Introductory Logic

Chapter 1: Beginning Logic

1.1: Introduction
	Logic is concerned with arguments. People give reasons for what they are claiming. What they are claiming is the *conclusion* of an argument The reasons they give are the *premisses* of it. Logic deals with the quality of support that premisses provide for a conclusion. In particular is concerned with the patterns internal to arguments.
1.2: Propositions
	The premisses and conclusions of arguments are propositions. Propositions are the sorts of things expressed in language which can be either true or false. Not all sentences express propositions: questions, commands exclamations, and wishes are not true or false.
1.3: Negation, Conjunction, and Disjunction
	Propositions can be basic or *compound. Compound propositions can be constructed from basic propositions by applying an operation* to the basic propositions. Some propositions are the negation of other propositions. They are constructed from other propositions by applying of the operation of negation to them. 'Jane is not happy' is the negation of 'Jane is happy' 'Jane is happy' is the negand of the proposition 'Jane is not happy': it is the thing negated. Negation is a *monadic* operation because it operates on just one proposition. 'Jane is happy and Bill is sad' is the *conjunction* of 'Jane is happy' and 'Bill is sad' They are its *conjuncts.* Conjunction operates on two propositions, so it is called a *dyadic* operator. The most common words to express this operation are: and, but, although, in spite of, ... 'Jane is happy or Bill is sad' is the *disjunction* of 'Jane is happy' and 'Bill is sad' They are its *disjuncts.* Disjunction is a *dyadic* operator. The most common words to express this operation are: either ... or, or, unless, ... There are two types of disjunction: inclusive ('A or B' means 'A or B or both'), and exclusive ('A or B' means 'A or B but not both')
1.4: Conditionals and Biconditionals
	'If it is raining then there are clouds' is a conditional. A conditional says that one proposition implies another. The operation applied is the dyadic operation of implication. The proposition which does the implying is the *antecedent: in the example it is 'It is raining'. The proposition which is implied is the consequent: in the example it is 'There are clouds'. The most common words to express this operation are 'if ... then ---', 'only if'. (Note that 'A only if B' means the same as 'if A then B'.) (Note that antecedent does not mean 'comes first in the sentence', for example 'A if B' is the same as 'if B then A' and so B is the antecedent.) 'We will go on a picnic if and only if it is sunny' is a biconditional: it is an abbreviation of 'We will go on a picnic if it is sunny and we will go on a picnic only if* it is sunny'. This is a conjunction of two conditionals. 1. 'We will go on a picnic if it is sunny' 2. 'We will go on a picnic only if it is sunny' These are equivalent to 1'. 'if it is sunny then we will go on a picnic ' 2'. 'if we will go on a picnic then it is sunny' So each proposition is both antecedent and consequent and they imply each other. In the biconditional 'A if and only if B', proposition A is the left hand expression (LHE) and B is the right hand expression (RHE). 'if and only if' is abbreviated 'iff'.
1.5: Basic and Compound Propositions
	Compound propositions can be constructed from basic propositions by applying an *operation* to the basic propositions. We have 6 kinds now: negation, conjunction, inclusive disjunction, exclusive disjunction, conditional and biconditional.
1.6: Contrary, Contradictory, and Equivalent
	There can be logical relationships between statements. The relationship of denial can be expressed in several ways. A pair of propositions are contradictory if and only if they cannot both be true and they cannot both be false A statement and its negation are negation contradictories. For example '2 is odd' and '2 is not odd'. But there are also non-negation contradictories: For example '2 is odd' and '2 is even'. Contradictories exclude each other and they do exhaust the possibilities. A pair of propositions are contraries if and only if they cannot both be true but they could both be false. For example. 'Bob is red' and 'Bob is blue'. Contraries exclude each other but they do not exhaust the possibilities. A pair of propositions are subcontraries if and only if they cannot both be false but they could both be true. For example. 'Bob is not red' and 'Bob is not blue'. Subcontraries do not exclude each other but they do exhaust the possibilities. --------------------------------------------------------------------------------------------- Here is the traditional square of oppositions All F are G No F are G Some F are G Some F are not G The statements that are diagonally opposed are contradictories. --------------------------------------------------------------------------------------------- Here is the *propos*itional square of oppositions Ra and S not R and not S Either R or S Either not R or not S (NB. inclusive disjunction is assumed.) The statements that are diagonally opposed are contradictories. The two top statements are contraries. The two bottom.statements are subcontraries. --------------------------------------------------------------------------------------------- A pair of propositions are *equivalent* if and only if the always have the same truth value. Given a pair of contradictories, the negation of one is the equivalent of the other.
1.7: Arguments and Logical Form
	In an argument the premisses may be marked by words or phrases called *premiss markers.* For example: because, for the following reason, ... The conclusion may be marked by words or phrases called *conclusion markers.* For example: therefore, so, hence, ... Some *link* premisses and conclusions. For example: since, Why? Because, ... If premisses are claimed to give conclusive support to the conclusion the argument is a *deductive* argument. If premisses are claimed to give support that is less than conclusive it is a *non-deductive* argument. If the claim that premisses give conclusive support to the conclusion is correct the argument is *deductively valid. If the claim that premisses give a high degree of likelihood to the conclusion is correct the argument is a strong non-deductive* one. Similarly for a *weak* one. The *standard form* of an argument containing n premisses and a conclusion is premiss 1 premiss 2 ... premiss n --------------- conclusion In analyzing arguments we abbreviate the basic propositions with capital letters. These abbreviations are listed in a *dictionary. Compound propositions can be seen as having logical form. This is usually displayed using letters p, q, r, ... For example: negations have the logical form not p* conditionals have the lf if p then q Arguments also have a logical form. For example: if p then q not q --------------- not p Since the p, q, r, variables may stand for compound propositions there may be several logical forms for any argument. For example an argument with this form if (p and q) then r not r --------------- not (p and q) could also be described as having the form if p then q not q --------------- not p or even, least usefully, the form p q --------------- r One of these forms has a special name: the *explicit propositional form* of any argument is the form of the argument in terms of its basic propositional constituents.
1.8: Validity and Soundness of Arguments
	An argument is *valid* if and only if the truth of the premisses guarantees the truth of the conclusion. If an argument has a valid form then it is a *valid argument. (NB: It only needs to have one* valid form to work.) An argument form is *invalid* if and only if there is an argument with that form with its premisses all true and its conclusion false. Here are four very useful valid forms with their traditional names. *Modus Ponens, or Affirming the Antecedent (AA) if p then q* p --------------- q *Modus Tollens, or Denying the Consequent (DC) if p then q* not q --------------- not p *Hypothetical Syllogism, or Chain Argument (CA) if p then q* if q then r --------------- if p then r *Disjunctive Syllogism, or Denying a Disjunct (DD) Either p or q* not p --------------- q An argument is *sound* if and only if it is both valid and has true premisses.