The Semantics of S

1. The Semantics of S Gregory Ch 2, pp 31 - 46: The Language of S

2. Syntax and Semantics • Syntax: the study of the signs of a language with regard only to their formal properties. • Vocabulary: grammatical categories • Constructing Well-Formed Formulae (‘WFFs’) • Semantics: the study of language with regard to meaningful interpretations or valuations of the components • Translation • Truth functions • Characteristic truth tables for the connectives

3. The Language S • Vocabulary: Grammatical Categories • Statement Letters A, B, C, . . . A1, B1, C1, . . . , Z­1­­, A2, . . . • Truth-Functional Connectives ¬ , ∧ , ∨ , → , ↔ • Punctuation Marks ( , ), [ , ] , { , } • Translating English sentences • Statement letters translate simple sentences, which have no proper parts that are themselves sentences • Connectives (‘sentence-forming operators’) build more complex sentences from simpler sentences

4. 2.2.1-2.2.2 Use and Mention • When one is talking about a language, the object language is the language being talked about. • When one is talking about a language, the metalanguage is the language in which one is talking about the object language. • Most of the time we simply use words or phrases, and they have their normal role in meaningful communication. At certain times, however, we talk about (or mention) specific words or phrases themselves, either by placing them between quotation marks or displaying them in a special manner • California is a state. • ‘California’ is the name of a state. ‘California’ names California.

5. Alice Through the Looking-Glass • "You are sad," the Knight said in an anxious tone: "let me sing you a song to comfort you…The name of the song is called 'Haddocks' Eyes.'"  • "Oh, that's the name of the song, is it?" Alice said, trying to feel interested.  • "No, you don't understand," the Knight said, looking a little vexed. "That's what the name is called. The name really is 'The Aged Aged Man.'"  • "Then I ought to have said 'That's what the song is called'?" Alice corrected herself.  • "No, you oughtn't: that's quite another thing! The song is called 'Ways And Means': but that's only what it's called, you know!"  • "Well, what is the song, then? " said Alice, who was by this time completely bewildered.  • "I was coming to that," the Knight said. "The song really is 'A-sitting On A Gate': and the tune's my own invention." 

6. 2.2.1Object Language and Metalanguage • When one is talking about a language, the object language is the language being talked about. • When one is talking about a language, the metalanguage is the language in which one is talking about the object language. • Metavariables are variables of the metalanguage that range over (take as possible values, serve as place-holders for) expressions in the object language. Metavariables: 𝔸, 𝔹, ℂ, . . . , ℤ, 𝔸1 . . . • NB: Metavariables are notjustreplacementsforsentenceletters! They can, and do, take more complexsentences are theirvalues!

7. 2.2.3 Metavariables and Substitution Instances • Metavariables are variables of the metalanguage that range over (take as possible values) expressions of the object language. We use uppercase Blackboard Bold • Note: Gregory uses ⇒ and ⇔ as metalinguistic expressions

8. Truth Functions The truth value of a sentence of S is a function of the truth value of its parts

9. 2.1 Truth Functional Logic • A simple sentence is a sentence that contains one subject and one predicate. • A compound sentence is a sentence that either contains one or more simple sentences and at least one compounding phrase, or contains a compound subject or a compound predicate. • A sentence is a truth-functional compound iff the truth value of the compound sentence is completely and uniquely determined by (is a function of) the truth values of the simple component sentences. Otherwise, the compound sentence is non-truth-functional. • Truth-functional logic is the logic of truth-functional combinations of simple sen- tences. It investigates the properties that arguments, sentences, and sets of sentences have in virtue of their truth-functional structure.

10. What is a Function? Function: relation where every object, a, from the domain, is related to precisely one object in the range, namely, to the value of the function at a. f One at a time, puleeze: this is a function of one argument! as f (a)s Domain: Range: We can think of a function as a black box, a machine that takes members of the domain as inputs and pumps out their values under that function

11. DomainCodomainRange

12. Functions • A function is a rule for linking members of the domain to members of the range. • It links them in such a way that for every member of the domain there is a unique member of the range to which it connects. • If you know the point at which you start, you know where you end up. Domain Range

13. Functions • But what about the links that go from more than one member of the domain to a single member of the range? • Is this ok? • Yes! Because even though in each of these cases we start at different places we still know where we end up! Domain Range

14. Not a Function! • But what if it were like this, with links from one member of the domain connecting to more than one member of the range? • Is this ok? • No! Because if it’s like this, then knowing where we start doesn’t tell us where we’re going to end up! Domain Range

15. f Functions & Other Relations ? If it’s not a function, it’s like a gumball machine: you put in your money but you don’t know exactly what will come out. If it’s a function, if you know what the function is, and know what goes in, then you know exactly what will come out.

16. Functions and other relations are sets! • A set is a well defined collection of distinct objects. The elements (members) of a set can be anything: numbers, people, letters of the alphabet, other sets…whatever. • Notation: we use curly brackets to designate sets, and can define them by • Enumeration: E.g. {January, February,..., December} Or • Set builder notation: E.g. {x : x is a month of the year}(pronounced ‘the set of x such that x is a month of the year’)

17. Ordered Sets • Ordinary sets are identical if they have have exactly the same members, so for example… • {1, 2, 3} = {3, 1, 2} • Sometimes however we want to talk about ordered sets, and use regular parentheses to signal that order matters, so for example… • (1, 2, 3) ≠ (3, 1, 2) • We call these sets ‘ordered n-tuples’: ordered pairs, ordered triples (as in the example above), etc.

18. Relations Annapolis Maryland Baltimore California Illinois Sacramento Cities Massachusetts Springfield being-the-capital-of relation States

19. A binary relation is a set oforderedpairs Relations Annapolis Maryland (Anapolis, Maryland) Baltimore California Illinois Sacramento (Sacramento, California) Cities Massachusetts Springfield (Springfield, Illinois) Being-the-capital-of relation States

20. A function is a special kind of relation Some elements are (-1,-2), (0,-3), (1,-2) and much, much more…

22. What are functions…really? • A function is a set of ordered pairs: <x, y> • A subset of the Cartesian Product of the domain and codomain.

23. What are functions…really? • Cartesian Product: all possible pairs of shape & color • Shape-Color function: {<triangle, red>, <bar, yellow>, <thingy, green>, <rectangle, red>}

24. What members? • The members of the domain and range of functions can be anything you please: • They can be numbers… • Or they can be truthvalues! FunctionlandEveryone Welcome!

25. f (x) = x + 1 +1 1 1 goes in…

26. f(x) = x + 1 +1 2 2 comes out…

27. f (x) = x + 1 +1 2 2 goes in…

28. f (x) = x + 1 +1 3 3 comes out…

29. f (x) = x + 1 +1 714 714 goes in…

30. f (x) = x + 1 +1 715 Boring. 715 comes out… Functions are predictable!

31. Truth Functions f T T F F Truth functions take truth values as inputs and output truth values. The 5 connectives of the language S represent 5 different truth functions

32. ^2 +1 A function is as a function does ¬ Some functionshave more thanone input → What makes a function the function it is (rather than some other function) is the characteristic pattern of input and output. We can display this by showing tables of values for functions.

33. ^2 f(x) = x2 x f(x)

34. ¬ f(p) =¬p p f (p) Each of the truth functions represented by the five connectives of propositional logic has a characteristic truth table that makes it the function it is.

35. The Logic of Truth Functions • Truth-Functional Compound: A sentence is a truth-functional compound iff the truth value of the compound sentence is completely and uniquely determined by (is a function of) the truth values of the simple component sentences. Otherwise, the compound sentence is non-truth-functional. • Truth-Functional Logic: Truth-functional logic is the logic of truth-functional combinations of simple sentences. It investigates the properties that arguments, sentences, and sets of sentences have in virtue of their truth-functional structure. • Ordinary English and the Language S: mind the gap! • Every compound sentence of Sis a truth-functional compound • Not every compound sentence of English is a truth-functional compound

36. Truth Tables for the Connectives Truth tables for the connectives give us the meanings of ‘not’, ‘and’, ‘or’, ‘if-then’, and ‘if and only if’ in the language of S-which may not be exactly the same as their meanings in ordinary English.

37. Truth Tables for the Connectives • ℙand ℚ are variables standing for sentences. • The columns underneath them represent every possible combination of truth values they may have.

38. Truth Tables for the Connectives Other columns represent the values of the truth functions at every combination of truth values.

39. Meet the Connectives! Characteristic Truth Tables for Negation, Conjunction, Disjunction, Conditional,and Biconditional

40. Truth Tables for the Connectives Negation reverses truth value: TrueFalse; FalseTrue Easy. Justlike English.

41. Truth Tables for the Connectives Conjunction is true when both conjuncts are true--otherwise false OK.

42. Truth Tables for the Connectives Disjunction is false when both disjuncts are false--otherwise true. It’s the inclusive ‘or’.

43. Truth Tables for the Connectives Biconditional is true when both sides have the same truth value--otherwise false. It expresses logical equivalence. New one oneon me, but OK.

44. Truth Tables for the Connectives Conditional is false when antecedent is true and consequent false--otherwise true. Hu? You’vegotta be kidding!

45. Conditionals • A conditional is an if-then sentence: ‘If .................. , (then) ...................’ • In a conditional the clause that follows the ‘if’ is the antecedent; the other clause is the consequent. • Example: If it rains then it pours. antececent consequent

46. The ‘Paradoxes’ of Material Implication Whenever the antecedent false the whole conditional is true! • ‘If the moon is made of green cheese then 2 + 2 = 4’ - TRUE • ‘If the moon is made of green cheese then 2 + 2 = 5’ - TRUE • ‘If the moon is made of green cheese then the moon is not made of green cheese - TRUE

47. The ‘Paradoxes’ of Material Implication Whenever the consequent is true the whole conditional is true! • ‘If the moon is made of green cheese then 2 + 2 = 4’ - TRUE • ‘If the moon is made of green cheese then 2 + 2 = 4’ - TRUE • ‘If 2 + 2 = 5 then 2 + 2 = 4 - TRUE

48. ? The ‘if-then’ of ordinary English is not truth-functional • All the connectives of propositional logic are truth functional. • But some of the connectives of ordinary English are not truth functional. • These include (among others) English connectives that concern certain mental states and the ordinary English if-then.

49. Non-Truth Functional Connectives China is more populous than India. Sambelievesthat… T What do Iknow?!?!! ? Sam believes that China is more populous than India.

50. Non-Truth Functional Connectives The moon is made of green cheese. Sambelievesthat… F Duh, mebbe…dunno. ? Sam believes that the moon is made of green cheese.