Arguments 1: Validity and Connectives of Propositional Logic

---

Arguments and Validity

• Many claims come to us as the conclusion of arguments. By "conclusion," we mean the claim that the argument is meant to defend.
• We will understand an argument as a list of claims, one of which is the conclusion, and the other claims of which are offered as reasons to believe the conclusion is true. We will call these other claims "premises."
• We need to determine what makes an argument good. Look at some examples.
  
  
  1. Premise 1: If Nixon was President, then Nixon was Commander in Chief.
  2. Premise 2: Nixon was President.
  3. Conclusion: Nixon was Commander in Chief.
  
  
  
  1. Premise 1: If Lincoln was an organism from deep in the sea, then Lincoln had three eyes.
  2. Premise 2: Lincoln was an organism from deep in the sea.
  3. Conclusion: Lincoln had three eyes.
  
  
  
  1. Premise 1: If Lincoln was President, then Lincoln had at least one portrait made of him.
  2. Premise 2: Lincoln had at least one portrait made of him.
  3. Conclusion: Lincoln was President.
     
  
  
  
• What is remarkable about the first argument is that if the premises are true it seems the conclusion must be true. That is an excellent standard to have for arguments, since it describes a clear relation between premises and a conclusion.
• Definition. A valid argument is an argument in which if the premises are true the conclusion must be true.
• Note that the second argument is absurd. Both premises and the conclusion are false. Does that mean it is a bad argument? Well, we could define a bad argument to be one where all the claims are false, but this would confuse the structure of the argument with the truth value of the claims that compose it. Our interest, right now, is argumentation itself. In that case, we must recognize that the second argument is valid. If the premises were true, the conclusion would have to be true. Valid arguments can have false conclusions if some of their premises are false.
• It is useful, therefore, to distinguish valid arguments with true premises from valid arguments with some false premises. We will call arguments like the first argument above "sound."
• Definition. A sound argument is an argument which is valid and which has true premises.
• Note that the third argument, even though each claim in it is true, is invalid. It is not the case that if one has a portrait one was President. But that is the reasoning that underlies the leap from premise 2 to the conclusion. Be aware that invalid arguments can contain all true statements. They are invalid because other arguments of the exact same form could have true premises and a false conclusion.

Propositional Logic and Connectives

• Logic is a formal method which provides a way to rigorously test arguments for validity. We will look at one part of logic -- called propositional logic -- in order to illustrate and clarify the nature of validity and good reasoning in arguments.
• Propositional logic is formulated out of propositions (what we have before now called claims) and "connectives." Connectives are ways of putting propositions together to make new propositions.
• We will represent propositions with letters P, Q, R...Z.
• Thus, the four arguments above could be represented
  
  
  1. Premise 1: If P, then Q.
  2. Premise 2: P.
  3. Conclusion: Q.
  
  
  
  1. Premise 1: If R, then S.
  2. Premise 2: R.
  3. Conclusion: S
  
  
  
  1. Premise 1: If T, then V.
  2. Premise 2: V.
  3. Conclusion: T.
  
  Assuming that we interpret our letters to be standing for the propositions:
  
  
  • P: Nixon was President
  • Q: Nixon was Commander in Chief.
  • R: Lincoln was an organism from deep in the sea.
  • S: Lincoln had three eyes.
  • T: Lincoln was President.
  • V: Lincoln had at least one portrait made of him.
  
  
• These arguments use the connective "if ... then ...." To further abbreviate our logic, we will replace these English words with a single arrow: -->. Thus, instead of "If P then Q" we will write "P --> Q". Our first argument would then look like:
  
  
  1. Premise 1: P --> Q.
  2. Premise 2: P.
  3. Conclusion: Q.
  
  
• Propositional logic is concerned only with the truth value of propositions and the relationships between propositions. This is why it is reasonable to replace all our propositions with letters. We do not need to know what P means, but rather just that it is either true or false.
• Given that we are concerned only with truth and falsity, we need to explain how we should understand the symbol -->. We call this a truth functional connective, because it relates two sentences and makes from them a new sentence. Since our concern with these sentences is only their true values, we need to explain just when some sentence formed with a "-->" is true or false. We do this with a true table.
• Let P and Q be any two propositions. There are four possible kinds of situations that concern us about these. We can represent these as rows of a table.
  
  

  P	Q
  T	T
  T	F
  F	T
  F	F

  
  
  
• For each of these kinds of situations we need to define whether "P-->Q" is true or false. Recall from our working definition of "claim" that we are interested in determinate claims -- claims that are true or false, and never both nor neither. That means each row of the table must have a value.
  
  

  P	Q	P-->Q
  T	T	T
  T	F	F
  F	T	T
  F	F	T

  
  
  
• Rows 1 and 2 are intuitive. If we have said "If P then Q" and P is true and Q is true, we have said something true. Instead, if we claim "If P then Q" and P is true but Q is false, this is clearly the very case we seem to be ruling out by our claim and we have spoken falsely. For the cases were P is false, we can think of these as cases where "If P then Q" is trivially true. "If P then Q" is clearly not refuted by such cases, and it must be either true or false, so we define it as true.
• This table tells us what "If P then Q" or "P --> Q" means. Once we have agreed on this table, we can continue with use the connective without any ambiguity.
• To make check an argument using these tables, you then need to
  1. Identify all the smallest elements (ideally, atomic sentences)
  2. For n smallest elements (atomic sentences), make a truth table with 2^n rows, one for each possible combination of truth values. Fill in a column for each smallest element.
  3. For each sentence of the argument, make a column and determine the truth value of that sentence
  4. Check to ensure that for every row where every premise is true, the conclusion is true; if this is so, the argument is valid.

Propositional Logic and Validity

• We can now use propositional logic to clarify our definition. We said that an argument is valid if, when the premises are true, the conclusion must be true. But what do we mean by "must"? We mean that it could not be any other way -- there is no situation in which the premises are true and the conclusion false.
• The table we used to define the "-->" can be extended. This is called a truth table. Let us look at the four kinds of situations that matter for the first argument above, but now add a column for each premise and for the conclusion. We get the truth values from the definition of "-->" and from just copying the earlier columns.
  
  

  P	Q	P-->Q	P	Q
  T	T	T	T	T
  T	F	F	T	F
  F	T	T	F	T
  F	F	T	F	F

  
  
  
• The first two columns here are just describing our possible situations. The next two are describing the truth values of our premises for those situations. The last column describes the truth value of our conclusion for those situations.
• Note that there is only one row in which all the premises are true. This row corresponds to any situation in which both of these claims are true. In any such situation -- we see by following the row to the conclusion -- the conclusion is also true. Thus, by saying "if the premises are true the conclusion must be true" we mean "in any situation in which all the premises are true, the conclusion is true."

Other Connectives

• Not all arguments in English contain sentences with "if...then..." in them. Arguments contain such words as "and," "but," "or," "not," and "just in case." Let us introduce a shorthand symbol for each of these, and define it using a truth table.
• "If... then...." We will call this the conditional. Alternative phrases for the conditional include "... provided that ..." and "... on the condition that ...." We will use the symbol "-->" for the conditional. It connects two and only two sentences, and forms from them a new sentence. Let P and Q be any sentences. The truth table definition is:
  
  

  P	Q	P-->Q
  T	T	T
  T	F	F
  F	T	T
  F	F	T

  
  
  
• "It is not the case that..." We will call this the negation. Alternative phrases for negation include "Not ...." We will use the symbol "~" for negation. It connects to one sentence and forms one new sentence. Let P be any sentence. Its truth table definition is:
  
  

  P	~P
  T	F
  F	T

  
  
  
• "... and ..."We will call this conjunction. Alternative phrases for the conjunction include "... but ..." and "... while ...." We will use the symbol "&" for the conditional. It connects two and only two sentences, and forms from them a new sentence. Let P and Q be any sentences. Its truth table definition is:
  
  

  P	Q	P&Q
  T	T	T
  T	F	F
  F	T	F
  F	F	F

  
  
  
• "...or..."We will call this the disjunction. Alternative phrases for the disjunction include "Either ... or ...." We will use the symbol "v" for the conditional. It connects two and only two sentences, and forms from them a new sentence. Let P and Q be any sentences. Its truth table definition is:
  
  

  P	Q	PvQ
  T	T	T
  T	F	T
  F	T	T
  F	F	F

  
  
  
• "...just in case...We will call this the biconditional. Alternative phrases for the biconditional include "... is necessary and sufficient for ...." We will use the symbol "<-->" for the biconditional. It connects two and only two sentences, and forms from them a new sentence. Let P and Q be any sentences. Its truth table definition is:
  
  

  P	Q	P<-->Q
  T	T	T
  T	F	F
  F	T	F
  F	F	T