Monday 2/3

Truth Tropic Language I: Defining PC

Readings

Notes

• The Propositional Calculus
• PC Entailments and Theorems (pdf)
• PC Translations (pdf)

Synopsis

We made a lot of progress today. Having revisited the distinction between inductive and deductive arguments from last time, we further distinguished between sound, valid, and invalid deductive arguments. We then defined a logic and specified the Propositional Calculus, our example of a logic this semester.

Permit me to emphasize that I do not expect, require, demand, or even believe that you understand every concept from this lecture. At best, the terminology of arguments is "in the air", as it were, and definitions are available for your repeated review. What I have discovered from previous classes is that once I start using the terminology on a regular basis, students steadily, if not rapidly, catch on to what is meant. If you feel completely lost, take heart: There are many, many others feeling the same way at this point.

Eventually you will be able (I promise!) to

• Explain the distinction between Truth-Tropic and Truth-Phobic Language.
• Explain the distinction between inductive and deductive arguments.
• Explain the distinction between a weak and a strong inductive argument.
• Explain the distinction between invalid, valid, and sound deductive arguments.

As we saw, validity and soundness are particularly slippery concepts, so this last will definitely take some work.

In any case, there are a few facts about arguments which are crucial. If you don't understand them at first, you should at least memorize them.

1. It is always possible for the conclusion of an inductive argument to be false, even when all the premises of the argument are true. (Remember the silly white raven!)
2. In a valid deductive argument, the conclusion must be true if the premises are all true.
3. If one or more of the premises in a valid deductive argument are false, it does not follow that the conclusion is false. The conclusion may still be true; the argument just doesn't give us any reason for thinking that it is true.
4. If the conclusion of a valid deductive argument is false, at least one of the premises must be false.
5. A valid argument may have all true premises and (necessarily) a true conclusion, a false conclusion and (necessarily) one or more false premises, false premises and a false conclusion, or false premises and a true conclusion.
6. The only situation in which the actual truth or falsity of the propositions in a deductive argument tell us anything at all about the validity of the argument is when the premises are all true but the conclusion is false: we then know that the argument is invalid. The validity of an argument is completely independent of the actual truth or falsity of the propositions in the argument in the sense that one can never find out whether the argument is valid based on the actual truth or falsity of the propositions in the argument.
7. A deductive argument is valid if it has the form of a valid argument; validity is a formal or syntactic feature of arguments. Where truth is a relation between language (propositions) and the world (states-of-affairs), validity is a relation between propositions (the premises) and another proposition (the conclusion) irrespective of the way the world happens to be (or which states-of-affairs obtain).
8. If a deductive argument is sound, then we know that its conclusion is true.
9. If a deductive argument is unsound, we know that it is either invalid, or it has at least one false premise.
10. Critically assessing deductive arguments requires that we first find out whether or not the argument is valid and then find out whether or not the premises are all true. If the argument is invalid or has at least one false premise, then it follows that we have no reason to think that the conclusion is true; it does not follow that we have any reason for thinking that the conclusion is false.

There are other facts, of course, but these are the most important ones for you to grasp from this lecture.

By way of investigating the truth-tropic property of language we know as validity, I continued today by introducing the concept of a formal logic.

As we see in specifying its vocabulary, syntax, and semantics, the Propositional Calculus (PC) is a lean, some might say simplistic, logic. Nevertheless, it provides us with lots of room to develop a more sophisticated understanding of validity.

Today we defined PC by specifying its vocabulary, syntax, and semantics. We used the semantics to develop a method of displaying in a table precisely how the truth of a Well Formed Formula (WFF) of PC--that is, a syntactically correct formula or sentence of PC--depends on the truth of its constitutive sentence letters.

Next time I will specify a fully mechanical method for filling out a truth table so as to systematically exhaust all the possible combinations of truth values of constitutive sentence letters no matter how many individual sentence letters there may be. (We will call the result a 'standard truth table', but more on that next time.) We will then show how to use standard truth tables to define validity on PC, which will also provide us, conveniently enough, with a method to test whether a given argument is valid. Reflect on how cool that is: We at once have a way of defining validity on the formal language constituting PC (thus, making it a full-blown logic!) which also becomes a universal test for validity. The resources we enjoy which Socrates, sadly, lacked, are tremendous, to say the least.