Skip to content

PHIL512 - Agenda and Notes

Agenda

  1. Review Ch1, pp.3-38

  2. Break

  3. A tiny bit of Ch2
    • Motivation for a formal language
    • Symbols
    • Syntax
    • Semantics
    • Translation

Questions from last time

  1. Think of an argument that "seems good" but is not good.
  2. Think of what makes an argument good or bad, e.g., intuition? consensus? science? God? Truth? Relevance? Form? Practical results?

Complexity of propositions

Propositions are complex because there is a tendency to think that all declarative sentences express propositions and interrogative (questions) and imperative (commands) do not. But this is not the case.

  1. Some questions express propositions, e.g., rhetorical questions like "Are you stupid"?
  2. An exclamation might express the propositions, e.g., "Ouch!" expresses the proposition "I am in pain". Or, "ugh" as "I don't like you".
  3. Some declarative utterances do not express propositions. So-called performative utterances like "I do" (said on your wedding day) or "I bet five dollars" (said when placing a bet). We cannot respond "true" or "false" to these utterances.
  4. "Don't devalue your experiences" (command) really says "Your experiences are important."
  5. Some declarative sentences are actually commands, e.g., "There is the door" (said in the context of an argument) is a command to leave.
  6. Sentences that might not be meaningful due to faulty presuppositions or meaningless terms ("God exists") or contain category mistakes, e.g., "colorless green ideas sleep furiously" (Chomsky).

The main problem is that we use context to determine whether S expresses a proposition but (1) it is hard to explain the context in which S is written and (2) how do you teach students to interpret contextual clues.

Validity

The first major conceptual hurdle for the student (and the instructor) involves validity. The two main definitions are the following:

  1. An argument is deductively valid if and only if it is impossible for the premises to be true and the conclusion false. An argument is deductively invalid iff it is possible for the premises to be true and the conclusion false.
  2. An argument is deductively valid if and only if it is necessarily the case that if the premises are true, then the conclusion is true. An argument is deductively invalid iff it is not necessarily the case that if the premises are true, then the conclusion is true.

Issues with explaining validity

There are four main problems on the instructional side:

  1. If a student presents a belief about validity / invalidity, can you explain (not just say) whether it is correct or incorrect
  2. Can you create an example of why the definition is incorrect
  3. Some students still won't get it.
  4. Whether you should test students whether arguments are valid or invalid.

Here are some incorrect definitions or misconceptions of validity, try to explain why they are incorrect (they get increasingly more difficult):

  1. if an argument has a true C, then it is valid.
  2. if an argument has all true propositions, then it is valid.
  3. if an argument has all false propositions, then it is invalid.
  4. if an argument has all true propositions and the premises and conclusion are topically-related, then the argument is valid.
  5. Validity is defined as follows: "if the premises are true, then the conclusion must be true."

Motivating tests for validity

There are two main issues relating to motivating tests for validity.

  1. Distinguishing the concept of validity from tests for validity
  2. Encouraging students to see that they need a test for validity.

First, people tend to believe that because they know the definition of something, they know how to pick that thing out. In other words, students tend to think that if you know the definition of a valid argument, you know how to identify a valid argument. But, after they reflect a moment, they realize that this is not the case. So, let's help them reflect.

Step 1 is to show prompt this reflection by creating some examples where (1) someone has a concept of X but (2) they might not be able to identify X (Of course, the reverse can be the case: you can id X but can't define X). I use the following examples:

  1. You might know gold has an atomic number of 79 but not know how to identify a gold watch.
  2. You might know the definition of a duck but not be sure whether some weird-goose (or maybe a duck) is a duck.

This takes us to our second issue: encouraging students to see that they need a test for validity. Consider this quote from American polymath Charles S. Peirce:

"Few persons care to study logic, because everybody conceives himself to be proficient enough in the art of reasoning already. But I observe that this satisfaction is limited to one's own ratiocination, and does not extend to that of other men." - Peirce

In other words, people tend to think that "logic" is important, and even that other people should improve their "ability to reason", but they think themselves to be sufficiently rational. They don't need to take a course in logic because they are more rational than most people.

Suppose the above is true and consider a student who (in our course) believes the above. The student may be able to conceptually distinguish the concept of validity from the method of identifying valid argument. However, even with this distinction, a student may think that they (1) always knew what "validity" was (but didn't know how to express in the "academic" way) and (2) they know how to identify good (valid) and bad (invalid) arguments. If they have this belief, then:

  1. They don't need to learn logic (even if everyone else does).
  2. They definitely don't need a new method for identifying valid / invalid arguments.

It is obvious that if you want students to learn, it is helpful to foster a will to learn. You can do this by making students curious about a subject but also by creating a dissatisfaction with their current state of knowledge. One way to accomplish both is to give students argument that they judge to be good but are actually bad (fallacies). The challenge is whether you can actually find one. I use two examples.

First, I ask students to answer this question quickly (Kahneman):

If a bat and a ball cost $1.10 and the bat costs $1 more than the ball, how much does the ball cost?

Second, I use this example (based on Wason):

  • P1: Some basketball players are millionaires.
  • P2: Some millionaries drive fancy cars.
  • C: Therefore, some basketball players drive fancy cars.

If you need more complex examples, you can use things like the Wason test or make use of different logic puzzles.

Informal tests for validity

So, a student might say:

"Ok. I have a better conceptual understanding of what makes an argument good or bad. I can distinguish truth from validity. I can define validity. But, I'm having a hard time of identifying valid arguments in every situation."

If the student is in this situation, you have primed them to reflect on informal methods for identifying validity.

I claim that people use two tests:

  1. Intuition test
  2. Imagination test

After defining these tests, then explaining why they don't work, the student will hopefully think like this:

"Same as before but now I'm thinking my method for determining whether an argument is good / bad is unreliable. Maybe I should find a better method."

And, if the student thinks the above, then you have set the stage for learning a formal language.

Ch2 at lightning speed

  • Two approaches to teaching PL:
    1. Bottom-up
    2. Top-down
  • PL is a formal language. The first logical language that students learn.
  • It consists of a set of symbols, syntax, semantics, and usually some methods for translating English into PL.
  • PL symbols (the alphabet): letters, operators, and parentheses
  • PL syntax (the grammar): how you take those symbols and put them together to form well-formed formulas (wffs).
  • PL semantics: how to give meaning to these wffs. Two functions! An interpretation function takes letters and assigns T and F to them. Valuation functions take truth values that are assigned to letters (or wffs) and use them to assign truth values to more complex wffs.
  • PL translation: How do we take English sentences that express propositions and translate them into PL?

Next Time

  1. Read Ch2, pp.41-98. I'll go through all of the material next time, so if you don't understand something, flag it!
  2. Quizzes: Complete Module 2 quizzes (I'll make these due on Wednesday evening).
  3. Question: How would you teach the same material? Even if you would mirror my approach, what are some other approaches you might imagine?
  4. Question: What part of the lesson did you find hardest to understand? What part do you think new students to logic would have the hardest part understanding?
  5. Suppose someone were to say "we can take all natural language reasoning and translate it into symbolic logic." To what extent is this true? To what extent is it false?
  6. Students always question translating "if P then Q" sentences (conditionals) into $P\rightarrow Q$. Several options are given in the textbook, but I usually only explain one of them. How would you go about answering this question?
  7. I always think about moving the section concerning "Complex translation" to the next chapter. What do you think?