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Date Created: 09/23/15
L22 Introduction to Symbolic Logic Page 1 of 1 L22 Intro to Symbolic Logic We will be starting this next topic in what may appear to be a very strange waymby studying some of the elements of logic as presented in philosophy courses Certainly no one would debate the fact that logic is of fundamental importance to science in general and physics in particular Indeed as mathematics is the tool of the physicist and logic is the foundation of mathematics one could argue that logic runs throughout the very core of the study of physics In this lecture we will be introduced to some fundamental concepts in the study of logic We will then motivate the introduction of a symbolism to represent logical statements The purpose of the symbols will be to remove ambiguities in logical statements and to allow us more fully to understand the structure of various statements This understanding will then allow us to progress with our study of logic in the next lecture which will introduce us to some of the concepts of Boolean Algebra Lul AAI Lnn n nn Aswan 1n T EIT any dLAA Luann 4 L nitAnna Truth Page 1 of 2 Truth and Validity Logic may be considered as thestudy of argument An argument consists of one or more statements called premises which are given in support of another statement called the conclusion Consider for example the following argument Only physics students take this class You are taking this class Therefore you are a physics student This argument has two statements acting as premises P P1 Only physics students take this class P2 You are taking this class It also has the following statement acting as the conclusion C C You are a physics student A similar but at the same time very different argument can be given as follows Physics students take classes You are taking classes Therefore you are a physics student The premises and conclusion of this argument are as follows Pl Physics students take classes P2 You are taking classes C You are a physics student It should be clear that these two arguments while almost identical are really quite different Indeed although all of the statements in both arguments are true only the reasoning in the rst argument is valid This brings us to an important and fundamental point in logic In logic truth is independent of validity In order to understand this statement we must understand how the concepts of truth and validity are applied This is a very straightforward matter We have already stated that an argument consists of one or more premises and a h n39Nmtmi Editnhvg7070lBCl lTESllg l25lect111 22Tmthb0dv tmthhtml 262008 Truth Page 2 of 2 conclusion These premises and conclusion may each be written as a statement Statements can be either true or false The statement Physics students take classes is true as is the statement You are a physics student At least if you are taking this class and not just reading this for fun However the statement You are a physics professor is most likelyfalse The reasoning used in drawing the conclusion from the premises can either be valid or invalid If the reasoning in an argument is valid then the truth of the premises guarantees the truth of the conclusion You may think of this as the de nition of valid reasoning if you wish On the other hand just because the premise is true and the conclusion is true we may not deduce that the reasoning is valid For example You are a student You are taking classes Therefore the earth is round or equivalently You are a student and you are taking classes therefore the earth is round are both examples in which the premises and conclusion are true but the reasoning is invalid An argument is said to be sound if its premises are true and its reasoning is valid It then follows that if an argument is sound then its conclusion must be true If an argument is not sound then it is said to be unsound h n39Hthn RAH11l1VQ7n7nTECfITR Tl 9l25Tech1re QQTnithbndv mithhtml 767008 Motivation Page 1 of 1 The Motivation for Symbols We are now going to introduce a new symbolic formalism into our study of logic The symbolic nature of this formalism avoids potential ambiguities that can creep into statements made using regular English The use of symbols better suited to the tasks ahead of us will not only make our job easier but can also allow us to develop strategies or applications that we might otherwise not think of As an example of how an adopted symbolism can aid or hinder our progress consider the following two simple problems in arithmetic We wish to add and multiply two integers Please perform the operations using the S lj n1 introduced in the pfgblem statement i XI iv ii XlxIV It is all but impossible for us to perform these simple arithmetic operations using Roman numerals What you will most likely have to do to solve these problems is to translate the problem into the symbolism of Arabic numerals the numerals you are used to using in case you didn t know and perform the operations i11415 iillx444 and then translate the problem and solution statement back into the desired Roman numerals i XI IV XV ii Xl x N XLIV Although there are de nite unambiguous rules for expressing integers using Roman numerals there are no clear rules for performing arithmetic operations using those numerals Indeed there is much evidence that the Romans did not perform arithmetic operations with their numerals resorting to the use of counting boards for such calculations Mathematics did not advance until an alternate symbolism was adopted that made the advance easier Much in the same way we will introduce a symbolism throughout this lecture that will make our advance into the realm of logic more natural and therefore easier It will be very important for you to understand completely the symbols that are introduced and the rules for working with them each step of the way as the formalism is introduced Lul4nn rinl Lxm n nT anhnnoT 10 T 0 pnfnrp oom n uah nnknr xr mnfivofinn 11ml quotHR0an Statement Types Page 1 of 1 Statement Types A simple statement is a statement that does not contain any other statement as a part We will use capital letters as symbols to represent statements For example the simple statement James is cool could be represented simply by the symbol A A James is cool A compound statement is a statement that contains two or more statements as parts or components A component of a compound statement is itself a statement either simple or compound For example James is cool and he is very tall is a compound statement containing the two component simple statements James is cool and He is very tall httnlmtgn ed11nhvs7n390T EChTPSTl Q TDSl nhire QQR mtement Tvnesbnrlv statement 767002 Conjunction Page 1 of 2 Conjunction The conjunction of two statements can be found by placing the word and between them to form a single compound statement Taking the conjunction of James is cool with He is very tall gives us the equivalent compound statement James is cool and he is very tall However the use of the word and for conjunction can be misleading For example the statement Susan and Frenika are teammates is not a conjunction as you cannot readily split it up into two separate statements that carry the same meaning as the statement above Since Susan is a teammate does not make any sense we can perhaps write this in two statements as Susan is a member of a team Frenika is a member of a team However these two statements together do not necessarily imply that they are teammates on the same team as is implied by the original statement On the other hand there are words other than and which can serve to form a conjunction For example the word but in the following compound statement acts the same as and and the resulting compound statement is a conjunction James is cool but he is very tall Use of the word but adds more depth to the conjunction but does not change the fact that both component statements are being made Other words that can possibly substitute for and to form a conjunction are yet still although however etc We can see the potential problems that can arise as a result of the use of regular English when trying to deal with logic statements The English language was not designed for the strict precision needed in a study of logic To avoid the problem of potential ambiguities in statements as the result of the use of English we now introduce a symbol which is de ned to signify conjunction A conjunction will be denoted by the symbol Thus if we let 1 n L 1nnnnrr mtnwml r 1n 1 r3le nah l nnn niwLnrlu nn11nn139n1a 1 rimran Conjunction Page 2 of 2 A James is cool and B He is very tall then the symbolic compound statement AB represents the conjunction James is cool and he is very tall Aul4L J AL r an Ant nan In T QCT nnhn n QII nninnnfinnnanrlxr nnninnr finn hf Truth Tables Page 1 of 2 Truth Tables We have already seen that statements can be either true or false Within the symbolic formalism that we are introducing we will use the number one 1 to represent true and the number zero 0 to representfalse 1 True 0 False We say that I or 0 are the possible truth values of a statement meaning simply that a statement with a truth value of I is true and one with a truth value of 0 is false Since we are dealing in a system in which there are only two possible truth values we say that this is a binary system of logic We say that the operation of taking the conjunction of two statements is a truth functional operation since the resulting compound statement is itself either true or false depending on the truth values of its component statements We will only be interested in truth functional operators in this course The conjunction of two statements is de ned to be true only if both of its components are true otherwise it is false Thus for example if A 1 and B 0 thenAB 0 sinceA B only LE4 131133 1 We can completely specify the conjunction operator by creating a table that shows all of the possible combinations of the truth values for the components of the conjunction along with the truth value of the result of the conjunction operation Such a table is called a truth table since it gives the truth value of every possible situation encountered by the conjunction operator The truth table for the conjunction operator is as follows I Conjunction Truth Table l A If B H AB l I IOHOH 0 l l 1 IlOJI 0 0 1 0 1 lllll We can determine in advance how many rows will be in the truth table for a given logic statement as follows This will not necessarily be an obvious thing in practice so it will 1 u HWLK 1 1nnnnn AntAT 1n 1 OET nanu O Tm1 TnklnnanAn 1quotqu nL1nn 1 omnnno Truth Tables Page 2 of 2 be important for us to be able to compute it in advance The quantities A and B above are called the components of the statement in what lies ahead we will call them the inputs to the logical statement If we let inputs represent the number of inputsin the given statement two in this case A and B then the number of rows in the truth table for that statement will be given by Zinpl ts that is the number 2 raised to the number of inputs to the statement For example in the conjunction statement considered above A B there are two inputs A and B so that inputs 2 The number of rows in a truth table for this statement which speci es the number of possible combinations of component values in the expression is given by 2mm In this case we therefore have that Zinputs224 which is seen to be the number of rows in the truth table above This may seem fairly obvious for this simple case but how many rows should you expect to have in a truth table for a statement containing 4 binary inputs nunsInst L 1 nnnnn Ant an 1n T 4 manna OOT mnq Tainankhan fwnf n hank L mm an EX21 Solution Page 1 of 2 Solution to Example 221 Consider the statement Karla is a woman s name along with the statement Tennessee is the name of a state that is south of Florida a Write the conjunction of these two statements in English The conjunction of two statements is obtained by inserting the word and or its equivalent between the two statements Thus the conjunction of the two statements could be written Karla is a woman s name and Tennessee is the name of a state that is south of Florida An alternate form for the conjunction could be obtained by using another word that serves the same purpose as the word and For example the statement below is also a conjunction of the two given statements Karla is a woman s name however Tennessee is the name of a state that is south of Florida Both of the statements above are claiming that Karla is a woman s name and that Tennessee is the name of a state that is south of Florida Both are examples of a conjunction of these two statements b Make appropriate de nitions and express the conjunction in symbolic form If we let just making up a letter D Karla is a woman s name and E Tennessee is the name of a state that is south of Florida then the conjunction of these two statements is expressed symbollically as DE This symbolic statement is equivalent to either of the conjunctions given in part a c What is the truth value of the conjunction formed above In order to nd the truth value of the conjunction of the two given statements we rst must evaluate the truth values of the two statements The rst statement Karla is a woman s name is certainly true Its truth value is therefore 1 The second statement Tennessee is the name of a state that is south of Florida is false so its truth value is 0 1 H II 1 I I nnnnrr A IT 1n 139 GEIT 4 lT39 1 quot 1 mvn 1 U1A39I ltl nno EX21 Solution Page 2 of 2 D Karla is a woman s narne l E Tennessee is the name of a state that is south of Florida 0 From the truth table for conjunctions it follows that DE 10 0 The truth value of the conjunction is equal to 0 1 I 1 AnnaT LHIY 1n T ET AA4 mnm1n quot 1ITTVquot 1 CA1L39AI nInnno Disjunction Page 1 of 2 Disjunction The disjunction of two statements is formed in English by inserting the word or between them The compound statement My friend has either forgotten our appointment or else he s lost can be rewritten in the equivalent form My friend has forgotten our appointment or my friend is lost which clearly shows that this compound statement is the disjunction of the two simple statements My friend has forgotten our appointment and My friend is lost As with conjunction we introduce a symbol to represent disjunction to avoid any of the ambiguities in the English language A disjunction will be denoted by the symbol Be careful It is tempting to read the sign as and which might lead us to think of this as a conjunction It is best to read the Sign as simply or If we make the de nitions S My friend has forgotten our appointment and U My friend is lost then the disjunction My friend has either forgotten our appointment or else he s lost can be expressed symbolically as SU As with conjunction the disjunction operator is a truthfunctional operator so that the truth value of the disjunction is dictated by the truth values of its component statements AAAAI39Y rr 1n 139 Arquot AAL nnrn1 rLA ILAJW 1r L LLH1 nirlnnnn Disjunction Page 2 of 2 The disjunction of two statements is false if both of its components are false otherwise it is true We can form a truth table for the disjunction operator just as we did for the conjunction operator As with conjunction the disjunction operator requires two components or inputs so inputs 2 The number of possible combinations of values of the components of the disjunction operation or the number of rows in our truth table for the disjunction operator is thus 2inputs224 The truth table for the disjunction operator is given below I Disjunction Truth Table I l A H B H AB l l 0 H 0 H 0 I l 1 ll 0 H 1 J l 0 lil H 1 l l 1 ll 1 H 1 l m Aav A n 1 1 u AiIAAAA Negation Page 1 of 1 Nega on The negation of a statement in English is formed by placing the word not in the original statement It is symbolically represented by placing a bar over the symbol representing the statement For example if we de ne the statement A quotJames is coolquot then the negation of this statement is K quotJames is not coolquot If A is true then the negation of A must be false and vice versa The negation operator is also sometimes called the not operator Since the not operator requires only one input so that inputs I it follows that the number of rows in the truth table for the not operator is 2ixiputs21239 The truth table for the negation operator is given below l Negation Truth Table I l A H A I l 0 ll 1 l l 1 ll 0 l 1 I 1 nnnnrr A IT 1n 139 firT L T A1I IL J L 14Ls1 f Innnn EX22 Solution Page 1 of 2 Solution to Example 222 Let A Rome is the capital of Spain B Paris is the capital of France and C London is the capital of England Evaluate the truth value of each of the following symbolic statements a A Since the statement Rome is the capital of Spain is obviously false the truth value of A must be 0 A 0 b B Paris is the capital of France is a true statement so B l c C London is the capital of England is also true so that C 1 d A B This is a disjunction Since one argument is false and the other is true the disjunction must be true as is speci ed in the truth table for disjunction AB011 In the following parts we will simply work out the answer in symbols without the discussion in words In each case the steps arejusti ea by the truth tablesfor the corresponding operation e A C A O and C 1 Therefore A C 0 l 0 EA c 0i1 001 00 0 g ABACj01i5l 06 011 AAAAI39I i Pr 1n T ET 4L A m1 16 GITTVG 0 14I ll nno EX23 Solution Page 1 of 2 Solution to Example 223 Write out the truth table for the quantity D which is de ned by the equation D A B C where A B and C are input statements and D is the output Hint How many rows must there be in the truth table for D We rst note that there are 3 inputs to the operation of forming the quantity D the inputs or components are A B and C so we should expect there to be 2 inputs 2 3 8 entries or rows in the truth table for D Let s work out the rst two rows in the truth table The entire table will then simply be shown but you should make sure that you can verify all of the entries in the table The rst row will consist of the input values A 0 B 0 and C 0 With these inputs we get the output DABC0OOOOO The second row will consist of the input values A 1 B 0 and C 0 The output D is then DABC1OO101 The other rows follow suit The results are shown in the following complete truth table for the quantity D Note that there are 8 rows in this table Make sure that you can verify all of the entries in this table I Truth Table for D A B C I A B C DABC I 0 II 0 II 0 II 0 I I 1 II 0 II 0 II 1 I I 0 II 1 II 0 II 0 I I 0 II 0 1 II 0 I I 1 II 1 0 II 1 I I 1 II 0 1 II 1 I I 0 II 1 II 1 II 1 I I II II II I h nlmfcn minn hva909nT pntnrpal39 10T9ltTPnh1rp 77Pynmnle 77 391PY7 Q Rnlntinn 767002 EX23 Solution Page 2 of 2 Immllnln 1 ll 1 I 1 nnnnrr L IT 1n T ln39 antna nqm r m in 0quot 1qu 2 Cnlnh39nv nmnno Sample Quiz 22 Page 1 of 1 Sample Quiz 22 1 In symbolic logic the symbolic statement A B represents the of A and B a conjunction b negation 0 addition d multiplication e disjunction 2 In symbolic logic the symbolic statement A B represents the of A and B a conjunction b negation 0 addition d multiplication e disjunction 3 In the binary logic system we will be working with the number 1 represents a a resistor b a capacitor c a high frequency voltage supply 1 true e false 4 One of the purposes of a symbolism for the study of logic is to a confuse the uninitiated b bore the masses c inspire the dolts of society 1 get rid of the potential ambiguities of the English language e be able to apply the normal rules of algebra 5 The special system of algebra used to work with a binary system of logic is called Algebra a Boolean b Binary 0 digital d Copernican e Pythagorean n 5w 3 4 i 1I rumsnaan AAA an 1n T fatT Mm dO knrln nn Lm1 oIonno SQ22Ans Page 1 of 1 Answers to Sample Quiz 22 1L4AILL Jl InmnnnnT nnhnnnT 10 T 7 nnfnrn OOQnQOQnQOAno qn xr Gavanno 11f OAKOhm Homework 22 Page 1 of 2 Homework 22 1 For each of the following arguments state the premises and conclusion state whether each is true or false state whether the reasoning is valid or invalid and state whether the argument is sound or unsound a Cats are mammals and tigers are cats Therefore tigers must be mammals b Cats are mammals and tigers are mammals therefore tigers must be cats c All dogs are black therefore all healthy dogs eat 1 All rectangles have 4 sides and all squares are rectangles so all squares must have 4 sides e All squares have right angles and all squares are parallelograms so all parallelograms must have right angles f All rectangles have 3 sides and all squares are rectangles so all squares must have 3 sides g Electrons are small and electrons are negatively charged so all electrons must reside in atoms 2 Evaluate the truth value of each of the following statements You may have to rewrite some of the statements to put them into a more standard form with simple statements a Cats are mammals and dogs are cats b Cats are mammals and dogs are not cats c Males are on the average taller than females but they are not necessarily smarter than them 1 All dogs are mammals even though all sh are blue or else this entire sentence is a frivolous compound statement 3 For each of the statements in problem 1 above replace the simple statements with letter symbols and then rewrite the statement using symbolic logic Also evaluate the truth value of each of the components of the statement as well as the truth value of the compound statement Egg You must use the following rule for this problem your statement de nitions for the letters you are using must not contain the word not Therefore for example you may not say Let C Dogs are not cats Instead you must say Let C Dogs are cats and then go from there httnmtsueduohv32020LecturesL19 L25Lecture 22HW2bodv hw2html 262008 Homework 22 Page 2 of 2 4 Let D The daytime sky is blue in sunny weather G The moon is always full as viewed from Earth and let R Water always naturally ows uphill Evaluate the truth value of each of the following symbolic logic statements a D R b R G D c 6 R D d m G D R 5 Let the quantity S be de ned by the equation 8 A 5 35 If we treat A D and E as inputs write out the truth table for the output S An awe rs I Mnlmt39nn nAnamnmon39mT nnfnrncT 10T 74H pr hrrp 99mW7hndv hut l1th 767009 HW22 Ans Page 1 of 2 Answers to Homework 22 l a Pl Cats are mammals T P2 Tigers are cats T C Tigers must be mammals T Reasoning valid Argument sound b P1 Cats are mammals T P2 Tigers are mammals T C Tigers must be cats T Reasoning invalid Argument unsound c P All dogs are black F C All healthy dogs eat T Reasoning invalid Argument unsound d P1 All rectangles have 4 sides T P2 All squares are rectangles T C All squares must have 4 sides T Reasoning valid Argument sound e P1 All squares have right angles T P2 All squares are parallelograms T C All parallelograms must have right angles F Reasoning invalid Argument unsound t P1 All rectangles have 3 sides F P2 All squares are rectangles T C All squares must have 3 sides F Reasoning valid Argument unsound g Pl Electrons are small T P2 Electrons are negatively charged T C All electrons must reside in atoms F Reasoning invalid Argument unsound 2 a false b true c true 1 true 3 a A Cats are mammals l B Dogs are cats 0 A B l 0 0 b A Cats are mammals l B Dogs are cats 0 A E 11 1 c A Males on the average are taller than females l B Males are necessarily smarter than females 0 a 3919 1 1 1 d A Dogs are mammals l B All sh are blue 0 C This sentence is a frivolous compound statement l ABC10101l 4DlG0R0 a0 b0 c1d0 5 Truth table for S h nlmfen minnl1vc7070iRnhn RQTJO175T nhn e QQHWQHW Anshndv hw 211R 767002