Welcome, dear students, to Class 11 Logic and Philosophy Chapter 9 Question Answer | Symbolic Logic | English Medium | ASSEB. This chapter introduces the modern face of formal logic, the system that grew out of the work of mathematicians and philosophers in the nineteenth and twentieth centuries and that today underpins not only philosophy but also computer science, mathematics, linguistics and the design of digital circuits. Symbolic logic replaces the long verbal argument of traditional logic with a compact and exact symbolism. By the end of this chapter you will be able to define symbolic logic, list its main characteristics, distinguish it from traditional Aristotelian logic, identify the basic logical constants and connectives, write the truth table of every standard truth-function, classify a propositional formula as a tautology, a contradiction or a contingency, translate ordinary English sentences into symbolic notation, and use truth tables to test the validity of common argument forms such as Modus Ponens, Modus Tollens, the Hypothetical Syllogism, the Disjunctive Syllogism and the Constructive and Destructive Dilemmas.
This article has been prepared strictly in accordance with the ASSEB (Assam State School Education Board) syllabus and the SCERT prescribed textbook for Higher Secondary First Year (Class 11) Logic and Philosophy. It contains a detailed summary, a complete textbook question-answer set with worked truth tables, additional important questions for examination practice, a glossary of technical terms, a connectives reference table and a master truth-table reference for the five basic truth functions.
Summary of Chapter 9: Symbolic Logic
Symbolic logic, also called mathematical logic or modern logic, is that branch of logic which uses special symbols, instead of ordinary words, to bring out the features of logical importance in arguments and to classify them into types to which general rules can be applied so that the validity of the argument can be tested mechanically. It is the logical descendant of the syllogistic logic of Aristotle, but it goes far beyond what Aristotle’s term-logic could analyse, because it can deal with relations and with arguments whose validity depends on the way whole propositions are joined together rather than on the internal structure of categorical propositions.
The roots of symbolic logic reach back to Gottfried Wilhelm Leibniz (1646-1716), who dreamed of a characteristica universalis or universal symbolic language and a calculus ratiocinator in which all reasoning could be reduced to calculation. The dream was first made into a working system by the English mathematician George Boole (1815-1864) in his books The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1854). Boole showed that the logic of classes and propositions could be treated algebraically; the binary algebra he created is still called Boolean algebra and it is the algebra used in computer circuits. After Boole the German philosopher and mathematician Gottlob Frege (1848-1925) in his Begriffsschrift (1879) created the first complete system of modern predicate logic with quantifiers. The work was carried to completion by Bertrand Russell (1872-1970) and Alfred North Whitehead (1861-1947) in their three-volume masterpiece Principia Mathematica (1910-1913), which sought to derive the whole of mathematics from a small number of purely logical axioms. Other major contributors include Augustus De Morgan, C. S. Peirce, Ernst Schroder, Giuseppe Peano, Ludwig Wittgenstein, David Hilbert and the Polish school of Lukasiewicz and Tarski.
Three features distinguish symbolic logic. First, the use of ideograms. An ideogram is a sign that stands directly for a concept rather than for the sound of a word; the symbols of mathematics, such as the plus sign and the equals sign, are ideograms. The symbols of symbolic logic, such as the dot for conjunction and the horseshoe for implication, are likewise ideograms. Second, the use of variables. A variable is a symbol that can stand for any one of a given range of values. In propositional logic the lower-case letters p, q, r are propositional variables that can stand for any proposition whatever. In predicate logic x, y, z are individual variables and F, G, H are predicate variables. Third, the application of the deductive method. By the deductive method we can derive an indefinite number of new statements from a small number of basic statements (axioms) by means of a small number of rules of inference.
Symbolic logic differs from traditional logic in several ways. Traditional logic is the logic of Aristotle as developed by his medieval and modern followers; it deals chiefly with the categorical proposition and the categorical syllogism. Symbolic logic uses symbols rather than ordinary language; it studies the logical form of propositions rather than only their categorical form; it employs variables; it is purely deductive whereas traditional logic was extended to inductive reasoning as well; and it can analyse arguments such as those involving relations that traditional logic could not handle. Yet symbolic logic does not throw away traditional logic; it includes the valid forms of categorical syllogism as a small subsystem of its much larger framework.
The vocabulary of propositional symbolic logic consists of logical variables (p, q, r, …) and logical constants. The variables stand for propositions; the constants are the connectives that join propositions to form compound propositions. There are five basic constants. Negation (‘not’, symbolised ~ or ¬) is a unary connective that reverses the truth-value of a proposition. Conjunction (‘and’, symbolised . or ∧) joins two propositions and is true only when both conjuncts are true. Disjunction (‘or’, symbolised ∨) joins two propositions; in its inclusive sense it is true when at least one disjunct is true; in its exclusive sense it is true when exactly one disjunct is true. Implication or material conditional (‘if … then’, symbolised ⊃ or →) joins two propositions and is false only when the antecedent is true and the consequent is false. Biconditional (‘if and only if’, symbolised ≡ or ↔) joins two propositions and is true when both have the same truth-value.
The truth table is the central tool of propositional logic. A truth table for a formula sets out, in a finite list, every possible assignment of truth-values (T or F) to the variables of the formula and computes the truth-value of the formula for each such assignment. If a formula comes out true under every assignment it is a tautology (logically true). If it comes out false under every assignment it is a contradiction (logically false). If it comes out true under some assignments and false under others it is a contingency (factually true or false). An argument is valid if and only if its corresponding conditional, formed by joining the conjunction of its premises to its conclusion by an implication sign, is a tautology; equivalently, an argument is valid if there is no row of the truth table in which all the premises are true and the conclusion is false.
Symbolic logic also identifies a small number of standard argument forms that occur again and again in reasoning. Modus Ponens: from p ⊃ q and p, infer q. Modus Tollens: from p ⊃ q and ~q, infer ~p. Hypothetical Syllogism: from p ⊃ q and q ⊃ r, infer p ⊃ r. Disjunctive Syllogism: from p ∨ q and ~p, infer q. Constructive Dilemma: from (p ⊃ q) . (r ⊃ s) and p ∨ r, infer q ∨ s. Destructive Dilemma: from (p ⊃ q) . (r ⊃ s) and ~q ∨ ~s, infer ~p ∨ ~r. Each of these forms can be shown valid by a truth-table test, and each invalid look-alike form (such as Affirming the Consequent or Denying the Antecedent) can be shown invalid in the same way.
Two important equivalences known as De Morgan’s laws, after the British mathematician Augustus De Morgan (1806-1871), allow us to interchange conjunctions and disjunctions through negation. They state that ~(p . q) is logically equivalent to ~p ∨ ~q, and ~(p ∨ q) is logically equivalent to ~p . ~q. These laws together with the laws of double negation, commutation, association, distribution, contraposition, exportation, tautology and equivalence form the standard rules of propositional calculus.
Textbook Questions and Answers
A. Very Short Answer Questions (1 mark)
1. What is symbolic logic?
Answer: Symbolic logic is that branch of logic which uses special symbols, instead of ordinary words, to bring out the features of logical importance in arguments and so to classify them into types to which general rules can be applied for testing validity.
2. By what other name is symbolic logic known?
Answer: Symbolic logic is also known as mathematical logic or modern logic.
3. What is an ideogram?
Answer: An ideogram is a sign which stands directly for a concept or idea, not for the sound of a spoken word. The symbols of arithmetic such as +, =, ×, and the symbols of symbolic logic such as ., ∨, ⊃, ≡, ~, are ideograms.
4. What is a variable?
Answer: A variable is a symbol which can stand for any one of a given range of values. In propositional logic p, q and r are propositional variables that can stand for any proposition.
5. What is a constant?
Answer: A constant is a symbol that has a fixed and definite meaning throughout a formal system. The connectives ~, ., ∨, ⊃ and ≡ are logical constants.
6. Who is regarded as the founder of modern symbolic logic?
Answer: The English mathematician George Boole (1815-1864) is regarded as the founder of modern symbolic logic.
7. Name the great work of Russell and Whitehead.
Answer: Their three-volume work Principia Mathematica (1910-1913).
8. Who wrote Begriffsschrift?
Answer: The German logician and philosopher Gottlob Frege wrote Begriffsschrift in 1879.
9. What is a truth-function?
Answer: A truth-function is a compound proposition whose truth-value is wholly determined by the truth-values of its component propositions.
10. Name the five basic truth-functions.
Answer: Negation (~), Conjunction (.), Disjunction (∨), Implication (⊃) and Biconditional or Equivalence (≡).
11. What is the symbol for negation?
Answer: The symbol for negation is the tilde ~ (or sometimes ¬).
12. What is the symbol for conjunction?
Answer: The symbol for conjunction is the dot . (or sometimes ∧).
13. What is the symbol for disjunction?
Answer: The symbol for disjunction is the wedge ∨.
14. What is the symbol for implication?
Answer: The symbol for material implication is the horseshoe ⊃ (or the arrow →).
15. What is the symbol for the biconditional?
Answer: The symbol for the biconditional is the triple bar ≡ (or the double-headed arrow ↔).
16. When is a conjunction true?
Answer: A conjunction p . q is true only when both p and q are true; in every other case it is false.
17. When is an inclusive disjunction false?
Answer: An inclusive disjunction p ∨ q is false only when both p and q are false; in every other case it is true.
18. When is a material implication false?
Answer: A material implication p ⊃ q is false only when the antecedent p is true and the consequent q is false; in every other case it is true.
19. When is a biconditional true?
Answer: A biconditional p ≡ q is true when both p and q have the same truth-value, that is, both true or both false.
20. What is a tautology?
Answer: A tautology is a propositional formula which is true under every possible assignment of truth-values to its variables.
21. What is a contradiction?
Answer: A contradiction is a propositional formula which is false under every possible assignment of truth-values to its variables.
22. What is a contingent proposition?
Answer: A contingent proposition is one that is true under some assignments and false under others; it is neither a tautology nor a contradiction.
23. State De Morgan’s first law.
Answer: ~(p . q) ≡ (~p ∨ ~q). The negation of a conjunction is logically equivalent to the disjunction of the negations.
24. State De Morgan’s second law.
Answer: ~(p ∨ q) ≡ (~p . ~q). The negation of a disjunction is logically equivalent to the conjunction of the negations.
25. What is the rule of double negation?
Answer: The rule of double negation states that ~~p is logically equivalent to p; two negations cancel each other.
26. State the form of Modus Ponens.
Answer: Modus Ponens has the form: p ⊃ q ; p ; therefore q.
27. State the form of Modus Tollens.
Answer: Modus Tollens has the form: p ⊃ q ; ~q ; therefore ~p.
28. What is the Hypothetical Syllogism?
Answer: Hypothetical Syllogism has the form: p ⊃ q ; q ⊃ r ; therefore p ⊃ r.
29. What is the Disjunctive Syllogism?
Answer: Disjunctive Syllogism has the form: p ∨ q ; ~p ; therefore q.
30. How many rows does the truth table for a formula with n distinct variables contain?
Answer: The truth table for a formula with n distinct variables contains 2 to the power n rows. For one variable there are 2 rows, for two variables 4 rows, for three variables 8 rows, and so on.
B. Short Answer Questions (2-3 marks)
31. State three differences between traditional logic and symbolic logic.
Answer: (i) Traditional logic uses ordinary language; symbolic logic uses ideographic symbols. (ii) Traditional logic studies chiefly the categorical proposition and the categorical syllogism; symbolic logic studies the logical form of propositions in general including compound and relational propositions. (iii) Traditional logic includes both deductive and inductive reasoning; symbolic logic is purely deductive.
32. Mention three advantages of using symbols in logic.
Answer: (i) Symbols make the logical form of an argument transparent so that its validity can be examined apart from the truth of its content. (ii) Symbols give clarity, conciseness and economy of expression and avoid the ambiguity, vagueness and equivocation of natural language. (iii) Symbols make it possible to formulate general rules of inference and to apply them mechanically and exactly to any argument that fits the form.
33. What are the three main characteristics of symbolic logic?
Answer: (i) The use of ideograms, that is, symbols that stand directly for concepts. (ii) The use of variables, that is, symbols that can stand for any one of a given range of values. (iii) The use of the deductive method, that is, the derivation of an unlimited number of theorems from a small number of axioms by a small number of rules of inference.
34. Distinguish between inclusive and exclusive disjunction.
Answer: An inclusive disjunction p ∨ q is true when at least one of the disjuncts is true, including the case in which both are true; it is false only when both are false. An exclusive disjunction is true when exactly one of the disjuncts is true; it is false when both are true and also when both are false. The Latin word vel expresses inclusive ‘or’ and the Latin aut expresses exclusive ‘or’. The standard logical symbol ∨ stands for inclusive disjunction.
35. Distinguish between a logical variable and a logical constant.
Answer: A logical variable, such as p or q, is a symbol that can stand for any proposition; its meaning is not fixed and changes from one occurrence to another. A logical constant, such as ., ∨, ⊃, ≡, ~, is a symbol with a fixed and definite meaning that does not change from one context to another. Variables are filled in by us; constants govern how variables combine into compound propositions.
36. Distinguish between proposition and propositional form.
Answer: A proposition is a particular statement that is either true or false, for example ‘The earth is round.’ A propositional form is a schema that contains variables and becomes a proposition only when the variables are replaced by specific propositions; for example, p ⊃ q is a propositional form, while ‘If it rains then the ground is wet’ is a proposition that has that form.
37. State the truth-conditions of negation and give its truth table.
Answer: The negation ~p is true when p is false, and ~p is false when p is true.
| p | ~p |
|---|---|
| T | F |
| F | T |
38. State the truth-conditions of conjunction and give its truth table.
Answer: The conjunction p . q is true only when both p and q are true; in every other case it is false.
| p | q | p . q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
39. State the truth-conditions of inclusive disjunction and give its truth table.
Answer: The inclusive disjunction p ∨ q is false only when both p and q are false; in every other case it is true.
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
40. State the truth-conditions of material implication and give its truth table.
Answer: The material implication p ⊃ q is false only when the antecedent p is true and the consequent q is false; in every other case it is true. In particular it is true whenever the antecedent is false (the so-called paradoxes of material implication).
| p | q | p ⊃ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
41. State the truth-conditions of the biconditional and give its truth table.
Answer: The biconditional p ≡ q is true when both p and q have the same truth-value (both true or both false) and false when they have opposite truth-values.
| p | q | p ≡ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
42. Give an example of a tautology and verify it by a truth table.
Answer: The Law of Excluded Middle p ∨ ~p is a tautology.
| p | ~p | p ∨ ~p |
|---|---|---|
| T | F | T |
| F | T | T |
The last column shows T in every row, so the formula is a tautology.
43. Give an example of a contradiction and verify it by a truth table.
Answer: The formula p . ~p is a contradiction.
| p | ~p | p . ~p |
|---|---|---|
| T | F | F |
| F | T | F |
The last column shows F in every row, so the formula is a contradiction.
44. Give an example of a contingent formula.
Answer: The formula p ⊃ q is contingent, since its truth table contains both T and F values: it is false in the row where p is T and q is F, and true in the other three rows.
45. Translate the following sentences into symbolic notation: (a) Ram is a student and Hari is a teacher. (b) Either it will rain or the match will be played. (c) If she works hard then she will pass. (d) The figure is a square if and only if it has four equal sides and four right angles.
Answer: Let s = ‘Ram is a student’, t = ‘Hari is a teacher’, r = ‘It will rain’, m = ‘The match will be played’, w = ‘She works hard’, p = ‘She will pass’, f = ‘The figure is a square’, e = ‘It has four equal sides’, a = ‘It has four right angles’. Then (a) s . t (b) r ∨ m (c) w ⊃ p (d) f ≡ (e . a).
46. Test whether the formula (p ⊃ q) ⊃ (~q ⊃ ~p) is a tautology.
Answer: We construct the truth table.
| p | q | p ⊃ q | ~q | ~p | ~q ⊃ ~p | (p ⊃ q) ⊃ (~q ⊃ ~p) |
|---|---|---|---|---|---|---|
| T | T | T | F | F | T | T |
| T | F | F | T | F | F | T |
| F | T | T | F | T | T | T |
| F | F | T | T | T | T | T |
The last column is T in every row, so the formula is a tautology. This is the law of contraposition.
47. Test the validity of Modus Ponens by truth table.
Answer: The argument form is p ⊃ q ; p ; therefore q. We must show that whenever both premises are T, the conclusion is also T.
| p | q | p ⊃ q | p | q |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | F | T |
| F | F | T | F | F |
Only in the first row are both premises (p ⊃ q) and p simultaneously true, and in that row the conclusion q is also true. Therefore Modus Ponens is valid.
48. Test the validity of Modus Tollens by truth table.
Answer: The argument form is p ⊃ q ; ~q ; therefore ~p.
| p | q | p ⊃ q | ~q | ~p |
|---|---|---|---|---|
| T | T | T | F | F |
| T | F | F | T | F |
| F | T | T | F | T |
| F | F | T | T | T |
Only in the fourth row are both premises (p ⊃ q) and ~q simultaneously true, and in that row the conclusion ~p is also true. Therefore Modus Tollens is valid.
49. Test the validity of the Disjunctive Syllogism by truth table.
Answer: The argument form is p ∨ q ; ~p ; therefore q.
| p | q | p ∨ q | ~p | q |
|---|---|---|---|---|
| T | T | T | F | T |
| T | F | T | F | F |
| F | T | T | T | T |
| F | F | F | T | F |
Only in the third row are both premises true, and in that row the conclusion q is also true. Therefore Disjunctive Syllogism is valid.
50. Test the Hypothetical Syllogism by truth table.
Answer: The argument form is p ⊃ q ; q ⊃ r ; therefore p ⊃ r.
| p | q | r | p ⊃ q | q ⊃ r | p ⊃ r |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | T | F | T | F | F |
| T | F | T | F | T | T |
| T | F | F | F | T | F |
| F | T | T | T | T | T |
| F | T | F | T | F | T |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
In every row in which both premises are T (rows 1, 5, 7, 8) the conclusion is also T. Therefore Hypothetical Syllogism is valid.
C. Long Answer Questions (5-7 marks)
51. Define symbolic logic and explain its main characteristics.
Answer: Symbolic logic, also called mathematical logic or modern logic, is that branch of logic which uses symbols, instead of ordinary words, to bring out the features of logical importance in arguments. It studies the logical form of propositions and arguments and provides exact mechanical procedures for testing the validity of arguments. The main characteristics of symbolic logic are three.
The first characteristic is the use of ideograms. An ideogram is a sign which stands directly for a concept rather than for the sound of a word. The numerals 1, 2, 3 are ideograms because they stand directly for the concepts of one, two, three, regardless of how those concepts are pronounced in different languages. The symbols of symbolic logic such as the dot for conjunction, the wedge for disjunction, the horseshoe for implication, the triple bar for the biconditional and the tilde for negation are likewise ideograms. By using ideograms symbolic logic frees itself from the ambiguity, vagueness and equivocation of ordinary language.
The second characteristic is the use of variables. A variable is a symbol that can stand for any one of a given range of values. In propositional logic the small letters p, q, r are propositional variables that can stand for any proposition; in predicate logic x, y, z are individual variables and F, G are predicate variables. By using variables symbolic logic states logical truths and rules of inference in their full generality, so that one rule covers infinitely many particular cases.
The third characteristic is the use of the deductive method. By the deductive method we can derive an unlimited number of new statements (theorems) from a small number of basic statements (axioms) by means of a small number of rules of inference. Logical reasoning is thus reduced, in principle, to a kind of calculation. Because of this character, symbolic logic is also called a calculus and has a close kinship with mathematics.
52. Distinguish between traditional logic and symbolic logic.
Answer: Traditional logic is the logic of Aristotle as developed by his medieval and modern followers. It deals chiefly with the categorical proposition (A, E, I, O), the four kinds of opposition, immediate inferences and the categorical syllogism. Symbolic logic is the modern formal logic created by Boole, Frege, Russell and others; it studies the logical form of propositions and arguments by means of an artificial symbolic language. The main differences are the following. (i) Language: Traditional logic uses ordinary language; symbolic logic uses an artificial language of symbols. (ii) Subject matter: Traditional logic is largely a logic of terms, that is, of subject and predicate; symbolic logic is primarily a logic of propositions and of relations. (iii) Scope: Traditional logic confines itself to the four categorical forms; symbolic logic can analyse compound propositions, multiply general propositions and relational propositions. (iv) Method: Traditional logic is partly deductive (the syllogism) and partly inductive (Mill’s methods); symbolic logic is purely deductive. (v) Procedure: Traditional logic tests validity by general rules and special rules of figures; symbolic logic tests validity by truth tables, rules of inference, natural deduction and other mechanical procedures. (vi) Variables: Traditional logic uses few variables; symbolic logic makes essential use of variables of several kinds. In spite of these differences symbolic logic does not contradict traditional logic; the valid moods of the categorical syllogism appear within symbolic logic as a small subsystem of the larger predicate calculus.
53. Explain the five basic logical connectives and write the truth table of each.
Answer: The five basic logical connectives or constants of propositional logic are negation, conjunction, disjunction, implication and the biconditional.
Negation (‘not’, symbol ~ or ¬) is the unary connective that reverses the truth-value of a proposition. ~p is true when p is false and false when p is true.
Conjunction (‘and’, symbol . or ∧) joins two propositions; p . q is true only when both p and q are true.
Disjunction (‘or’, symbol ∨) in its inclusive sense joins two propositions; p ∨ q is true when at least one of p, q is true and is false only when both are false.
Implication (‘if … then’, symbol ⊃ or →) joins two propositions; p ⊃ q is false only when p is true and q is false, and true in every other case. The proposition p is the antecedent and q the consequent.
Biconditional (‘if and only if’, symbol ≡ or ↔) joins two propositions; p ≡ q is true when both have the same truth-value and false when they differ.
The combined truth table of all five connectives is:
| p | q | ~p | p . q | p ∨ q | p ⊃ q | p ≡ q |
|---|---|---|---|---|---|---|
| T | T | F | T | T | T | T |
| T | F | F | F | T | F | F |
| F | T | T | F | T | T | F |
| F | F | T | F | F | T | T |
54. Explain the method of testing validity by truth table, with one example each of a valid and an invalid argument.
Answer: The truth-table test for validity rests on the following definition: an argument is valid if and only if it is impossible for all its premises to be true and the conclusion false at the same time. To apply the test, we list the variables of the argument, construct a truth table that displays every possible assignment of T and F to the variables, compute under each variable the truth-value of every premise and of the conclusion, and finally inspect the rows. If there is even one row in which all the premises come out T and the conclusion comes out F, the argument is invalid; if there is no such row, the argument is valid.
Example of a valid argument (Modus Ponens): ‘If it is raining then the ground is wet. It is raining. Therefore the ground is wet.’ Symbolically: p ⊃ q ; p ; therefore q. The truth table has only one row in which both p ⊃ q and p are T, namely the row where p is T and q is T; and in that row q is also T. Hence there is no row of T-T-F (premises true, conclusion false), so the argument is valid.
Example of an invalid argument (Affirming the Consequent): ‘If it is raining then the ground is wet. The ground is wet. Therefore it is raining.’ Symbolically: p ⊃ q ; q ; therefore p. In the third row of the truth table p is F and q is T, so p ⊃ q is T and q is T (both premises true) but the conclusion p is F. This counter-example row shows that the argument is invalid.
55. Define and illustrate tautology, contradiction and contingency.
Answer: A tautology is a propositional formula that is true under every possible assignment of truth-values to its variables. The classical examples are the Law of Identity p ⊃ p, the Law of Excluded Middle p ∨ ~p, and the Law of Non-contradiction ~(p . ~p). The truth table of every tautology has T in every row of the principal column.
A contradiction is a propositional formula that is false under every possible assignment. The simplest contradiction is p . ~p, which says that p is true and not true at the same time. The truth table of every contradiction has F in every row of the principal column.
A contingent formula is one that is true under some assignments and false under others; that is, neither a tautology nor a contradiction. The simple variable p is contingent. The compound formula p ⊃ q is contingent because in the second row (p T, q F) it is false but in the other three rows it is true. Most propositions of ordinary discourse are contingent; tautologies and contradictions are special.
56. State De Morgan’s laws and verify them by truth tables.
Answer: De Morgan’s laws are two logical equivalences named after the British mathematician Augustus De Morgan (1806-1871). They state: (i) ~(p . q) ≡ (~p ∨ ~q); the negation of a conjunction is the disjunction of the negations. (ii) ~(p ∨ q) ≡ (~p . ~q); the negation of a disjunction is the conjunction of the negations.
Verification of (i):
| p | q | p . q | ~(p . q) | ~p | ~q | ~p ∨ ~q |
|---|---|---|---|---|---|---|
| T | T | T | F | F | F | F |
| T | F | F | T | F | T | T |
| F | T | F | T | T | F | T |
| F | F | F | T | T | T | T |
The columns ~(p . q) and ~p ∨ ~q are identical, so the equivalence holds.
Verification of (ii):
| p | q | p ∨ q | ~(p ∨ q) | ~p | ~q | ~p . ~q |
|---|---|---|---|---|---|---|
| T | T | T | F | F | F | F |
| T | F | T | F | F | T | F |
| F | T | T | F | T | F | F |
| F | F | F | T | T | T | T |
The columns ~(p ∨ q) and ~p . ~q are identical, so the equivalence holds.
57. State and illustrate the six standard rules of inference of propositional logic.
Answer: The six standard rules of inference taught in the first course of symbolic logic are the following.
(i) Modus Ponens (M.P.): from p ⊃ q and p, infer q. Example: ‘If it rains the road is wet; it rains; therefore the road is wet.’
(ii) Modus Tollens (M.T.): from p ⊃ q and ~q, infer ~p. Example: ‘If he is in Delhi he is in India; he is not in India; therefore he is not in Delhi.’
(iii) Hypothetical Syllogism (H.S.): from p ⊃ q and q ⊃ r, infer p ⊃ r. Example: ‘If it rains the match is cancelled; if the match is cancelled the players go home; therefore if it rains the players go home.’
(iv) Disjunctive Syllogism (D.S.): from p ∨ q and ~p, infer q. Example: ‘Either the gem is genuine or it is fake; it is not genuine; therefore it is fake.’
(v) Constructive Dilemma (C.D.): from (p ⊃ q) . (r ⊃ s) and p ∨ r, infer q ∨ s. Example: ‘If we go by air we shall be tired and if we go by train we shall be late; we go either by air or by train; therefore we shall either be tired or be late.’
(vi) Destructive Dilemma (D.D.): from (p ⊃ q) . (r ⊃ s) and ~q ∨ ~s, infer ~p ∨ ~r. Example: ‘If he is honest he tells the truth and if he is wise he keeps silent; either he does not tell the truth or he does not keep silent; therefore either he is not honest or he is not wise.’ Each rule is shown valid by a truth-table test as in the previous answers.
58. Test the validity of the Constructive Dilemma by truth table.
Answer: The argument form is (p ⊃ q) . (r ⊃ s) ; p ∨ r ; therefore q ∨ s. With four variables there are sixteen rows. We give the abbreviated table showing only the relevant columns. In the long table the only rows in which both premises are true are those in which at least one of p, r is true and the corresponding implication is true. Inspection of every such row shows that q ∨ s is true in each. There is no row in which both premises are true and the conclusion is false. Therefore Constructive Dilemma is valid.
| Row | p | q | r | s | p ⊃ q | r ⊃ s | (p ⊃ q) . (r ⊃ s) | p ∨ r | q ∨ s |
|---|---|---|---|---|---|---|---|---|---|
| 1 | T | T | T | T | T | T | T | T | T |
| 2 | T | T | T | F | T | F | F | T | T |
| 3 | T | T | F | T | T | T | T | T | T |
| 4 | T | T | F | F | T | T | T | T | T |
| 5 | T | F | T | T | F | T | F | T | T |
| 9 | F | T | T | T | T | T | T | T | T |
| 13 | F | F | T | T | T | T | T | T | T |
| 15 | F | F | T | F | T | F | F | T | F |
| 16 | F | F | F | F | T | T | T | F | F |
In every row where both premises are T, the conclusion q ∨ s is also T. Therefore the Constructive Dilemma is valid.
59. Translate the following argument into symbols and test its validity by truth table: ‘If Ram is honest then he tells the truth. Ram does not tell the truth. Therefore Ram is not honest.’
Answer: Let p = ‘Ram is honest’, q = ‘Ram tells the truth’. The argument becomes p ⊃ q ; ~q ; therefore ~p. This is the form of Modus Tollens.
| p | q | p ⊃ q | ~q | ~p |
|---|---|---|---|---|
| T | T | T | F | F |
| T | F | F | T | F |
| F | T | T | F | T |
| F | F | T | T | T |
The only row in which both premises are T is row four, and in that row the conclusion ~p is also T. Hence the argument is valid.
60. Translate and test: ‘Either the suspect was at home or he was at the office. He was not at home. Therefore he was at the office.’
Answer: Let p = ‘The suspect was at home’, q = ‘The suspect was at the office’. The argument is p ∨ q ; ~p ; therefore q. This is Disjunctive Syllogism. The truth table has only one row in which p ∨ q and ~p are both T, namely p F, q T, and in that row the conclusion q is also T. Hence the argument is valid.
Additional Important Questions
D. Multiple Choice Questions (1 mark)
1. Symbolic logic is also called:
(a) Inductive logic (b) Mathematical logic (c) Empirical logic (d) Formal logic of terms
Answer: (b) Mathematical logic.
2. The founder of modern symbolic logic is:
(a) Aristotle (b) John Stuart Mill (c) George Boole (d) Francis Bacon
Answer: (c) George Boole.
3. Principia Mathematica was written by:
(a) Frege alone (b) Russell and Whitehead (c) Boole and De Morgan (d) Peano and Hilbert
Answer: (b) Russell and Whitehead.
4. The author of Begriffsschrift (1879) is:
(a) Frege (b) Boole (c) Russell (d) Wittgenstein
Answer: (a) Frege.
5. The symbol ~ stands for:
(a) Conjunction (b) Disjunction (c) Implication (d) Negation
Answer: (d) Negation.
6. The symbol . (dot) stands for:
(a) Conjunction (b) Disjunction (c) Implication (d) Biconditional
Answer: (a) Conjunction.
7. The symbol ∨ stands for:
(a) Conjunction (b) Disjunction (c) Implication (d) Negation
Answer: (b) Disjunction.
8. The symbol ⊃ stands for:
(a) Conjunction (b) Disjunction (c) Implication (d) Biconditional
Answer: (c) Implication.
9. The symbol ≡ stands for:
(a) Implication (b) Biconditional (c) Conjunction (d) Negation
Answer: (b) Biconditional.
10. A conjunction is true:
(a) When at least one conjunct is true (b) Only when both conjuncts are true (c) Only when both conjuncts are false (d) Never
Answer: (b) Only when both conjuncts are true.
11. An inclusive disjunction is false:
(a) Only when both disjuncts are true (b) Only when both disjuncts are false (c) When the first disjunct is false (d) Always
Answer: (b) Only when both disjuncts are false.
12. A material implication p ⊃ q is false:
(a) Only when p is T and q is F (b) Only when both are F (c) Always (d) Never
Answer: (a) Only when p is T and q is F.
13. A biconditional p ≡ q is true:
(a) When p and q have the same truth-value (b) When p and q have different truth-values (c) When both are T (d) When both are F
Answer: (a) When p and q have the same truth-value.
14. A formula that is true under every assignment of truth-values is called:
(a) Contingency (b) Contradiction (c) Tautology (d) Theorem
Answer: (c) Tautology.
15. A formula that is false under every assignment is called:
(a) Tautology (b) Contradiction (c) Contingency (d) Axiom
Answer: (b) Contradiction.
16. The truth table of a formula with three variables has:
(a) 4 rows (b) 6 rows (c) 8 rows (d) 16 rows
Answer: (c) 8 rows.
17. The argument form ‘p ⊃ q ; p ; therefore q’ is called:
(a) Modus Tollens (b) Modus Ponens (c) Hypothetical Syllogism (d) Disjunctive Syllogism
Answer: (b) Modus Ponens.
18. The argument form ‘p ⊃ q ; ~q ; therefore ~p’ is called:
(a) Modus Ponens (b) Modus Tollens (c) Hypothetical Syllogism (d) Constructive Dilemma
Answer: (b) Modus Tollens.
19. The argument form ‘p ⊃ q ; q ; therefore p’ is:
(a) Valid by Modus Ponens (b) Invalid; the fallacy of affirming the consequent (c) Valid by Modus Tollens (d) A tautology
Answer: (b) Invalid; the fallacy of affirming the consequent.
20. The argument form ‘p ⊃ q ; ~p ; therefore ~q’ is:
(a) Valid (b) Invalid; the fallacy of denying the antecedent (c) Valid by Modus Tollens (d) A contradiction
Answer: (b) Invalid; the fallacy of denying the antecedent.
21. ~(p . q) is logically equivalent to:
(a) ~p . ~q (b) ~p ∨ ~q (c) p ∨ q (d) p . q
Answer: (b) ~p ∨ ~q (De Morgan’s law).
22. ~(p ∨ q) is logically equivalent to:
(a) ~p . ~q (b) ~p ∨ ~q (c) p ⊃ q (d) p ≡ q
Answer: (a) ~p . ~q (De Morgan’s law).
23. The Law of Excluded Middle is expressed as:
(a) p . ~p (b) p ∨ ~p (c) p ⊃ p (d) p ≡ p
Answer: (b) p ∨ ~p.
24. The Law of Non-contradiction is expressed as:
(a) ~(p . ~p) (b) p . ~p (c) p ∨ ~p (d) ~p ⊃ p
Answer: (a) ~(p . ~p).
25. The Law of Identity is expressed as:
(a) p ≡ ~p (b) p ⊃ p (c) p . ~p (d) ~~p
Answer: (b) p ⊃ p.
E. Fill in the Blanks
1. Symbolic logic is also called ____________ logic. Answer: mathematical (or modern).
2. ____________ is regarded as the founder of modern symbolic logic. Answer: George Boole.
3. Russell and Whitehead’s three-volume work is called ____________. Answer: Principia Mathematica.
4. A symbol that stands directly for a concept is called an ____________. Answer: ideogram.
5. A symbol that can stand for any one of a given range of values is a ____________. Answer: variable.
6. The symbol for negation is ____________. Answer: ~ (or ¬).
7. The symbol for conjunction is ____________. Answer: . (or ∧).
8. The symbol for disjunction is ____________. Answer: ∨.
9. The symbol for material implication is ____________. Answer: ⊃ (or →).
10. The symbol for the biconditional is ____________. Answer: ≡ (or ↔).
11. A conjunction is true only when both conjuncts are ____________. Answer: true.
12. An inclusive disjunction is false only when both disjuncts are ____________. Answer: false.
13. A material implication p ⊃ q is false only when p is true and q is ____________. Answer: false.
14. A biconditional is true when both components have the ____________ truth-value. Answer: same.
15. A formula true in every row of its truth table is a ____________. Answer: tautology.
16. A formula false in every row of its truth table is a ____________. Answer: contradiction.
17. A truth table for n distinct variables has ____________ rows. Answer: 2 to the power n.
18. The rule ‘from p ⊃ q and p infer q’ is called ____________. Answer: Modus Ponens.
19. The rule ‘from p ⊃ q and ~q infer ~p’ is called ____________. Answer: Modus Tollens.
20. The equivalence ~(p . q) ≡ (~p ∨ ~q) is called ____________ law. Answer: De Morgan’s.
F. True or False
1. Symbolic logic is older than Aristotelian logic. Answer: False.
2. Symbolic logic uses ideograms. Answer: True.
3. Symbolic logic is purely deductive. Answer: True.
4. Symbolic logic includes induction. Answer: False.
5. The symbol ≡ stands for implication. Answer: False (it stands for the biconditional).
6. A conjunction is true when at least one conjunct is true. Answer: False.
7. An inclusive disjunction is true when both disjuncts are true. Answer: True.
8. p ⊃ q is true when p is false. Answer: True (vacuously).
9. A tautology can be false in some row. Answer: False.
10. Modus Ponens is invalid. Answer: False.
11. Affirming the Consequent is a valid argument form. Answer: False.
12. Denying the Antecedent is a valid argument form. Answer: False.
13. ~~p is logically equivalent to p. Answer: True.
14. ~(p ∨ q) is logically equivalent to ~p ∨ ~q. Answer: False; it is equivalent to ~p . ~q.
15. The principle of contraposition states that p ⊃ q is equivalent to ~q ⊃ ~p. Answer: True.
G. Worked Examples and Long Practice
61. Test by truth table whether the formula (p . q) ⊃ p is a tautology.
Answer:
| p | q | p . q | (p . q) ⊃ p |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |
Every row in the principal column is T, so the formula is a tautology. This is the rule of Simplification.
62. Test by truth table whether the formula p ⊃ (p ∨ q) is a tautology.
Answer:
| p | q | p ∨ q | p ⊃ (p ∨ q) |
|---|---|---|---|
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | T |
Every row is T, so the formula is a tautology. This is the rule of Addition.
63. Test the validity of: ‘If a number is divisible by 6 it is divisible by 3. 18 is divisible by 6. Therefore 18 is divisible by 3.’
Answer: Let p = ‘The number is divisible by 6’, q = ‘The number is divisible by 3’. The argument is p ⊃ q ; p ; therefore q (Modus Ponens). The truth table shows that whenever both premises are T, q is also T; the argument is valid.
64. Test the argument: ‘If she studies then she passes. She does not study. Therefore she does not pass.’
Answer: Let p = ‘She studies’, q = ‘She passes’. Form: p ⊃ q ; ~p ; therefore ~q.
| p | q | p ⊃ q | ~p | ~q |
|---|---|---|---|---|
| T | T | T | F | F |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | T | T |
In row 3 both premises (p ⊃ q) and ~p are T but the conclusion ~q is F. This counter-example shows the argument is invalid; it commits the fallacy of denying the antecedent.
65. Test the argument: ‘If it rains the road is wet. The road is wet. Therefore it has rained.’
Answer: Form: p ⊃ q ; q ; therefore p. In row 3 of the truth table, p is F, q is T; so p ⊃ q is T and q is T (both premises T), but the conclusion p is F. The argument is invalid; it commits the fallacy of affirming the consequent.
66. Translate and test: ‘Either he goes by bus or he goes by train. He does not go by bus. Therefore he goes by train.’
Answer: Let b = ‘He goes by bus’, t = ‘He goes by train’. Form: b ∨ t ; ~b ; therefore t. This is Disjunctive Syllogism, which is valid as shown in question 49.
67. Translate and test: ‘If we win we shall celebrate; we shall not celebrate. Therefore we did not win.’
Answer: Let w = ‘We win’, c = ‘We celebrate’. Form: w ⊃ c ; ~c ; therefore ~w (Modus Tollens). Valid.
68. Show by truth table that p . q is logically equivalent to q . p (commutative law of conjunction).
Answer:
| p | q | p . q | q . p |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | F | F |
| F | F | F | F |
The two columns are identical, so p . q ≡ q . p.
69. Show by truth table that p ⊃ q is logically equivalent to ~p ∨ q.
Answer:
| p | q | ~p | p ⊃ q | ~p ∨ q |
|---|---|---|---|---|
| T | T | F | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |
The columns p ⊃ q and ~p ∨ q match in every row. This equivalence is called the rule of Material Implication.
70. Show by truth table that p ≡ q is logically equivalent to (p ⊃ q) . (q ⊃ p).
Answer:
| p | q | p ⊃ q | q ⊃ p | (p ⊃ q) . (q ⊃ p) | p ≡ q |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | T | F | F |
| F | T | T | F | F | F |
| F | F | T | T | T | T |
The two principal columns match. This equivalence is the rule of Material Equivalence.
Reference Tables
Logical Connectives at a Glance
| Connective | English Word | Symbol | Alternative | Read As | Type |
|---|---|---|---|---|---|
| Negation | not | ~ | ¬ | ‘It is not the case that p’ | Unary |
| Conjunction | and | . | ∧ | ‘p and q’ | Binary |
| Disjunction (inclusive) | or | ∨ | — | ‘p or q (or both)’ | Binary |
| Implication | if … then | ⊃ | → | ‘If p then q’ | Binary |
| Biconditional | if and only if | ≡ | ↔ | ‘p if and only if q’ | Binary |
Master Truth Table for the Five Basic Connectives
| p | q | ~p | ~q | p . q | p ∨ q | p ⊃ q | p ≡ q |
|---|---|---|---|---|---|---|---|
| T | T | F | F | T | T | T | T |
| T | F | F | T | F | T | F | F |
| F | T | T | F | F | T | T | F |
| F | F | T | T | F | F | T | T |
Standard Argument Forms (Rules of Inference)
| Name | Premises | Conclusion | Status |
|---|---|---|---|
| Modus Ponens | p ⊃ q ; p | q | Valid |
| Modus Tollens | p ⊃ q ; ~q | ~p | Valid |
| Hypothetical Syllogism | p ⊃ q ; q ⊃ r | p ⊃ r | Valid |
| Disjunctive Syllogism | p ∨ q ; ~p | q | Valid |
| Constructive Dilemma | (p ⊃ q) . (r ⊃ s) ; p ∨ r | q ∨ s | Valid |
| Destructive Dilemma | (p ⊃ q) . (r ⊃ s) ; ~q ∨ ~s | ~p ∨ ~r | Valid |
| Simplification | p . q | p (or q) | Valid |
| Conjunction | p ; q | p . q | Valid |
| Addition | p | p ∨ q | Valid |
| Affirming the Consequent | p ⊃ q ; q | p | Invalid (Fallacy) |
| Denying the Antecedent | p ⊃ q ; ~p | ~q | Invalid (Fallacy) |
Standard Logical Equivalences
| Name | Equivalence |
|---|---|
| Double Negation | ~~p ≡ p |
| Commutation | p . q ≡ q . p ; p ∨ q ≡ q ∨ p |
| Association | (p . q) . r ≡ p . (q . r) ; (p ∨ q) ∨ r ≡ p ∨ (q ∨ r) |
| Distribution | p . (q ∨ r) ≡ (p . q) ∨ (p . r) ; p ∨ (q . r) ≡ (p ∨ q) . (p ∨ r) |
| De Morgan’s First Law | ~(p . q) ≡ ~p ∨ ~q |
| De Morgan’s Second Law | ~(p ∨ q) ≡ ~p . ~q |
| Contraposition (Transposition) | p ⊃ q ≡ ~q ⊃ ~p |
| Material Implication | p ⊃ q ≡ ~p ∨ q |
| Material Equivalence | p ≡ q ≡ (p ⊃ q) . (q ⊃ p) |
| Exportation | (p . q) ⊃ r ≡ p ⊃ (q ⊃ r) |
| Tautology (Idempotence) | p ≡ p . p ; p ≡ p ∨ p |
Three Classifications of Propositional Formulae
| Type | Definition | Truth Table | Example |
|---|---|---|---|
| Tautology | True under every assignment | All T | p ∨ ~p |
| Contradiction | False under every assignment | All F | p . ~p |
| Contingency | Sometimes T, sometimes F | Mixed T and F | p ⊃ q |
Glossary of Key Terms
| Term | Meaning |
|---|---|
| Symbolic Logic | Branch of logic that uses symbols (ideograms) to study the form of propositions and arguments and test their validity. |
| Mathematical Logic | Another name for symbolic logic, reflecting its kinship with mathematics. |
| Traditional Logic | Aristotelian term-logic studying categorical propositions and the syllogism in ordinary language. |
| Ideogram | A sign that stands directly for a concept rather than for the sound of a word. |
| Variable | A symbol that can stand for any one of a given range of values. |
| Constant | A symbol with a fixed and definite meaning, e.g. ~, ., ∨, ⊃, ≡. |
| Proposition | A declarative sentence that is either true or false but not both. |
| Propositional Form | A schema containing variables that becomes a proposition when the variables are replaced by particular propositions. |
| Truth-Function | A compound proposition whose truth-value depends solely on the truth-values of its components. |
| Truth Table | A finite table listing every possible truth-value combination of the variables of a formula and the resulting truth-value of the formula. |
| Negation | The unary truth-function ‘not p’, symbolised ~p, that reverses the truth-value of p. |
| Conjunction | The binary truth-function ‘p and q’, symbolised p . q, true only when both p and q are true. |
| Disjunction (inclusive) | The binary truth-function ‘p or q (or both)’, symbolised p ∨ q, false only when both are false. |
| Disjunction (exclusive) | The truth-function true when exactly one of p, q is true. |
| Implication | The binary truth-function ‘if p then q’, symbolised p ⊃ q, false only when p is true and q is false. |
| Biconditional | The binary truth-function ‘p if and only if q’, symbolised p ≡ q, true when p and q have the same truth-value. |
| Antecedent | The ‘if’ part of an implication; the p in p ⊃ q. |
| Consequent | The ‘then’ part of an implication; the q in p ⊃ q. |
| Tautology | A formula true under every assignment of truth-values to its variables. |
| Contradiction | A formula false under every assignment of truth-values to its variables. |
| Contingency | A formula true under some assignments and false under others. |
| Validity | An argument is valid if it is impossible for all its premises to be true and the conclusion false at the same time. |
| Modus Ponens | Valid argument form: p ⊃ q ; p ; therefore q. |
| Modus Tollens | Valid argument form: p ⊃ q ; ~q ; therefore ~p. |
| Hypothetical Syllogism | Valid argument form: p ⊃ q ; q ⊃ r ; therefore p ⊃ r. |
| Disjunctive Syllogism | Valid argument form: p ∨ q ; ~p ; therefore q. |
| Constructive Dilemma | Valid form: (p ⊃ q) . (r ⊃ s) ; p ∨ r ; therefore q ∨ s. |
| Destructive Dilemma | Valid form: (p ⊃ q) . (r ⊃ s) ; ~q ∨ ~s ; therefore ~p ∨ ~r. |
| Affirming the Consequent | Invalid form: p ⊃ q ; q ; therefore p. |
| Denying the Antecedent | Invalid form: p ⊃ q ; ~p ; therefore ~q. |
| De Morgan’s Laws | ~(p . q) ≡ ~p ∨ ~q ; ~(p ∨ q) ≡ ~p . ~q. |
| Double Negation | ~~p ≡ p. |
| Contraposition | p ⊃ q ≡ ~q ⊃ ~p. |
| Boolean Algebra | The algebra of two values (1 and 0, T and F) created by George Boole, used in symbolic logic and digital circuits. |
| Principia Mathematica | The three-volume work of Russell and Whitehead (1910-1913) that systematised modern symbolic logic and tried to derive mathematics from logic. |
Pioneers of Symbolic Logic at a Glance
| Logician | Period | Country | Main Contribution |
|---|---|---|---|
| Gottfried Wilhelm Leibniz | 1646-1716 | Germany | Conceived the idea of a universal symbolic language and a calculus of reasoning. |
| George Boole | 1815-1864 | England | Founded modern symbolic logic; created Boolean algebra. |
| Augustus De Morgan | 1806-1871 | England | De Morgan’s laws; logic of relations. |
| Charles Sanders Peirce | 1839-1914 | USA | Quantification theory; truth-table method. |
| Gottlob Frege | 1848-1925 | Germany | Begriffsschrift (1879); first complete predicate logic with quantifiers. |
| Giuseppe Peano | 1858-1932 | Italy | Modern logical notation; axioms for arithmetic. |
| Bertrand Russell | 1872-1970 | England | Co-author Principia Mathematica; theory of types. |
| Alfred North Whitehead | 1861-1947 | England | Co-author Principia Mathematica. |
| Ludwig Wittgenstein | 1889-1951 | Austria/England | Truth-table method in Tractatus Logico-Philosophicus. |
| Jan Lukasiewicz | 1878-1956 | Poland | Many-valued logic; Polish notation. |
| Alfred Tarski | 1901-1983 | Poland/USA | Semantic theory of truth; metalogic. |
| Kurt Godel | 1906-1978 | Austria/USA | Incompleteness theorems. |
This concludes Class 11 Logic and Philosophy Chapter 9: Symbolic Logic | English Medium | ASSEB. Read the chapter together with Chapter 8 on Categorical Syllogism so that you can see how the modern symbolic apparatus generalises and extends the older Aristotelian framework. Practise drawing truth tables of every example you encounter; once the mechanical procedure is mastered, the validity or invalidity of any propositional argument can be determined with certainty. For the next chapter you will apply these skills to the formal proof of validity using rules of inference and rules of replacement, building short formal derivations of arguments whose validity has already been confirmed by truth-table.