Symbolic and Modern Logic: Bridging Deductive Reasoning

1. Symbolic logic Modern logic

2. To fully understand deductive reasoning we need a general theory of deduction to • Explain the relations between premises and conclusions. • Provide techniques for discriminating between valid and invalid.

3. Early on, we examined classical logic (Aristotelian logic): Square + Syllogisms • We now look at modern logic: Symbolic Logic

4. These two systems have similar aims, but proceed in very different ways. Modern logic uses an artificial symbolic language. Why? Because natural languages—like English—often use vague words, double meaning, metaphors, etc.

5. Traditional logic is not versatile enough to capture the subtleties of language. • A symbolic language overcomes these issues—symbols facilitate our thinking and keep us focused on the logical structure of arguments. For example: The Indo-Arabic numerals we use today is an improved symbolic language. Think about multiplying Roman numerals: CXIII multiplied by IV. 113 x 4 is much easier to multiply, right?

6. Classical logic has a serious limitation: To test arguments, you must translate them into syllogisms. Symbolic logic on the other hand, by using symbols, makes this task much easier. Let’s look into the logical symbols: As usual, we shall use relatively simple arguments such as, The boy has a red hat or he has a white hat. The boy does not have a white hat. Therefore the boy has a red hat. …or If the moon is made of cheese, then Mr. Robinson is an elf. Mr. Robinson is not an elf. Therefore, the moon is not made of cheese.

7. Arguments of this type may contain compound statements. Statements are either simple or compound. Simple Statement: A statement that contains no other statement as a component. Example: “Charlie is tall” is a simple subject-predicate statement.

8. Compound Statement: A statement that contains one or more statements as components. Example: “Charlie is tall and Charlie is intelligent” is a compound statement. Two statement in one: “1. Charlie is tall and 2. Charlie is intelligent.” This particular statement is conjoined by the word “and” but there are different compound statements, each of which requires its own logical symbol.

9. symbols

10. CONJUNCTION (AND) In the example just given, the compound statement is formed by way of conjunction, by using the word “and”. The compound statement, “Charlie is tall and Charlie is intelligent” is a conjunction. In a conjunction the first conjunct is “Charlie is tall” and the second conjunct is “Charlie is intelligent”.

11. BUT BE CAREFUL! One might suppose that “Lincoln and Grant were contemporaries” is a conjunction. Nope! In this case the word “and” simply expresses a relationship between two nouns. Also, if I say, “Marc took off his shoes and got into bed” I intend to express a temporal succession. But if I say, “Marc went to bed and (Marc) took off his shoes” I am expressing a conjunction.

12. The symbol we use to connect statements conjunctively is the symbol “+” Thus the previous conjunction can be written as “Charlie is tall + Charlie is intelligent”.

13. The point of symbolic logic is to translate these sorts of statements into symbols so that we can express arguments more simply and then test their validity by constructing a truth table. Just as a preview: The compound statement above may be a premise to an argument: Charlie is tall and Charlie is intelligent. Therefore, Charlie is intelligent.

14. This argument can be stated symbolically by using a letter to stand for each conjunct like this: T = “Charlie is tall.” • and • I = “Charlie is intelligent.” • Then we can rewrite the argument symbolically as such (the three dots mean “therefore”): • T + I • ∴I

15. Truth Value Since every statement is either true or false, we say that every statement has a truth value. The entire compound statement also has a truth value. The truth value of a compound statement is determined by the truth value of its components: if both its conjuncts are true, the conjunction is true, otherwise it is false.

16. For that reason, a conjunction is said to be a truth-functional compound statement. And its conjuncts are truth-functional compounds. And the symbol “+” is a truth-functional connective.

17. Given any two statements there are only 4 possible sets of truth values they can have. Let’s take our Charlie argument to illustrate: T = “Charlie is tall.” • I = “Charlie is intelligent.” • When T is true and I is true = “T + I” is true. • When T is false and I is true = “T + I” is false. • When T is true and I is false = “T + I” is false. • When T is false and I is false = “T + I” is false. • …in other words, a conjunction is true if and only if both its conjuncts are true.

18. Another way to represent the argument using symbols is to construct a truth table. By truth table it is meant something like this: • P Q P+Q • ________________ • T T T • F T F • T F F • F F F • This table illustrates all possible combinations or sets of truth values of compound statements.

19. NOTE: There are other words that can be used to conjoin two statements into a compound statement, and therefore they can be represented by the “+” symbol. But Yet Also Still Although However Moreover Nevertheless Comma Semicolon

20. NEGATION (NOT) The negation of a statement is typically indicated with the word “not”. Negating a statement also can be done by prefixing a sentence by “it is false that…” • Modern logicians use the symbol minus “-” This is an easy one: If a statement is true, then its negation must be false, and vice versa.

21. Example: Let’s use the letter P to represent the statement “popcorn is good” and -P (Not P) to negate that statement. If it is true that popcorn is good, then it must be false that popcorn is not good, right? And if it is false that popcorn is good, then the negation of the statement is true. This is how it looks like if we visualize it using a table: • P –P • __________ • T F • F T

22. DISJUNCTION (OR) The disjunction of two statements is formed by adding the word “or” between them. Each statement is called disjunct. To symbolize the disjunction we use the lower case letter “v” • A disjunction as a compound is false only in the case that both its disjuncts are false. • P Q P v Q • ____________________ T T T • F T T • T F T • F F F

23. CONDITIONAL STATEMENT (IF…THEN) When two statements are combined by the words “if” and “then”, the resulting compound is a conditional statement. Example: “If I graduate, then I will get a job.” Here, the statement “ The statement that follows the word “if” is called the antecedent. Thus, “I graduate” is the antecedent. And the sentence that follows the word “then” is called the consequent. The sentence “I will get a job” is the consequent.

24. A conditional statement asserts that in any case in which its antecedent is true, its consequent is also true. It does not claim that its antecedent is true, but only that if it is true, then the consequent is also true. NOTE: A conditional statement and its truth value must be determined by the implication. The implication is the relation between the antecedent and the consequent.

25. Consider the following conditional statements, each of which asserts different implications. 1. If all humans are mortal and Socrates is a human, then Socrates is mortal: The consequent here follows by logical necessity. 2. If Marc is a bachelor, then Marc is unmarried: Here the consequent follows by virtue of meaning. 3. If you place a bar of soap into a tub filled with water, then the bar of soap will float on the water: Here the consequent follows inductively and the truth must be empirically determined. 4. If I lose the game, then I will eat my hat: Here the consequent is merely a decision stipulated by the speaker.

26. Any conditional statement is false if its antecedent is true and its consequent false. T ⊃ F = F In any other case the compound statement is true. • The truth functional connective used for the conditional is the horseshoe, “⊃” • P Q P ⊃ Q • _________________ • T T T • F T T • T F F • F F T

27. You noticed that in when P and Q are false, the conditional is true! And when P is false and Q true, the conditional is true! • Let me explain: Say that our conditional statement is the following: • “If you graduate, then I will buy you a car.” (If G then C) Now think about it: If you graduate and I’ll buy you a car, the compound is true. If you do not graduateand I buy you a car, the compound is still true. I did not lie to you. However, I you graduate and I won’t buy you a car, then I did not keep my promise, so the compound is false. Finally, If you do not graduate and I don’t buy you a car, so both conditions do not happen, it is nevertheless the case that if you graduate then I’ll buy you a car.

28. MATERIAL EQUIVALENCE (IF AND ONLY IF) Two statements are said to be materially equivalent when they have the same truth value, that is, when they are both true or both false. The truth-functional connective used for material equivalence is the three-bar sign, “≡” and is pronounced “if and only if” often abbreviated as “iff”. P Q P ≡ Q • _________________ • T T T • F T F • T F F • F F T • As you can see, the compound is true if and only if both statements have the same truth value. Otherwise, the statement is false.

29. Punctuation Just like in normal language, in symbolic logic we need punctuation. Consider the following: “The teacher said John is intelligent.” vs. “The teacher, said John, is intelligent.” • by inserting commas to a sentence we can create a totally different meaning. • Even in math: “2x3+5” vs. “2x (3+5)” • In symbolic logic we use parentheses, brackets, braces, etc.to create different meanings.

30. For example, if I say, • “It is not true that if Max runs, then Bill swims and John fences.” I mean • “If Max runs, then Bill swims and John fences” is false. (The entire conditional statement is false) • But notice that “Bill swims and John fences” is a conjunction. And this conjunction is the consequent of the entire conditional statement. • To translate the statement into symbols, we may underline full statements and circle connectives: • “it is not true that if Max runs then Bill swims and John fences.” First step: Second step: Assign an uppercase letter to each statement. Max runs = M Bill Swims = B John Fences = F

31. Third step: rewrite the sentence using symbols. “it is not true that if Max runs then Bill swims and John fences.” B - M + F ⊃ • -M⊃B+F Fourth step: use parenthesis, brackets, etc. if needed. • -[M ⊃ (B+F)]

32. Max runs = M Bill swims = B John fences = F • “it is not true that if Max runs and Bill swims then John fences, and • it’s not true that Max runs or Bill swims.” • “it is not true that if M and B then F, and it’s not true that M or B.” -M+B ⊃ F + - M v B - (M+B) ⊃ F + - (M v B) - [(M+B) ⊃ F] + - (M v B)

33. If the President vetoes the bill or if Congress balks and the • people protest, then the chances of passing this year’s new tax amendment are • not good.” The President vetoes the bill = V Congress balks = C The people protest = P The chances of passing this year’s new tax amendment are good = G Notice I did not include “not” because it is a negation and I want only the simple sentence.

34. IfV or if C and P, then not G C and P should be grouped together, so (C + P) • If V or if (C and P), then not G • Now, the whole statement is a conditional: • “IfV or if (C and P)” is the antecedent. • “then not G” is the consequent. • Therefore, they should be connected with the horseshoe ⊃

35. If V or if (C and P) ⊃ not G • To further indicate that the antecedent and consequent are separated we use brackets (because we already used parentheses). • [If V or if (C and P)] ⊃ not G • Now we need to get rid of words and convert them into connectives: [V v(C + P)] ⊃ –G

36. It is not true that if I + - A then J I am innocent = I I have an alibi = A I go to jail = J • It’s not true that If I am innocent and do not have an alibi, then I go to jail. It is not true that (if I + - A) then J - (I + - A) ⊃J - [(I + - A) ⊃J]