Ch 2 Syntax and Semantics Of Propositional Logic

Question 1

By enclosing an expression in quotation marks once can…

Answer 1

talk about that expression

Quotation makes allow one to designate, that is, to refer to single expressions.

Question 2

If * and # are English sentences then..

Answer 2

‘* and #’ is an English sentence

Greek letters used for * and # are metavariables it metalinguistic variables

Question 3

What is a sentence letter in L1

Answer 3

P,Q,R,P1 ect are sentence letters

Question 4

What is a sentence of L1

Answer 4

All sentence letters are sentences of L1

All the rules combining sentence letters are L1

Nothing else is a sentence of L1

Question 5

What are connectives?

Answer 5

The 5 symbols
Conjunction 
Disjunction
Negation
Arrow
DoubleArriw

Question 6

What is Bracketing convention 1?

Answer 6

The outer brackets may be omitted from a sentence that is not part of another sentence

Question 7

What is bracketing convention 2?

Answer 7

The inner set of brackets may be omitted from a sentence in the form ((P^R)^Q) and analogous convention applied to v

This could be abbreviated to (P^R^Q)

Question 8

Which logical operations bind stronger than others?

Answer 8

^ and v bind more strongly than arrow or double arrow

Question 9

What is the sentence P Q^R an abbreviation of?

Answer 9

Since ^ binds more strongly than , ^ gains the upper hand and (P (Q^R)) is the correct reading

Question 10

What is Bracketing convention 3?

Answer 10

P,Q,R are sentences of L1
* is either ^ or v and # is either -> or

Then if (P#(QR) or ((PQ)#R) occurs as part of the sentence that is to be abbreviated, the inner set of brackets may be omitted.

Question 11

The L1 sentence P -> Q is false iff

Answer 11

P is true and Q is false, otherwise it is true

Question 12

Instead of saying that a sentence is true, logicians say that

Answer 12

the sentence has the truth-value True

Question 13

What is an L1 structure?

Answer 13

An assignment of exactly one truth-value (T or F) to every sentence letter of L1

Question 14

Truth in an L1 structure

Let A be some L1-structure. Then |…|A assigns either T or F to every sentence of L1 in the following way

Answer 14

If P is a sentence letter, |P|A is the truth value assigned to P by the L1-Structure A

Question 15

The main column

Answer 15

Is the column with the truth value of the entire sentence

Question 16

A sentence # of L1 is logically true if and only if

Answer 16

is true is all L1-structures

Question 17

Any sentence # of L1 is a contradiction iff

Answer 17

is not true in any L1 structure

Question 18

A sentence # and a sentence * are logically equivalent iff

Answer 18

and * are true in exactly the same L1 structures

Question 19

The logically true sentence of L1 are also called

Answer 19

Tautologies

Question 20

Let ¥ be a set of sentences in L1 and # a sentence of L1. The argument with all sentences in ¥ as premises and # as conclusion is valid iff

Answer 20

There is no L1 structure in which all sentences in ¥ are true and # is false

Thus an L1-argument is not valid iff there is a structure that makes all premisses true and the conclusion false

Question 21

An L1 structure is a counterexample to the argument with ¥ as the set of premisses and # as conclusion iff |$|A = T for all $€¥ and |#|A = F

Answer 21

There fore an argument in L1 is valid iff it does not have a counterexample

Question 22

What is the definition for semantic consistency?

Answer 22

A set ¥ of of L1-sentences is semantically consistent iff there is an L1 structure A such that |¥|A =T for all sentence # of ¥. Semantic inconsistency is just the opposite of semantic consistency: a set ¥ of L1-sentences is semantically inconsistent iff ¥ is not consistent

Question 23

If # and all elements of ¥ are L1-sentences, then the following obtains:

Answer 23

¥ |= # iff the set containing all sentences in ¥ and -# is semantically inconsistent