Chapter 2: A New Language

2.1: Introduction

We define an artificial languge PL (for Propositional Language)

We will translate compound statements into PL.

2.2: Symbols and Translation

Negation is written with a tilde. Thus not P is written ~P,

Conjunction is written with an ampersand. P and Q is written (P & Q). Note the parentheses.

Disjunction (inclusive) uses the symbol vel. P or Q or both is written (P v Q).

Disjunction (exclusive): eks: P or Q but not both is written (P x Q). (Actually the textbook uses slashed tribar which is º with a vertical bar through it, but I can't make that sign.)

Implication: hook: if P then Q is written (P É Q). This statement is a material conditional.

Equivalence: tribar: P if and only if Q is written (P º Q). This statement is a bi-conditional or a material equivalence

There are some unusual operators.

The Sheffer stroke sign is used for the nand operator. Thus not P and not Q is written (P | Q).

The Peirce arrow sign is used for the nor operator. Thus neither P nor Q is written (P ¯ Q).

A, B, C, ... are propositional constants. They abbreviate particular propositions.

p, q, r, ... are propositional variables. We use them to stand in the place of any proposition.

The logical form of a proposition or argument is given using propositional variables.

For example: ~A, ~~(A & B) both have the form ~p.

We also say that ~A, ~~(A & B) are instances of ~p.

The explicit form of a proposition is where each propositional variable takes just one propositional constant for its instance.

2.3: Syntax for PL

PL has a grammar. The sentences of PL are called well formed formulae of PL (WPL, wffs of PL.)

Wffs using connectives and variables are propositional forms. For example (p & q).

Wffs using connectives and constants are propositional sentences. For example (A & B).

Wffs using connectives, variables, and constants are propositional hybrids. For example (A & q).

The rules for wffs for PL are as follows:

A. Define a set of elements

1. Take a defined set of propositional constants: A, B, ...

2. Take a defined set of propositional variables: p, q, ...

(constants and variables together we call propositional letters)

3. Take the eight defined logical symbols (or incomplete symbols): ~, &, v, x, É, º, (, ).

letters and logical symbols together form the primitive symbols.

B. Set up a recursive definition for the WPL.

1. basis clause: Any propositional letter is a WPL

2. recursive clause.

    i.       If X is a WPL then ~X is a WPL;

    ii.      If X and Y are WPLs, and * is one of &, v, º, É, then (X * Y) is a WPL.

3. terminal clause: Nothing is a WPL unless it is a WPL because of the preceding clauses.

The last of the operators added to a formula in its construction is the main operator. To mark out the main operator (when we need to) we point an arrow at it from below.

Note that a WPL has exactly the same number of left and right parentheses. If a formula doesn't satisfy this then it isn't a WPL.

Many abbreviations for WPLs involve omitting parentheses.

Parentheses which surround the WPL can be omitted.

One pair of parentheses may be replaced by square brackets.

2.4: Truth and PL

We now give a semantics for PL. This means that we give a definition of the meanings of the sentences of the language.

The definitions are in terms of the truth-values: true and false.

We'll use numeral 1 and 0 to stand for true and false. (Read them as 'true' and 'false'.)

The definitions are by truth tables.

In a truth table we set out all the possible combinations of truth values for the parts of the compound proposition in the left hand side of the table (the matrix) and the truth value of the resulting compund is in the right hand side.

Negation

Interpret this as saying: when p is valued as 1, then ~p is valued as 0, and when p is valued as 0, then ~p is valued as 1.

Conjunction

Interpret this as saying:

when p is valued as 1 and q is valued as 1, then p & q is valued as 1, and

when p is valued as 1 and q is valued as 0, then p & q is valued as 0, and

when p is valued as 0 and q is valued as 1, then p & q is valued as 0, and

when p is valued as 0 and q is valued as 0, then p & q is valued as 0.

Conjunction

Inclusive Disjunction

Exclusive disjunction

Material Equivalence

Material Conditional

Notice that for hook the order of the parts matters, because p É q  may have a different truth value from q É p. This isn't the case for he other operators listed. For this reason they are said to be commutative.

With these definitions, given the truth-values of basic propositions, we can calculate the truth-value of any compound proposition in PL.

If p, q, and r are given t-values, our definitions give just one t-value to any compound proposition constructed from those parts. Thus our definitions define a truth-functional semantics.

Because the values can be calculated with these definitions, our language together with this semantics is now a logical calculus.

Previously we had a language PL, but with this semantics we have the  logical system which we shall call Propositiona Calculus (PC)

There is a curiosity in the semantics that we have given for material implication. If the antecedent is false then the truth-value of the consequent is irrelevant: the material conditional is always true. The only time the material conditional can be false is when its antecedent is true and its consequent is false.