PHIL1110 Study Guide - Final Guide: Argument Map, Deductive Reasoning, Propositional Calculus
Logic
CHAPTER 7
Logic 1
7.1. Validity and Soundness
In our last lecture, we completed our introduction to argument mapping. While argument mapping can be very
useful, it has some deep limitations. We were able to get a clearer idea of the structure of an argument, but
• argument mapping is completely silent on the nature and quality of support. In the next two lectures we are
going to focus on a particular kind of support: deductive
support.
Let’s look at an example to get started.
If both of the premises are true, then the conclusion is forced to be true.
• This doesn’t mean that that the premises are true, we’re just saying what would
happen if they were true. The support here is deductive.
• Moreover, the support works and this give us a valid argument. Let’s review/give our proper definition of
validity.
Definition 24. An argument is valid, if it is necessary that whenever the premises are all true, then so is the
conclusion.
We can also put it another way:
Proposition 25. An argument is valid if and only if it is impossible for the premises to
be true and the conclusion false.
Question: Can you explain why the other definition also works? Here’s a clue:
68
7.1. VALIDITY AND SOUNDNESS 69
• It’s necessary that ... is true. means the same thing as
• It’s not possible that ... is not true.
Remark: We can see that the use of the terms, “necessary” and “possible” have allowed
us to make our idea of forcing the conclusion to be true more precise. But what kind of necessity are we talking
about here?
Example: Is the following argument valid?
(1) Sophie is a person.
(2) Therefore, Sophie is not a robot.
Remark: No, but booking at our definition, I can see why one might be tempted to say that it it.
• Let’s try to imagine a possibility where the premises are true and the conclusion is false.
• This would be a possibility where Sophie is a person and Sophie is a robot.
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• This doesn’t seem like a real possibility, so it could seem that the argument is
valid.
Remark: No, but booking at our definition, I can see why one might be tempted to say that it it.
• Let’s try to imagine a possibility where the premises are true and the conclusion is false.
• This would be a possibility where Sophie is a person and Sophie is a robot.
• This doesn’t seem like a real possibility, so it could seem that the argument is
valid.
Question: What has gone wrong here?
Answer: The notion of possibility we’re using here is too narrow.
• So we’ll want to consider possibilities where Sophie is a person and a robot!
• That might seem a little confronting. In fact, we’ve already talked about this.
• Recall my Defense of Nonsense.
• I argued that the use of nonsense allowed us to see what was good in the structure
of a deductive argument.
• We saw that the argument structure remained good regardless of the meaning
of the terms used.
7.1. VALIDITY AND SOUNDNESS 70
• This is the notion of possibility we need here: possible regardless of the meanings of the parts.
But which parts?
• Over the next two weeks, you’ll develop a clearer idea of this.
One might also worry that I’m saying that valid argument are – in some sense – good, but they could have false
conclusions.
For example:
(1) The moon is a satellite.
(2) If the moon is a satellite, then it is made of cheese. (3) Therefore, the moon is made of cheese.
This is a valid argument.
But the conclusion is false. So do we really want to say it’s good.
This brings us to our second important definition regarding deductive arguments:
Definition 26. An argument is sound if it is valid and the premises are true. Remark. So a sound argument is
good in the very simple sense that you ought to accept
its conclusion.
• But in logic, we aren’t generally that interested in soundness. Why?
• Because we’re interested in studying what in the structure of the argument
makes it good.
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• This is the subject matter of logic (as a study of reason).
• We’ll leave the truth of the premises to the experts in other fields.
Remark. We don’t have to worry about valid arguments with false conclusions. • Such argument
always have a false premises.
• So there’s not compulsion to accept the conclusion.
7.1.1. Exercises. Answer the following questions (take your time, they’re tricky):
1. (1) Can an unsound argument have true premises and a true conclusion?
2. (2) Can a valid argument have a true conclusion and false premises?
3. (3) If an argument has premises which cannot all be true and a false conclusion, is
it valid?
7.2. IMPROVING ARGUMENT MAPPING 71
7.2. Improving argument mapping
Let’s recall some examples of common moves and limitation in argument maps.
We’ll then spend the rest of the next lectures showing how logic can do a better job in these cases.
One common question about argument mapping is about what do when we have a propo- sition like the
following:
We should invest in transport infrastructure and we should have higher taxation.
We might represent this as follows in an argument map:
But it’s pretty obvious that we could also think of this as two different propositions. So instead we might write:
Question: Which one is right?
Remark: In some sense it doesn’t matter. This is because the following hold:
Here’s another issue. Recall how we usually resolve the issue of suppressed premises. We start with an
argument like the following:
7.3. THE LANGUAGE OF PROPOSITION LOGIC 72
and then add a premise:
This raises a couple of questions:
(1) Why makes this support work?
(2) How could we defend this new premise?
We’ll now develop some logical tools to answer these questions. 7.3. The language of proposition logic
This should hopefully give us some motivation for the development of logic. In the rest of this lecture, I’m
going to introduce propositional logic.
(1) I’ll start by setting up the language.
(2) Then we’ll show how to do some simple proofs in this system.
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Document Summary
In our last lecture, we completed our introduction to argument mapping. While argument mapping can be very useful, it has some deep limitations. We were able to get a clearer idea of the structure of an argument, but: argument mapping is completely silent on the nature and quality of support. In the next two lectures we are going to focus on a particular kind of support: deductive support. Let"s look at an example to get started. If both of the premises are true, then the conclusion is forced to be true: this doesn"t mean that that the premises are true, we"re just saying what would happen if they were true. The support here is deductive: moreover, the support works and this give us a valid argument. An argument is valid, if it is necessary that whenever the premises are all true, then so is the conclusion.
