Complex Statements
- One that contains another statement as a component part.
Simple Statements
- One that does not contain any other statement as a part.
Truth table construction
A truth table starts with a row containing all of the simple statement and truth-functional statement letters and then has a number of rows equal to 2n, where n = the number of simple statements involved.1) Start in the left-most column. Fill in half of the rows with T and the remaining half with F.2) Move to the next column to the right. Fill in half as many rows with T, and half as many with F, but double the number of times you do this.3) Continue filling in Ts and Fs in each column until you have completed the truth table.
Truth Functional Statements
• A truth-functional statement's truth is determined by the truth or falsity of its component (simple) statements• Truth-functional statements are called logical operators or logical connectives when treated symbolically. We will usethese terms synonymously.
Negation
The statement p is false is a __________. (Sometimes, __________ are written as not-p.) A _________ is true when its component statement is false, and false when its component statement is true. Thus, the statement p is false is true when p is false, and false when p is true.
Conjunction
- The statement p and q is a conjunction. It is true only when p is true and q is true; it is false when p is false, or when q is false, or when both p and q are false.- Conjunction is typically represented by "and" in English. However, conjunction is a logical representation of any two true statements. This relation is often represented by other words such as "but" and "however".• We represent it symbolically with "∧" or "&".
Disjunction
- The statement p or q is a _________. It is true when p is true, or when q is true, or when p and q are both true; it is false when both p and q are false.- A ___________ claims that at least one of p or q is true, or that both are true. Hence, the ___________ is false only if both p or q is false.- It is important to note that in a [true] disjunctive truth function the two disjuncts are exhaustive but not exclusive. This means that the disjuncts cannot both be false, since they are exhaustive, but both may be true, because they are not exclusive.
Implication
- The statement if p then q is an implication. The two component sentences in an implication have different names since, unlike conjuncts and disjuncts, each plays a different role: the fist is called the ANTECEDENT, and the second is called the CONSEQUENT. An implication such as if p then q is false only when p is true and q is false; inall other cases (i.e., when p is false, or when q is true), it is true.- An implication only claims that, if the antecedent is true, then the consequent will be true. I.e. An implication statement only claims that p is a sufficient condition for q.- An implication statement does not make a claim about anything beyond this sufficiency relation, such as what is the case when the antecedent is false or the consequent true.
Sufficient and Necessary Conditions
- "(p → q) ∧ (-p → -q)" claims that (p is a sufficient condition for q) and (if not-p then not-q, i.e. p is a necessary condition for q, because if p is false, q will also be false)
Exclusive Disjunction
(p V q) ∧ -(p ∧ q)" claims that (either p or q is true) and (it is not the case that p and q are both true)
Antecedent
a thing or event that existed before or logically precedes another.
Consequent
following as a result or effect.
Formally Valid
- the argument's structure is such that the truth of the premises guarantees (100%) the truth of the conclusion.• Deductive arguments need to be formally valid to be sound- truth-preserving, i.e. the truth (or falsity) of the premises is preserved through each step of the argument: truth in, truth out; false in, false out
Formally Invalid
- the truth of the premises does not guarantee the truth of the conclusion.• Inductive arguments are _____________________, but are nevertheless acceptable forms of argumentation and may be sound
Truth Tables for Validity
- If there is a row where the premises are true and the conclusion is false, the argument is formally invalid.- If, in every row with all true premises, the conclusion is true, then the argument is formally valid.- If the argument is formally valid and containstrue premises, then the conclusion must be true as well.
Chain Argument
- (or hypothetical syllogism) joins a number of affirmed implication statements together and allows us to conclude that a particular implication statement is true. E.g.:1) If p then q2) If q then r3) So, if p then r
Disjunctive Syllogism
denies one of two disjuncts in a disjunctive statement, allowing us to conclude that the other disjunct is true. E.g.:1) Either p or q2) -p3) So q
Affirming the Consequent
- Occurs when we affirm the consequent of an affirmed implication statement and then conclude on that basis that the antecedent may be affirmed as well. E.g.:1) If it is raining, then it is pouring (p → q)2) It is pouring (q - affirming the consequent)3) Therefore, it is raining (p)
Denying the Antecedent
- Occurs when an implication statement is affirmed and its antecedent denied, and then, on that basis, the consequent is denied as well.E.g.:1) If it is raining, then it is pouring (p → q)2) It is not raining pouring (-p - denying the antecedent)3) Therefore, it is not pouring (-q)