Revisit summer... go to the Fitzwilliam Museum!
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1 Revisit summer... go to the Fitzwilliam Museum!
2 Faculty of Philosophy Formal Logic Lecture 5 Peter Smith Peter Smith: Formal Logic, Lecture 5 2
3 Outline Propositional connectives, and the assumption of bivalence Propositions and worlds Complex propositions Two kinds of symbols? Peter Smith: Formal Logic, Lecture 5 3
4 Propositional connectives, and the assumption of bivalence Introducing the language PL We are going to be examining how we can systematically evaluate arguments which depend for their validity or invalidity on the distribution of and, or, and not in premisses and conclusions. Peter Smith: Formal Logic, Lecture 5 4
5 Propositional connectives, and the assumption of bivalence Introducing the language PL We are going to be examining how we can systematically evaluate arguments which depend for their validity or invalidity on the distribution of and, or, and not in premisses and conclusions. This isn t because learning how to deal with such simple arguments is intrinsically exciting. It isn t! But we can introduce various key ideas and techniques in this baby case without having to tangle with distracting complexities. Peter Smith: Formal Logic, Lecture 5 5
6 Propositional connectives, and the assumption of bivalence Introducing the language PL We are going to be examining how we can systematically evaluate arguments which depend for their validity or invalidity on the distribution of and, or, and not in premisses and conclusions. This isn t because learning how to deal with such simple arguments is intrinsically exciting. It isn t! But we can introduce various key ideas and techniques in this baby case without having to tangle with distracting complexities. Vernacular and, or, and not are subject to various (lexical) semantic complexities and ambiguities and their use can create (structural) scope ambiguities. Peter Smith: Formal Logic, Lecture 5 6
7 Propositional connectives, and the assumption of bivalence Introducing the language PL We are going to be examining how we can systematically evaluate arguments which depend for their validity or invalidity on the distribution of and, or, and not in premisses and conclusions. This isn t because learning how to deal with such simple arguments is intrinsically exciting. It isn t! But we can introduce various key ideas and techniques in this baby case without having to tangle with distracting complexities. Vernacular and, or, and not are subject to various (lexical) semantic complexities and ambiguities and their use can create (structural) scope ambiguities. So following the divide and rule strategy, we first need to characterize a formal language PL for regimenting arguments clearly and without ambiguities. Then we discuss how to assess arguments once regimented. Peter Smith: Formal Logic, Lecture 5 7
8 Propositional connectives, and the assumption of bivalence We introduced the three basic connectives and are called propositional connectives, for obvious reasons. is also treated as an honorary connective. φ ψ (φ ψ) (φ ψ) T T T T T F F T F T F T F F F F φ φ T F F T Peter Smith: Formal Logic, Lecture 5 8
9 Propositions and worlds Propositional connectives, and the assumption of bivalence Propositions and worlds Complex propositions Two kinds of symbols? Peter Smith: Formal Logic, Lecture 5 9
10 Propositions and worlds Propositions and bivalence We are going to be assuming that any proposition φ that we are concerned with is either true or false... Peter Smith: Formal Logic, Lecture 5 10
11 Propositions and worlds Propositions and bivalence We are going to be assuming that any proposition φ that we are concerned with is either true or false i.e. has one of the two truth-values, true and false... Peter Smith: Formal Logic, Lecture 5 11
12 Propositions and worlds Propositions and bivalence We are going to be assuming that any proposition φ that we are concerned with is either true or false i.e. has one of the two truth-values, true and false is bivalent. Peter Smith: Formal Logic, Lecture 5 12
13 Propositions and worlds Propositions and bivalence We are going to be assuming that any proposition φ that we are concerned with is either true or false i.e. has one of the two truth-values, true and false is bivalent. So we are going to set aside cases where there are arguably truth-value gaps : Peter Smith: Formal Logic, Lecture 5 13
14 Propositions and worlds Propositions and bivalence We are going to be assuming that any proposition φ that we are concerned with is either true or false i.e. has one of the two truth-values, true and false is bivalent. So we are going to set aside cases where there are arguably truth-value gaps : 1. vague propositions (?? it is neither true nor false that the borderline bald man is bald). Peter Smith: Formal Logic, Lecture 5 14
15 Propositions and worlds Propositions and bivalence We are going to be assuming that any proposition φ that we are concerned with is either true or false i.e. has one of the two truth-values, true and false is bivalent. So we are going to set aside cases where there are arguably truth-value gaps : 1. vague propositions (?? it is neither true nor false that the borderline bald man is bald). 2. liar propositions (?? This proposition is not true is neither true nor false). Peter Smith: Formal Logic, Lecture 5 15
16 Propositions and worlds Propositions and bivalence We are going to be assuming that any proposition φ that we are concerned with is either true or false i.e. has one of the two truth-values, true and false is bivalent. So we are going to set aside cases where there are arguably truth-value gaps : 1. vague propositions (?? it is neither true nor false that the borderline bald man is bald). 2. liar propositions (?? This proposition is not true is neither true nor false). 3. propositions involving denotationless names (?? Mr Sviatolak Brintangle is married is neither true nor false if there is no such person as Mr B.) Peter Smith: Formal Logic, Lecture 5 16
17 Propositions and worlds Propositions and possible worlds A bivalent proposition φ partitions all the possible worlds all the ways that things could in fact be into two classes: the worlds where φ would be true, and those would be false. Peter Smith: Formal Logic, Lecture 5 17
18 Propositions and worlds Propositions and possible worlds A bivalent proposition φ partitions all the possible worlds all the ways that things could in fact be into two classes: the worlds where φ would be true, and those would be false. Conversely, given a set of possible worlds, ways that things could be, there is the proposition that the actual world is one of those. Peter Smith: Formal Logic, Lecture 5 18
19 Propositions and worlds Propositions and possible worlds A bivalent proposition φ partitions all the possible worlds all the ways that things could in fact be into two classes: the worlds where φ would be true, and those would be false. Conversely, given a set of possible worlds, ways that things could be, there is the proposition that the actual world is one of those. So given a proposition φ let s write Φ for the corresponding set containing all possible worlds where φ is true. Peter Smith: Formal Logic, Lecture 5 19
20 Propositions and worlds Propositions and possible worlds A bivalent proposition φ partitions all the possible worlds all the ways that things could in fact be into two classes: the worlds where φ would be true, and those would be false. Conversely, given a set of possible worlds, ways that things could be, there is the proposition that the actual world is one of those. So given a proposition φ let s write Φ for the corresponding set containing all possible worlds where φ is true. Then φ is equivalent to the proposition that the actual world is in Φ, i.e. the actual world is one of the worlds where φ is true, i.e. φ is true. Peter Smith: Formal Logic, Lecture 5 20
21 Propositions and worlds The connectives again Suppose φ and ψ are propositions, and that Φ and Ψ are the corresponding sets of possible worlds where φ and ψ respectively are true. Peter Smith: Formal Logic, Lecture 5 21
22 Propositions and worlds The connectives again Suppose φ and ψ are propositions, and that Φ and Ψ are the corresponding sets of possible worlds where φ and ψ respectively are true. The proposition (φ ψ) corresponds to (Φ Ψ), the intersection of Φ and Ψ. Peter Smith: Formal Logic, Lecture 5 22
23 Propositions and worlds The connectives again Suppose φ and ψ are propositions, and that Φ and Ψ are the corresponding sets of possible worlds where φ and ψ respectively are true. The proposition (φ ψ) corresponds to (Φ Ψ), the intersection of Φ and Ψ. The proposition (φ ψ) corresponds to (Φ Ψ), the union of Φ and Ψ. Peter Smith: Formal Logic, Lecture 5 23
24 Propositions and worlds The connectives again Suppose φ and ψ are propositions, and that Φ and Ψ are the corresponding sets of possible worlds where φ and ψ respectively are true. The proposition (φ ψ) corresponds to (Φ Ψ), the intersection of Φ and Ψ. The proposition (φ ψ) corresponds to (Φ Ψ), the union of Φ and Ψ. The proposition φ corresponds to Φ, the complement of Φ. Peter Smith: Formal Logic, Lecture 5 24
25 Propositions and worlds The connectives again Suppose φ and ψ are propositions, and that Φ and Ψ are the corresponding sets of possible worlds where φ and ψ respectively are true. The proposition (φ ψ) corresponds to (Φ Ψ), the intersection of Φ and Ψ. The proposition (φ ψ) corresponds to (Φ Ψ), the union of Φ and Ψ. The proposition φ corresponds to Φ, the complement of Φ. (This happy alignment of notation is not an accident!) Peter Smith: Formal Logic, Lecture 5 25
26 Complex propositions Propositional connectives, and the assumption of bivalence Propositions and worlds Complex propositions Two kinds of symbols? Peter Smith: Formal Logic, Lecture 5 26
27 Complex propositions Using more than one connective We can build up complex propositions by using more than one connective, as in: (P Q) (P Q) (R (P Q)) (R (P Q)) Peter Smith: Formal Logic, Lecture 5 27
28 Complex propositions Using more than one connective We can build up complex propositions by using more than one connective, as in: (P Q) (P Q) (R (P Q)) (R (P Q)) NB, to reiterate: 1. Every occurrence of the connectives and always comes with an accompanying pair of brackets, which make the scope of the connectives entirely clear. 2. The connective never introduces extra brackets the rule (in rough terms) is that a negation governs what immediately follows it. Peter Smith: Formal Logic, Lecture 5 28
29 Complex propositions Interpreting examples Suppose: Then: P = Felix is on the mat. Q = Fido is on the mat. 1. (P Q) = Felix is on the mat and Fido isn t. Peter Smith: Formal Logic, Lecture 5 29
30 Complex propositions Interpreting examples Suppose: Then: P = Felix is on the mat. Q = Fido is on the mat. 1. (P Q) = Felix is on the mat and Fido isn t. 2. (P Q) = It isn t the case that either Felix or Fido are on the mat = Neither Felix nor Fido are on the mat. Peter Smith: Formal Logic, Lecture 5 30
31 Complex propositions Interpreting examples Suppose: Then: P = Felix is on the mat. Q = Fido is on the mat. 1. (P Q) = Felix is on the mat and Fido isn t. 2. (P Q) = It isn t the case that either Felix or Fido are on the mat = Neither Felix nor Fido are on the mat. 3. ( P Q) = Either Felix isn t on the mat or Fido is. Peter Smith: Formal Logic, Lecture 5 31
32 Complex propositions Calculating truth values Suppose: P is false. Q is true. Then: P is true. Q is false. Peter Smith: Formal Logic, Lecture 5 32
33 Complex propositions Calculating truth values Suppose: P is false. Q is true. Then: P is true. Q is false. So: 1. (P Q) is false. Peter Smith: Formal Logic, Lecture 5 33
34 Complex propositions Calculating truth values Suppose: P is false. Q is true. Then: P is true. Q is false. So: 1. (P Q) is false. 2. (P Q) is false. Peter Smith: Formal Logic, Lecture 5 34
35 Complex propositions Calculating truth values Suppose: P is false. Q is true. Then: P is true. Q is false. So: 1. (P Q) is false. 2. (P Q) is false. 3. ( P Q) is true. Peter Smith: Formal Logic, Lecture 5 35
36 Complex propositions Calculating truth values Suppose: P is false. Q is true. Then: P is true. Q is false. So: 1. (P Q) is false. 2. (P Q) is false. 3. ( P Q) is true. NB, to calculate the truth-values of the complex ( molecular ) propositions we only need to know the truth-values of the simple ( atomic ) ones. You don t need to remember what they mean! Peter Smith: Formal Logic, Lecture 5 36
37 Complex propositions Some more examples φ ψ (φ ψ) (φ ψ) T T T T T F F T F T F T F F F F φ φ T F F T Suppose that P is True, Q is False, R is True. Then, what are the truth-values of 1. (Q R) 2. ((Q R) P) 3. (Q (R P)) 4. (P Q) Peter Smith: Formal Logic, Lecture 5 37
38 Complex propositions Some more examples P is T, Q is F, R is T. 1. (Q R) is F. Peter Smith: Formal Logic, Lecture 5 38
39 Complex propositions Some more examples P is T, Q is F, R is T. 1. (Q R) is F. 2. ((Q R) P) is T. Peter Smith: Formal Logic, Lecture 5 39
40 Complex propositions Some more examples P is T, Q is F, R is T. 1. (Q R) is F. 2. ((Q R) P) is T. 3. (Q (R P)) is F. Peter Smith: Formal Logic, Lecture 5 40
41 Complex propositions Some more examples P is T, Q is F, R is T. 1. (Q R) is F. 2. ((Q R) P) is T. 3. (Q (R P)) is F. Note the importance of bracketing here: Q R P would be scope-ambiguous (and have different truth-values depending on how we disambiguate it): hence the need to insist on the brackets with the connectives and. Peter Smith: Formal Logic, Lecture 5 41
42 Complex propositions Some more examples P is T, Q is F, R is T. 1. (Q R) is F. 2. ((Q R) P) is T. 3. (Q (R P)) is F. Note the importance of bracketing here: Q R P would be scope-ambiguous (and have different truth-values depending on how we disambiguate it): hence the need to insist on the brackets with the connectives and. 4. (P Q) is F because (P Q) is T. Peter Smith: Formal Logic, Lecture 5 42
43 Two kinds of symbols? Propositional connectives, and the assumption of bivalence Propositions and worlds Complex propositions Two kinds of symbols? Peter Smith: Formal Logic, Lecture 5 43
44 Two kinds of symbols? An aside on P, Q versus ϕ, ψ Why have we been using two kinds of symbols? Peter Smith: Formal Logic, Lecture 5 44
45 Two kinds of symbols? An aside on P, Q versus ϕ, ψ Why have we been using two kinds of symbols? Roughly: P, Q, etc. are shorthand for particular propositions (maybe we don t specify which propositions or we can forget which, but they are now informally, soon formally apt to convey particular messages). Peter Smith: Formal Logic, Lecture 5 45
46 Two kinds of symbols? An aside on P, Q versus ϕ, ψ Why have we been using two kinds of symbols? Roughly: P, Q, etc. are shorthand for particular propositions (maybe we don t specify which propositions or we can forget which, but they are now informally, soon formally apt to convey particular messages). Roughly: ϕ, ψ are devices for generalization about propositions, as in: whatever propositions ϕ, ψ are in play, the result of writing ( followed by ϕ followed by followed by ) is true just when one of ϕ, ψ is true. Peter Smith: Formal Logic, Lecture 5 46
47 Two kinds of symbols? An aside on P, Q versus ϕ, ψ Why have we been using two kinds of symbols? Roughly: P, Q, etc. are shorthand for particular propositions (maybe we don t specify which propositions or we can forget which, but they are now informally, soon formally apt to convey particular messages). Roughly: ϕ, ψ are devices for generalization about propositions, as in: whatever propositions ϕ, ψ are in play, the result of writing ( followed by ϕ followed by followed by ) is true just when one of ϕ, ψ is true. More on this in due course (IFL, Ch. 10). Peter Smith: Formal Logic, Lecture 5 47
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