Download Logical Arguments and Counterexamples: Understanding Forms and Validity and more Study notes Reasoning in PDF only on Docsity! 1.2 Forms and Counterexamples Consider the following arguments: 1. All oaks are trees. 2. All trees are plants. So, all oaks are plants. 1. All monauli are flageolets. 2. All flageolets are fipple-flutes. So, all monauli are fipple-flutes. These arguments have the same form, namely: Form 1 1. All A are B. 2. All B are C. So, all A are C. ‘A’, ‘B’, and ‘C’ here stand for terms: Definition: A term is a word or phrase that stands for a class, i.e., a collection or set of things. General diagram of the logic of this argument form: C (Plants) B (Trees) A (Oaks) Another valid form: 1. All emeralds are gems. 2. Some rocks are not gems. So, some rocks are not emeralds. 1. All collies are dogs. 2. Some animals are not dogs. So, some animals are not collies. 2 Here is a diagram for the counterexample: B C A Showing invalidity by counterexample 1. Identify the form of the argument 2. If the validity of the argument is suspect, attempt to produce a substitution instance of the argument form in which the premises are obviously true and the conclusion obviously false. 3. Conclude that the argument is invalid. 5 Example 1. All determinists are fatalists. 2. Some fatalists are not Calvinists. So, some Calvinists are not determinists. Form of the argument: 1. All A are B. 2. Some B are not C. So, some C are not A. Counterexample: 1. All dogs are animals. 2. Some animals are not collies. So, some collies are not dogs. Example 1. No capitalists are philanthropists. 2. All philanthropists are altruists. So, no capitalists are altruists. 6 Form of the argument: 1. No A are B. 2. All B are C. So, no A are C. Counterexample (2 steps): Begin with an obviously false conclu- sion and work backwards — use simple, well understood concepts, e.g., biological kinds: 1. No dogs are B. 2. All B are animals. So, no dogs are animals. Notice we start with our obviously false conclusion and fill in for the terms A and C. Now all we need to do is find an appropriate term for B to complete our counterexample: 1. No dogs are cats. 2. All cats are animals. So, no dogs are animals. Limitations of the method of Counterexamples 1. The method cannot show that a valid form is valid. Finding a substitution instance with true premises and a true conclusion does not show validity! 7
