Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14atomic sentences (i.e., the most basic type of sentence)Examples from English (a natural language): Max ran.Max saw Claire.Claire gave Scruffy to Max.Claire gave Scruffy to Max.subject verb direct object indirect object predicateTranslated as sentences of FOL:Ran(Max)Saw(Max, Claire)Gave(Claire, Scruffy, Max)Cube(a)LeftOf(a,b)Between(a,b,c)Gave (Claire, Scruffy, Max)predicate arguments or logical subjects (in relation to the predicate)individual constants(in terms of reference)To summarize, sentences of FOL usually contain . . .predicates (e.g., Cube, Larger, Between, Give, Own ...)individual constants (e.g., a, b, c, Max, Scruffy…)A predicate plus one or more individual constants gives you an atomic sentence.Arity of the predicate:1 argument = an arity of 1 = unary predicateexamples: Tet(a)Dog(Scruffy)2 arguments = an arity of 2 = binary predicateexamples: RightOf(a,b)Love(Max,Claire)3 arguments = an arity of 3 = ternary predicateexamples: Between(b,c,d)Gave(Max,Scruffy,Claire)Table 1.1 on p. 22 of the textbook provides definitions of the predicates and individual constants of the ‘blocks world’ we’ll be using (see next two slides).Sometimes the definitions can be peculiar . . . (e.g., how does Between as a predicate in the blocks world differ from ‘between’ in English?)Note too that the order of arguments matters!(e.g., LeftOf(b,c) versus LeftOf(c,b))Table 1.1: Blocks language predicates.Atomic Sentence InterpretationTet(a) a is a tetrahedronCube(a) a is a cubeDodec(a) a is a dodecahedronSmall(a) a is smallMedium(a) a is mediumLarge(a) a is largeSameSize(a, b) a is the same size as bSameShape(a, b) a is the same shape as bLarger(a, b) a is larger than bSmaller(a, b) a is smaller than bSameCol(a, b) a is in the same column as bSameRow(a, b) a is in the same row as bAtomic Sentence InterpretationAdjoins(a, b) a and b are located on adjacent (but not diagonally) squaresLeftOf(a, b) a is located nearer to the left edge of the grid than bRightOf(a, b) a is located nearer to the right edge of the grid than bFrontOf(a, b) a is located nearer to the front of the grid than bBackOf(a, b) a is located nearer to the back of the grid than bBetween(a, b, c) a, b and c are in the same row, column, or diagonal, and a is between b and cEach atomic sentence of FOL expresses a claim that has a truth-value of either True (T) or False (F).To know whether a sentence is T or F, you typically have to look at a world and determine whether the sentence is T or F in that world. For example, the FOL sentence Cube(b) (“b is a cube”) may be T in some worlds (i.e., worlds in which the object labeled ‘b’ is a cube) but F in others (worlds in which the object ‘b’ is not a cube).• Launch Tarski’s World• Complete exercise 1.4 on p. 26• Complete exercise 1.7 on p. 271.7 (answers)Orig 90 180 2701 F F F F2 F F F F3 T F F T4 F F T T5 T F F F6 F F T FDifferent FOLs will differ in their choices of individual constants and predicates, depending on what is needed to describe the world or domain in question . . .GaveScruffy(claire,max)Gave(claire,scruffy,max)Notice that the above two predicate-argument approaches can convey the same meanings, but using different grammars.Why would we normally prefer the second approach to the predicate instead of the first?GaveScruffy(claire,max)Gave(claire,scruffy,max)Usually
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