Preliminary Matters. Semantics vs syntax (informally):

Transcription

1 Preliminary Matters Semantics vs syntax (informally): Yuri Balashov, PHIL/LING 4510/6510 o Syntax is a (complete and systematic) description of the expressions of a language (e.g. French, PYTHON, or SL), the (structural, formal) relations among them, the rules for constructing more complex expressions from simpler ones, and the rules allowing one to make a transition from one or several expressions to another expression or several expressions all without regard to questions of meaning and interpretation. o Semantics is a systematic account of the meaning (interpretation) of the expressions of a (formal or natural) language of their connection to the world: Reference (denotation) Truth o Compositionality: the semantic value of a composite expression is determined by the semantical values of its part and the syntax of the language. o Recursive (inductive) style/format 1

2 Ordered Pairs, Relations, Cartesian Products, Functions o Ordered Pair (<x,y>) An ordered pair (<x,y>) is a pair of objects with an order imposed on them. Two ordered pairs <x,y> and <z,w> are identical iff x=z and y=w. Set-theoretic definition of ordered pair (Kuratowski 1921): <x,y> = df {{x},{x,y}} Ordered triple: <x,y,z> = df <<x,y>,z>. Ordered quadruple: <x,y,z,w> = df <<x,y,z>,w> 2

3 o Cartesian Product (X Y) Yuri Balashov, PHIL/LING 4510/6510 The set of all ordered pairs <x,y>, where x X and y Y, is called the Cartesian Product of X and Y: X Y = df {<x,y>: (x X, y Y)} 3

4 o Relations Yuri Balashov, PHIL/LING 4510/6510 A binary relation R from a set X to a set Y is a set of ordered pairs <x,y> where x X and y Y. Domain of R (dom (R)) is the set of all x standing in R to some y: dom (R) = {x: y Rxy} Range of R (ran (R)) is the set of all y to which some x (one or more) stand in R: ran (R) = {y: x xry} A binary relation R from X to Y is a subset of Cartesian product X Y. Digraph (= directed graph): a diagram composed of vertices (nodes) and arcs (arrows) connecting vertices. More formally: a digraph G is an ordered pair of sets G = <V, A>, where V is a set of vertices and A is a set of ordered pairs (arcs) of vertices from V. 4

5 Digraphs provide a convenient way to illustrate/model binary relations. E.g. this digraph (G Phil ) F S K H is simply <{F, H, S, K}, {<F,H>, <H,S>, <S,K>, <F,S>, <H,H>, <K,H>, <S,H>}>. 5

6 o Functions Yuri Balashov, PHIL/LING 4510/6510 A function F is a special type of relation where every object x from a domain is related to one and only one object F(x) called the value of the function at x. A function (mapping, correspondence) F from a set X to (into) a set Y (F: X Y) is a relation from X to Y, such that: (i) x X y (= F(x)) Y <x, y > F; (ii) x 1, x 2 X, if x 1 = x 2 ) then [F(x 1 ) = F(x 2 )] X: the domain of F. {F(x): x X} Y: the range of F One-to-one (Injection): F is one-to-one (injective) iff [F(x 1 ) = F(x 2 )] (x 1 = x 2 ) Onto (Surjection): F is onto (surjective) iff y Y x [F(x) = Y] Bijection: F is a bijection iff F is both onto and one-to-one. [Figure source: 6

7 o Turning (finally) to the Semantics of SL We shall not deal with truth-tables explicitly. Yuri Balashov, PHIL/LING 4510/6510 Instead, we shall study the theory behind truth-tables. But you should have a good grasp of truth-table techniques ( ). 7

8 1 Yuri Balashov, PHIL/LING 4510/6510 Def: A truth-value assignment (TVA) is a function A from the set of all sentence letters (atomic sentences) S atom of SL into the set {T, F}: A: S atom {T, F} Def: Truth and falsity on a TVA 2 1. If P S atom then P is true on A if A(P) = T, and P is false on A if A(P) = F. 2. ~P is true on A if P is false on A; ~P is false on A if P is true on A; 3. (P&Q) is true on A if P is true on A and Q is true on A; otherwise (P&Q) is false on A. 4. (P Q) is true on A if P is true on A or Q is true on A; otherwise (P Q) is false on A. 5. (P Q) is true on A if P is false on A or Q is true on A; otherwise (P Q) is false on A. 6. (P Q) is true on A if P and Q are both true on A, or P and Q are both false on A; otherwise (P Q) is false on A. 1 The definitions, other statements, notation, and conventions adopted here and in other notes for PHIL/LING 4510/6510 reflect, with some modifications, the formalism developed in TLB and, especially, by professor Charles Cross in his handouts and other supplementary materials for the previous versions of PHIL/LING 4510/6510 and PHIL/LING 4520/6520 he taught at UGA many times in the last three decades. These materials are used here with his kind permission. 2 Here P and Q range over (the members of the set of) sentences of SL: P, Q S. Satom S. 8

9 Def: A sentence P of SL is truth-functionally true ( SLP, or simply P where only SL is under discussion) iff P is true on every TVA. Def: A sentence P of SL is truth-functionally false (P ) iff P is false on every TVA. Def: A sentence P of SL is truth-functionally indeterminate iff it is neither truth-functionally true nor truth-functionally false. Def: Sentences P and Q of SL are truth-functionally equivalent (P Q) iff P and Q have the same truth value on every TVA. Def: A set of sentences of SL is truth-functionally consistent ( ) iff there is a TVA on which every member of is true. A set of sentences of SL is truth-functionally inconsistent ( ) iff it is not truth-functionally consistent. Def: A set of sentences of SL truth-functionally entails a sentence P ( P) iff there is no TVA on which every member of is true and P is false. Def: An argument of SL is truth-functionally valid iff there is no TVA on which all its premises are true and the conclusion is false. Corollary: An argument of SL is truth-functionally valid iff the set consisting of its premises truth-functionally entails its conclusion. 9

10 Principles of Truth-Functional Entailment in SL 3 I. Structural principles (Reflexivity of ) If P, then P. (Monotonicity of ) If and P, then P. (Transitivity of ) If {P} Q and P, then Q. 3 Here and below: the subscript SL is dropped from the notation SL where only SL is under consideration. The definitions, other statements, notation, and conventions adopted here and in other notes for PHIL/LING 4510/6510 reflect, with some modifications, the formalism developed in TLB (Chs. 3 and 6) and, especially, by professor Charles Cross in his handouts and other supplementary materials for the previous versions of PHIL/LING 4510/6510 and PHIL/LING 4520/6520 he taught at UGA many times in the last three decades. These materials are used here with his kind permission. 10

11 II. Principles for connectives (&I ) If P and Q, then P&Q. (&E ) If P&Q, then P and Q. ( I ) If {P} Q, then P Q. ( E ) If P and P Q, then Q. (~I ) If {P} Q and {P} ~ Q, then ~ P. (~E ) If {~ P} Q and {~ P} ~ Q, then P. ( I ) If P, then P Q and Q P. ( E ) If P Q and {P} R and {Q} R, then R. ( I ) If {P} Q and {Q} P, then P Q. ( E ) If P and either P Q or Q P, then Q. 11

12 III. Truth-functional entailment and inconsistency ( Incons1) iff there is a P such that P and ~ P. ( Incons2) P iff {~ P}. 12

13 (a) Prove (Monotonicity of ): Exercises If and P, then P. Yuri Balashov, PHIL/LING 4510/

14 (b) Prove (~I ): Yuri Balashov, PHIL/LING 4510/6510 If {P} Q and {P} ~ Q, then ~ P. 14

15 (c) Prove ( Incons1): iff there is a P such that P and ~ P. 15

16 (d) 3.2E: 6b.* Suppose P and Q are truth-functionally indeterminate sentences. Does it follow that P&Q is truthfunctionally indeterminate? 16

17 (e) Without using truth-tables or the Def. of P, establish the following fact by a proof with numbered steps: {M, ~N} ~ (M N) 17

18 (f) Without using truth-tables or the Def. of P, establish the following fact by a proof with numbered steps: If {P Q} R and Q, then R 18