Phil Introductory Formal Logic
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1 Phil Introductory Formal Logic Lecture 4: Formal Semantics In this lecture we give precise meaning to formulae math stuff: sets, pairs, products, relations, functions PL connectives as truth-functions PL situations and truth-tables tautology, contradiction, contingency, consequence QL interpretations
2 Sets a set is several things considered together as one thing
3 Sets a set is several things considered together as one thing there are two ways of specifying a set:
4 Sets a set is several things considered together as one thing there are two ways of specifying a set: list the items in it
5 Sets a set is several things considered together as one thing there are two ways of specifying a set: list the items in it {Sydney, Toulouse, London}
6 Sets a set is several things considered together as one thing there are two ways of specifying a set: list the items in it {Sydney, Toulouse, London} {2, 3, 15, 328}
7 Sets a set is several things considered together as one thing there are two ways of specifying a set: list the items in it {Sydney, Toulouse, London} {2, 3, 15, 328} {}
8 Sets a set is several things considered together as one thing there are two ways of specifying a set: list the items in it {Sydney, Toulouse, London} {2, 3, 15, 328} {} describe its members using a property
9 Sets a set is several things considered together as one thing there are two ways of specifying a set: list the items in it {Sydney, Toulouse, London} {2, 3, 15, 328} {} describe its members using a property { the wheels on Greg s car }
10 Sets a set is several things considered together as one thing there are two ways of specifying a set: list the items in it {Sydney, Toulouse, London} {2, 3, 15, 328} {} describe its members using a property { the wheels on Greg s car } { students currently enrolled in Phil134 }
11 Sets a set is several things considered together as one thing there are two ways of specifying a set: list the items in it {Sydney, Toulouse, London} {2, 3, 15, 328} {} describe its members using a property { the wheels on Greg s car } { students currently enrolled in Phil134 } { goldfish currently enrolled in Phil134 }
12 Sets a set is several things considered together as one thing there are two ways of specifying a set: list the items in it {Sydney, Toulouse, London} {2, 3, 15, 328} {} describe its members using a property { the wheels on Greg s car } { students currently enrolled in Phil134 } { goldfish currently enrolled in Phil134 } a set s members have no order: {1, 2} = {2, 1}
13 Sets a set is several things considered together as one thing there are two ways of specifying a set: list the items in it {Sydney, Toulouse, London} {2, 3, 15, 328} {} describe its members using a property { the wheels on Greg s car } { students currently enrolled in Phil134 } { goldfish currently enrolled in Phil134 } a set s members have no order: {1, 2} = {2, 1} things can not be in the set more than once: {1, 1} = {1}
14 Set operations and relations membership we write x {x, y, z} to say that x is in the set
15 Set operations and relations membership we write x {x, y, z} to say that x is in the set subset if every member of A is also in B we write A B
16 Set operations and relations membership we write x {x, y, z} to say that x is in the set subset if every member of A is also in B we write A B union we write A B for the set containing everything in A and everything in B
17 Set operations and relations membership we write x {x, y, z} to say that x is in the set subset if every member of A is also in B we write A B union we write A B for the set containing everything in A and everything in B intersection we write A B for the set of things that are in both A and B
18 Set operations and relations membership we write x {x, y, z} to say that x is in the set subset if every member of A is also in B we write A B union we write A B for the set containing everything in A and everything in B intersection we write A B for the set of things that are in both A and B subtraction A B is the elements from A that are not in B
19 Exercise: Set operations Let A = {1, 2, 3} and B = {true, false}. Which of the following are correct? 1 {2} A 2 2 B 3 {2, 3} (A B) 4 2 (A B) 5 2 (A {1, 3})
20 Ordered Pairs an ordered pair is two things in order: a first thing and a second thing
21 Ordered Pairs an ordered pair is two things in order: a first thing and a second thing (15, turnip )
22 Ordered Pairs an ordered pair is two things in order: a first thing and a second thing (15, turnip ) (Melbourne, true)
23 Ordered Pairs an ordered pair is two things in order: a first thing and a second thing (15, turnip ) (Melbourne, true) the order matters: (1, 2) (2, 1)
24 Ordered Pairs an ordered pair is two things in order: a first thing and a second thing (15, turnip ) (Melbourne, true) the order matters: (1, 2) (2, 1) the same thing can be in a pair twice: (1, 1) is an ordered pair
25 Products The product operation takes an ordered pair of sets, and gives a set of ordered pairs.
26 Products The product operation takes an ordered pair of sets, and gives a set of ordered pairs. It is all the pairs with the first thing from the first set, and the second thing from the second set.
27 Products The product operation takes an ordered pair of sets, and gives a set of ordered pairs. It is all the pairs with the first thing from the first set, and the second thing from the second set. A B = {ordered pairs (a, b) where a A and b B}
28 Products The product operation takes an ordered pair of sets, and gives a set of ordered pairs. It is all the pairs with the first thing from the first set, and the second thing from the second set. A B = {ordered pairs (a, b) where a A and b B} eg. {Sydney, Paris} {2, 4, 11} = {(Sydney, 2), (Sydney, 4), (Sydney, 11), (Paris, 2), (Paris, 4), (Paris, 11)}
29 Relations and Functions a relation is a subset of a product
30 Relations and Functions a relation is a subset of a product a function A B is a relation A B which has exactly one pair (a, b) for each a A
31 Relations and Functions a relation is a subset of a product a function A B is a relation A B which has exactly one pair (a, b) for each a A do not confuse the function arrow with the if... then... arrow
32 Relations and Functions a relation is a subset of a product a function A B is a relation A B which has exactly one pair (a, b) for each a A do not confuse the function arrow with the if... then... arrow if f is a function A B and (a, b) f then we say f (a) = b
33 Exercise: functions Let A = {1, 2, 3} and B = {true, false}. Which of the following are functions A B 1 {(1, true), (1, false)} 2 {(1, 2), (2, 3), (3, 1)} 3 {(1, true), (2, false), (3, true), (1, false)} 4 {(1, true), (2, false), (3, true)} 5 {(1, true), (3, true)}
34 PL Situations Now we are ready to give precise meaning to our formal languages. recall that a statement is true or false depending on the situation so, a PL situation must make each PL formula true or false the situation only needs to give truth values for the atomic formulae, because the truth of compound formulae are determined by the truth of their components therefore, a PL situation is a function {P, Q, R,...} {true, false}
35 Propositional connectives as truth functions I am rich and she is poor is true when I am rich is true and she is poor is true, otherwise it is false if P = I am rich and Q = she is poor, then P & Q is true in a situation if P and Q are both true there, false otherwise. so, the meaning of & is the function {T, F} {T, F} {T, F}, {((T, T ), T ), ((T, F ), F), ((F, T ), F), ((F, F), F)} we can write this much more conveniently as a truth-table
36 Truth tables (careful: now A and B stand for PL formulae, not sets!)
37 Truth tables (careful: now A and B stand for PL formulae, not sets!) The truth-function for & is defined by this truth-table. A B A & B T T T T T T F T F F F T F F T F F F F F we have one row for each possible input
38 Truth tables (careful: now A and B stand for PL formulae, not sets!) The truth-function for & is defined by this truth-table. A B A & B T T T T T T F T F F F T F F T F F F F F we have one row for each possible input the value of the formula is written under its main connective
39 Truth tables (careful: now A and B stand for PL formulae, not sets!) The truth-function for & is defined by this truth-table. A B A & B T T T T T T F T F F F T F F T F F F F F we have one row for each possible input the value of the formula is written under its main connective if A and B were atoms, say P and Q, the rows would be each possible situation
40 Truth tables as a validity test Since a truth-table shows us the truth value for some formulae in every possible situation, we can use it to check validity of sequents.
41 Truth tables as a validity test Since a truth-table shows us the truth value for some formulae in every possible situation, we can use it to check validity of sequents. For example, is P & Q, P, : Q a valid sequent? P Q P & Q, P : Q T T T T T F T T T F T F F F T F F T F F T T F T F F F F F T F F Now we can easily check: are there any situations in which every premise is true but the conclusion false? No, so the sequent is valid.
42 QL Semantics More on this next time, but briefly, a situation for QL is a set D for the domain, a subset of the domain for each property letter, a relation D D for each binary relation letter..., and a member of D for each name letter. Atomic formulae are true when the named things are in the named property or relation.
43 Exercises Homework Assignment Tomassi, Chapter evens (for the formulae you worked in 4.1) 4.3: 2, 4, 6
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