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Table of Entries Addition Affirming the Consequent Alternation Alternatives And Antecedent Argument Argument Form Association Assumption Biconditional Commutation Component Compound Statement Conclusion Conditional Proof Conditional Statement Conjunct Conjunction Connective Consequent Consistent Constructive Dilemma Contingent Contradiction Contradictory Contrapositive Contrary De Morgan's Laws Deduction Deductive Argument Deductive System Denying the Antecedent Destructive Dilemma Discharge Disjunct Disjunction Disjunctive Syllogism Distribution Distributive Laws Double Negation Entails Exclusive Or Exportation Fallacy False Follows From Formal Formal Proof of Validity Grounds Hook Horseshoe Hypothetical Syllogism If ... then ... Implication Implies Inclusive Or Inconsistent Indirect Proof Induction Inductive Argument Infer Inference Inferring Invalid Logical Consequence Logically Equivalent Material Implication Material Equivalence Materially Equivalent Modus Ponens Modus Tollens Natural Deduction Natural Deductive System Necessary Condition Negation ... only if ... Or Paradoxes of Material Implication Parentheses Partial Common Meaning Premise Proof by Contradiction Proposition Propositional Calculus Propositional Connective Propositional Operator Reasoning Reductio Ad Absurdum Relation Rule of Inference Rule of Replacement Scope Sentence Simple Statement Simplification Sound Argument Soundness Specific Form Statement Statement Form Statement Letter Statement Variable Strengthened Conditional Sufficient Condition Proof Substitution Instance Symbolic Logic Tautologous Tautology Transposition True Truth-Functional Component Truth-Functional Compound Truth Table Truth Value Under Construction Valid Argument Validity Wedge Statement A linguistic expression that asserts something to be the case, and that therefore has a truth-value (can be true or false). "Roses are red or violets are blue" is a statement (in the English language). "Je voudrais une chambre" is a statement (in the French language). "Roses" is an English language expression but it is not a statement, because it does not assert anything to be the case and so does not have a truth value. Argument An argument consists of a non-empty set of statements called premises, a possibly empty sequence of statements called intermediate steps, and a single statement called the conclusion. Furthermore, each intermediate step is claimed to follow logically from one or more premises and/or preceding intermediate steps, and the conclusion is claimed to follow logically from one or more premises and/or intermediate steps. Here is an argument with two premises and no intermediate steps: All men are mortal. Socrates Therefore, Socrates is mortal. is a man. The word "therefore" indicates that an argument is being presented. "All men are mortal" and "Socrates is a man" are not claimed to follow from any other statements, so they must both be premises. "Socrates is mortal" is being claimed to follow from other statements. an argument is being presented. serves to claim that the last statement follows from one or more preceding statements, so is the conclusion of an argument. Neither of the first two statements is claimed to follow from any other, so they must both be premises. Here is an argument with three premises and one intermediate step: 1. 2. 3. 4. 5. All men are mortal Socrates is a man Everything mortal will cease to exist So, Socrates is mortal (by 1 and 2) Therefore, Socrates will cease to exist (by 3 and 4) Again, the word "therefore" indicates that the sequence is argument with statement 5 as its conclusion. Statement 4 claimed to follow from statements 1 and 2, so is intermediate step. None of statements 1, 2 and 3 is claimed follow from any other, so they must be all be premises. an is an to Here is a sequence of statements that is not an argument: 1. All men are mortal 2. Socrates is a man 3. Socrates is mortal It is not an argument, because no statement in the sequence is claimed to be a logical consequence of preceding ones. Lines 1 and 2 are premises, line 3 is the conclusion. An argument will not always be written as a sequence in the way described. For example, the second argument above may be written like this: Socrates is mortal, because Socrates is a man and all men are mortal. But everything mortal will cease to exist, so Socrates will cease to exist. Nevertheless, it is always possible to write an argument as a sequence in the way described (although it might take some work). An argument is still an argument even if it has a false premise, and even if one of the intermediate steps or the conclusion is not a logical consequence in the way that it is claimed to be. For example: 1. All men are mortal 2. Socrates is not a man 3. Therefore, Socrates is not mortal This is an argument, even though the second premise is false, and even though the conclusion is not a logical consequence of the premises. Evaluating an argument for the truth of its premises and conclusion and for its logical consequence is the topic of soundness and validity. Statement Letter A letter that stands for a particular statement, usually used for purposes of abbreviation. For example, we can define statement letters R and V by letting R stand for "Roses are red" and V stand for "Violets are blue". Then we can abbreviate the argument 1. If roses are red then violets are blue. 2. Roses are red. 3. Violets are blue. 1. If R then V. 2. R. 3. V. Or if we define a statement letter I by letting it stand for "If roses are red then violets are blue", then we can abbreviate the argument to this: 1. I. 2. R. 3. V. It is usual to use UPPER CASE letters for statement letters. Note that a statement letter is different to a statement variable. They ought not to be confused. Statement Variable A placeholder in a statement form or argument form into which can be substituted a particular statement or statement letter. The statement form "p or q" contains two statement variables p and q. We can substitute the statement "Roses are red" into p and the statement "Violets are blue" into q to get the statement "Roses are red or violets are blue". Or we can substitute the statement letter R into p and the statement letter V into q to get the statement "R or V". The argument form modus ponens 1. If p then q. 2. p. 3. q. contains two statement variables - p and q. We can substitute the statement "Roses are red" into p and the statement "Violets are blue" into q to get the argument: 1. If roses are red then violets are blue. 2. Roses are red. 3. Violets are blue. Or we can substitute the statement letter R into p and the statement letter V into q to get the argument: 1. If R then V. 2. R. 3. V. It is usual to use lower case letters for statement variables. Note that a statement variable is different to a statement letter. A statement letter stands for one particular statement. A statement variable, however, can be substituted into by many different statement letters (or statements). Substitution Instance A statement is a substitution instance of a statement form if it can be obtained by substituting statements into the statement variables of the form. The statement "Roses are red or violets are blue" is a substitution instance of the statement form "p or q", because it can be obtained by substituting "Roses are red" for p and "Violets are blue" for q. It is not a substitution instance of the statement form "p and q", because there are no statements that can be substituted for p and q to get "Roses are red or violets are blue". Nor is it a substitution instance of the statement form "p or p", because the same statement or statement letter must be substituted into every occurrence of the same statement variable. An argument is a substitution instance of an argument form if it can be obtained by substituting statements into the statement variables of the form. The argument 1. If roses are red then violets are blue. 2. Roses are red. 3. Violets are blue. is a substitution instance of the argument form 1. If p then q. 2. p. 3. q. because it can be obtained by substituting "Roses are red" for p and "Violets are blue" for q. It is not a substitution instance of the argument form 1. p or q. 2. Not p. 3. q. because it cannot be obtained from this form by substituting statements for p and q. Validity If it is impossible for an argument to have true premises and a false conclusion then it is valid, otherwise it is invalid. The argument: All men are mortal Socrates is a man Therefore, Socrates is mortal is valid, because it is impossible for the premises ('All men are mortal', and 'Socrates is a man') to be true and the conclusion ('Socrates is mortal') to be false. The argument: All cats are mortal Socrates is not a cat Therefore, Socrates is not mortal is invalid, because it is possible for the are mortal', and 'Socrates is not a cat) conclusion ('Socrates is not mortal') to the premises are true and the conclusion is premises ('All cats to be true and the be false (in fact, false). Note that the validity of an argument is determined by the possible truth-values of its premises and conclusion, not by their actual truth values. An argument might have true premises and a true conclusion and yet be invalid: All men are mortal Socrates is a man Therefore, Socrates is Greek An argument might have false premises and a false conclusion and yet be valid: No men have red hair Socrates has red hair Therefore, Socrates is not a man For this reason, the validity of an argument is no measure of the truth of its conclusion. Some valid arguments have a true conclusion, some have a false one. Some invalid arguments have a true conclusion, some have a false one. The only relation between the validity of an argument and the actual truth values of its premises and conclusion is this: if an argument is valid and has true premises, then its conclusion must be true. Or: if an argument has true premises and a false conclusion then it must be invalid.