Download Elementary Logic: Understanding Valid Arguments and Truth Functions and more Study notes Philosophy in PDF only on Docsity!
Phil P
Elementary Logic
What is this course about?
Two arguments:
All cats love liver.
Peter is a cat. So Peter loves liver.
All cats love liver.
Peter loves liver.
So Peter is a cat.
Two arguments: All cats love liver. Valid Argument Peter is a cat.
- Peter loves liver. All cats love liver. Peter loves liver. Invalid Argument Peter is a cat. An argument is valid just when there is no possible circumstance in which its premises (the sentences above the horizontal line) come out true and the conclusion (the sentence below the horizontal line) comes out false.
Another argument to consider: All agents with a double-0 rating are dangerous. Anyone who is a double agent is either dangerous or devious. James Bond is both dangerous and devious. Either James Bond has a double-0 rating or he is a double agent. Remember: An argument is valid just when there is no possible circumstance in which its premises (the sentences above the horizontal line) come out true and the conclusion (the sentence below the horizontal line) comes out false. An argument is invalid just when it has a counterexample: that is, a possible circumstance in which the argument’s premises come out true, and its conclusion comes out false. Moral: What determines whether an argument is valid is not whether its premises are true or not, not whether its conclusion is true or not. An argument can be valid even if (as is the case with
argument 1) it is not clear whether its premises or its conclusion are true. An argument can be valid even if it is clear (as in the case of argument 2) that not all of its premises are true, or clear that its conclusion is false. What makes an argument valid is the impossibility of a circumstance in which its premises are all true and its conclusion is false.
A sentence constructed out of one or more other sentences by the use of one or more sentential operators is a compound sentence; otherwise it’s simple. A sentence out of which a compound sentence is composed in this way is a component of that sentence. This component may be simple or compound. Some examples:
Snow is white. [simple]
- Snow is white and grass is green. [compound; constructed from two simple components, “Snow is white”, “Grass is green”, each inserted into a blank in the sentential operator, “_____ and _____”]
- Either I’m color-blind, or snow is white and grass is green.
[compond; one simple component, “I’m color-blind”, one compound one, sentence 2, each inserted into a blank in the sentential operator, “Either____ or _____”]
- A symbolic Representation of this structure: We can build a symbolic language containing formulas that mirror the structure we have observed in English sentences. Let single capital letters represent simple English sentences. And let’s employ five operators: ( ____ ∙ ____ ) the dot ( ____ ____ ) the wedge ( ____ ____ ) the horseshoe ( ____ ____ ) the triple bar ( ____ ~ ____ ) the tilde
- The Grammar of this language (How to recognize a well-formed formula—a wff.)
- Any single capital letter is a wff;
- The result of putting a dot between two wffs and enclosing the result in parentheses is a wff;
- Likewise if a wedge is used instead of a dot;
- Likewise if a horsehoe is used instead of a dot;
- Likewise if a triple bar is used instead of a dot;
A conjunction is true just when both of its conjuncts are true. To express this via the truth-table for the dot: where^ p^ and^ q^ are any wffs, p q (p ∙ q) T T T T F F F T F F F F The dot behave just as the operator “____ and ____” does in English.
A negation is true just when its component is false; a negation is false just when its component is true. Here is the truth table for the tilde: p ~p T F F T The tilde behaves as “It’s not the case that ____” does in English.
- We can safely inter-translate the dot, wedge and tilde and the English sentential operators whose behavior they imitate.
- But the last two operators are not good translations of any ordinary English sentential operator. We treat their translations as useful jargon. A conditional is false only when its antecedent is true and its consequent is false; in all other cases, it is true. The truth-table for the horseshoe: p q (p q) (q p) T T T T T F F T F T T F F F T T We translate the horseshoe by “If___, then ”. What is wrong with taking our translation of the horseshoe seriously as an adequate translation of “If , then___ ”?
- Truth Value Computation
Given the foregoing information, we can compute the truth-value of any wff from the truth-values of its components. Example: Suppose we are informed that A , B , and C are true. Then we can compute the truth-value of
((~A ~B) C)
and (~(A ∙ B) C)
And (surprisingly) we can compute the truth-value of ((A ~B) D) and ((A ~D) B) Even though we don’t know the truth-value of D.
Moral: We don’t always need to know the truth-values of all its simple components to compute the truth-value of a wff. That’s because (for example)
Any conditional with a false antecedent is true;
Any conditional with a true consequent is true;
- Symbolizing English Sentences--How it’s Done (The overview) Example: If the sun doesn’t shine, the grass will die. Step 1: Identify the simple components of the sentence and assign each a different capital letter. In this case:
S The sun will shine. G The grass will die. (In its use for this purpose, the triple-bar doesn’t mean anything apart from “symbolizes”.)
Example:
_____but ….. _____although….. _____even though….. _____despite the fact that….. _____; moreover ….. _____; nevertheless …..
Symbolizing with the tilde: Let A Alice is guilty. B Butch is guilty. (i) They're not both guilty. (~A v ~B) or ~(A B) (ii) They're both not guilty. (~A ~B) or ~(A v B) and (ii) do not say the same thing.
