko Scharer Biconditional – If and only if How would we symbolize this? Simon will fail if and only if he doesn‘t study. This is the same as a conjunction of two conditionals: Simon will fail if he doesn‘t study and Simon will fail only if he doesn‘t study. If it is not the case that Simon will study then Simon will fail and if Simon will fail then it is not the case that Simon will study. We know how to express conditionals and conjunction with just ~ and . Conditional (if  then ):    Conjunction ( and ): ~ (  ~ ) The two conditionals would be: (~ S T) and (T  ~ S) So the entire sentence, expressed with just negation and conditional would be: ~ [ (~ S T)  ~ (T  ~ S)] It‘s much easier to use a symbol for „if and only if‟:  Simon will fail if and only if he doesn‘t study. T  ~S All the sentences that we will be able to symbolize in our expanded language for SL can be symbolized just with negation and conditional (and you might want to try this just for fun). However, by adding in a few more logical connectives, we can make the symbolization a little easier! In the meantime, it would be useful to learn more about the three new logical connectives: conjunction, disjunction and biconditional. Logic Unit 2: Symbolization 15 ©2011 Niko Scharer 2.7 CONJUNCTION  This is the characteristic truth table for conjunction – ‗and‘. P Q P  Q It defines the use of the conjunction sign:  T T T The truth table shows the truth-value of the complex sentence T F F given the truth value of the atomic sentences. For instance, when both P and Q are assigned the truth-value T, then the sentence P  F T F Q also has the truth-value T (true). You can see that on every F F F other truth-value assignment the sentence is false. A conjunction is true if and only if both conjuncts are true. If you join two atomic sentences with a conjunction, then the resulting molecular sentence is true if and only if both conjuncts are true. Plato was a Greek philosopher and Socrates was a Greek philosopher. (P  S) ―P‖ and ―S‖ are the conjuncts, and are both atomic sentences. ―(P  S)‖ is the resulting molecular sentence. Plato and Socrates were Greek philosophers. can be paraphrased ... Plato was a Greek philosopher and Socrates was a Greek philosopher. and can be symbolized ... P  S Stylistic variances The following can all be symbolized using the conjunction sign: Socrates was a Greek philosopher and so was Plato. (S  P) Plato was a Greek philosopher but Turing was a mathematician. (P  T) Plato was a Greek philosopher as well as Socrates. (P  S) Although Turing was a mathematician, Socrates was a Greek philosophe(T  S) Plato and Socrates were Greek philosophers; however, Turing was a mathematician. (P  S  T) Other conjunctions include although, yet, also, in addition to, moreover, even though … Logic Unit 2: Symbolization 16 ©2011 Niko Scharer 2.8 DISJUNCTION  This is the characteristic truth table for disjunction – ‗or‘. P Q P  Q It defines the use of the conjunction sign:  T T T The truth table shows that the truth-value of the complex sentence T F T given the truth-value of the atomic sentences. For instance, when both P and Q are assigned the truth-value T, then the sentence P  Q F T T also has the truth-value T (true). You can see that the only truth- F F F value assignment on which the sentence is false is when both atomic sentences are false. A disjunction is true if and only if at least one disjunct is true. If you join two atomic sentences with a disjunction, then the resulting molecular sentence is : true if and only if at least one of the disjuncts is. Russell was a mathematician or Turing was a mathematician. ―R‖ and ―T‖ are the disjuncts and both are atomic sentences. ―(R  T)‖ is the resulting molecular sentence. Disjunction is equivalent to ‗or‘ used in the inclusive sense. The inclusive sense of ‗or‘ includes either disjunct or both – the molecular disjunctive sentence is true if and only if at least one of the disjuncts is. People use the inclusive sense of ‗or‘ when they ask if you would like cream or sugar in your coffee. You can have cream or sugar You can have cream or you can have sugar. R  S You can have cream, sugar or both. In contrast, when we invite a person to come for dinner on Monday or Tuesday, we are using ‗or‘ in the exclusive sense  we only mean to invite them for one meal, and mean to exclude the possibility they will come both nights! Stylistic variances: use  to symbolize: or, unless, either/or, else, otherwise … Tom will fail unless he studies. T  S (T: Tom will fail. S: Tom studies) Tom will study else he will fail: S  T Sara or Tom or Uri or Vanna will pass the courseS  T  U  V (S: Sara will pass the course. T: Tom will pass. U: Uri will pass. V: Vanna will pass.) The order in which the conjuncts are listed doesn‘t matter. (S  T) is logically equivalent to (T  S) Logic Unit 2: Symbolization 17 ©2011 Niko Scharer 2.9 BICONDITIONAL  This is the characteristic truth table for the biconditional (also called P Q P  Q the material biconditional or equivalence) – if and only if. T T T It defines the use of the biconditional sign:  T F F The truth table shows that the truth-value of the complex sentence F T F given the truth-value of the atomic sentences. Here, when both P and Q are assigned the same truth-value (as on the first and last line), F F T then the sentence P  Q has the truth-value T (true). Whenever P and Q are assigned different truth-values (as on the second and third line) the sentence is false. A biconditional is true if and only if the two sides have the same truth-value. It is called a biconditional because it is equivalent to the conjunction of two conditionals: (P  Q)  (Q  P) P  Q: P only if Q Q  P: P if Q P  Q: P if and only if Q It also means: either both P and Q or neither P nor Q. Which can be symbolized: (P  Q)  (~P  ~Q) Tom passes if and only if he studies. T  S ―If and only if‖ is often abbreviated: IFF. Thus, ―P iff Q‖ is symbolized: P  Q. One might also symbolize the following sentences with the biconditional: For Tom to pass the course it is necessary and sufficient that he study. Either Tom studies and passes or he does neither. Tom passes when and only when he studies. Tom passes exactly on condition that he studies. Tom passes just in case he studies. (Although logicians talk this way, ‗just in case‘ is rarely used this way in ordinary language.) Stylistic variances: sentences symbolized with the biconditional sign may include expressions such as: is equivalent to, is necessary and sufficient for, exactly on the condition that, exactly when... The biconditional is often used for definitions. A closed figure is a triangle if and only if it has three sides. Logic Unit 2: Symbolization 18 ©2011 Niko Scharer 2.9 E1 Symbolize each of the following sentences using the abbreviation scheme provided: P: I exist. S: I think U: Determinism is true. Q: God exists. T: The bible is the word of God. V: I am free. R: Angels exist. (a) Determinism is true but I am free. (b) Either determinism is true or I am free. (c) Determinism is false, however I am not free. (d) Angels exist if, but only if, God does. (e) I exist if I think. (f) I am free just in the case that determinism is false. (g) God exists if and only if the bible is the word of God, but it is not. (h) Either the bible is the word of God or God doesn’t exist. (i) Although God exists, determinism is true. (j) God’s existence is a necessary and sufficient condition for the existence of angels. (k) Determinism is true or I am free, but not both. (l) Both angels and God exist; however, the bible is not the word of God. (m) The bible is the word of God, who exists. (n) If determinism is true, then neither am I free nor does God exist. (o) Neither angels nor God exists. (p) I am not free unless determinism is false. (q) Provided both that I think and if I think then I exist, I exist. (r) I am not free unless God exists, and in that case, only if determinism is false am I free. NOW YOU CAN TRY LOGIC 2010 SYMBOLIZATION EXERCISES FOR CHAPTER 2 … but more tips about complex symbolization in the next section. Logic Unit 2: Symbolization 19 ©2011 Niko Scharer 2.10 SYMBOLIZING COMPLEX SENTENCES We use connectives to join atomic sentences (sentences expressed with a single letter). But we can also use them to join molecular sentences (sentences that already conta
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