A proposition is a complete thought.
An atomic proposition is a single, simple thought.
A molecular proposition is a single, complex thought made out of one or more simple thoughts.
Propositions are expressed by statements.
Statements are sentences that have truth-values i.e. they are either true or false, but never both. Commands, questions, exhortations, and exclamations are all sentences that do not express statements. Commands and questions are neither true nor false. Logic is concerned only with statements and the relationships that obtain between statements.
One and the same proposition can be expressed in different sentences and in different languages.
A simple statement is one that does not contain any other statement as a part. We will use the upper-case letters, P, Q, R, ..., as symbols for simple statements.
A compound statement is one or more simple statements (as parts or what we will call components) plus at least one logical operator (logical connective). A component of a compound is any whole statement that is part of a larger statement; components may themselves be compounds.
An operator (or connective) joins simple statements into compounds, and joins compounds into larger compounds. We will use the symbols, ▼, ● , →, and ↔ to designate the dyadic sentential connectives. They are called dyadic sentential connectives because they join exactly 2 sentences (or what we are calling statements). The symbol, ~, is a monadic operator; it affects single statements only, and does not join statements into compounds.
The symbols for statements and for operators comprise our notation or symbolic language. Parentheses serve as punctuation.
|P||"P is true"||assertion|
Compounds and connectives
|~P||"P is false" or "not P"||negation|
|P ▼ Q||"either P is true, or Q is true, or both"||disjunction|
|P ● Q||"both P and Q are true"||conjunction|
|P → Q||"if P is true, then Q is true" or " P only if Q'||conditional|
|P ↔ Q||"P and Q are either both true or both false"||biconditional|
It is important to note that "not P" is a compound statement, not a simple one. It is a compound statement because negation is a logical operation, not a fact about the world.
To indicate that a compound is to be taken as a whole or single statement, we put it in parentheses. For example, P ▼ Q is a compound. Its negation is ~(P ▼ Q). When the negation sign is outside the parentheses, it affects the entire compound, not just the first component, P. Its conjunction with the compound, Q → R, would be expressed, (P ▼ Q)●(Q → R). If we wanted to say that the whole latter statement was false, we would write, ~[(P ▼ Q)●(Q → R)]. If we wanted to say that either the whole latter statement was true, or that S ↔ T was true, we would write, ~[(P ▼ Q)(●Q → R)] ▼ (S ↔ T). And so on.
The truth value of a statement is its truth or falsity. All meaningful statements have truth values, whether they are simple or compound, asserted or negated. That is, P is either true or false, ~P is either true or false, P Q is either true or false, and so on.
Every compound statement has one logical connective (or operator) that governs what we will call the form of the statement. There are exactly five (5) basic statement forms, one for each of the five truth-functional connectives that we have studied. The five statement forms for our five logical operations are:
|Logical Operation||Statement Form|
|Conjunction||p ● q|
|Disjunction||p ▼ q|
|Conditional||p → q|
|Biconditional||p ↔ q|
in which the lower case letters p and q are sentential variables (place holders for simple statements). We obtain a compound statement from a statement form by replacing all of the sentential variables in the statement form with a statement (either a simple statement or compound one). The operator (the connective) which determines the form of a statement is called the main connective (or dominant operator). The dominant operator in a formula is the one with the fewest parentheses surrounding it.
A compound statement is truth-functional if its truth value as a whole can be figured out solely on the basis of the truth values of its parts or components. A connective is truth-functional if it makes only compounds that are truth-functional. For example, if we knew the truth values of p and of q, then we could figure out the truth value of the compound, p ▼ q. Therefore the compound, p ▼ q, is a truth-functional compound and disjunction is a truth-functional connective.
All five of the connectives we are studying (negation, disjunction, conjunction, implication, and equivalence) are truth-functional. With these five connectives we can express all the truth-functional relations among statements.
A truth table is a complete list of the possible truth values of a statement. We use "T" to mean "true", and "⊥" to mean "false".
For example, p is either true or false. So its truth table has just 2 rows:
But the compound, p ▼ q, has 2
components, each of which can be true or false. So there are 4 possible
combinations of truth values. The disjunction of p with q will be true
as a compound whenever p is true, or q is true, or both:
|p||q||p ▼ q|
If a compound has n distinct simple components, then it will have 2n rows in its truth table.
The truth table columns that define the basic connectives are as follows:
|p||q||~p||~q||p ▼ q||p ● q||p → q||p ↔ q|
It would be a good idea to memorize this table.
Most statements will have some combination of T's and ⊥'s in their truth table columns; they are called contingencies. Some statements will have nothing but T's; they are called tautologies. Others will have nothing but ⊥'s; they are called contradictions. Obviously these three types of propositions exhaust the possibilities for statements that have truth table columns --which means for all truth-functional statements.
In the conditional, p → q, the first statement or "if- clause" (here p) is called the antecedent and the second statement or "then-clause" (here q) is called the consequent. Of course in more complicated conditionals, the antecedent and consequent could be compounds rather than (as here) simple statements.
Any argument can be expressed as a compound statement in the following way. Take all the premises, conjoin them, and make that conjunction the antecedent of a conditional; make the conclusion the consequent. This statement is called the corresponding conditional of the argument. Every argument has a corresponding conditional. Note that because the corresponding conditional of an argument is a statement, it is therefore either a tautology, a contradiction, or a contingency. If the corresponding conditional is a tautology, then the argument to which it corresponds is valid.
An argument is valid if and only if its corresponding conditional is a tautology. There are other tests for validity using truth tables. The chief alternative test searches for a counterexample or invalidating row: a universe in which all the premises are true and the conclusion is false. If there are no counterexamples, the argument is valid; if there is even one, it is invalid.
Two statements are consistent if and only if their conjunction is not a contradiction, that is, if it is possible for both statements to be simultaneously true.
Two statements are logically equivalent if and only if their truth table columns are identical . that is, if and only if the statement of their equivalence using "↔" is a tautology.
Obviously truth tables are adequate to test validity, tautology, contradiction, contingency, consistency, and equivalence. This is important because truth tables require no ingenuity or insight, just patience and the mechanical application of rules. No matter how dumb we are, truth tables correctly constructed will always give us the right answer.
Can you see why a truth table can be constructed for every
truth-functional statement, and only for truth-functional statements?
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