Philosophy 110 Introduction to Logic and Reasoning

Philosophy 110Introduction to Logic and Reasoning

Nicolas FillionAssistant Professor

Simon Fraser University

October 28th, 2013

Transition Systematizing things a bit

1 Transition

2 Systematizing things a bit

Return on the exam and upcoming assignments

Now that you wrote the midterm,you must be wondering what yourgrade was!

They will be posted on Canvas in aday or two. We’ve worked hard allweekend on grading!

Your TA will go over the details in your next tutorial. For now. . .

Know that the average is rather good! Around 75%.

Some people scored very well on deductions, others, not somuch. . .

If you’re in the latter group, it should ring a bell! It’s won’tget any easier in for the weeks to come. . .

But even if you did well, don’t becarried away!

We elected to grade in a verycharitable way! (Details in tutorials)

Do not expect us to grade socharitably on the final!

Now you know where you stand. If it went wrong, there are stillpossibilities for you.

If you failed, then you can still get a B or even better! Don’t bediscouraged! But get to work! And talk to me or your TA!

What’s next?

Firstly, assignment 4 is due this week on Friday before11:59pm. (Hint: it’s never the funny answer in the multiplechoices.)

Then two more assignments and participation in your tutorial.

And then the final exam, which is going to be mostly (butnot uniquely) about predicate logic (50% of the total grade).

What’s next?

Where we’re at, and where we’re going

What is this predicate logic thing anyway? Why do we need that?

Consider this example:

All Hipsters are anti-conformists.All hipsters are Posers.

Some Anti-conformists are posers.

Are there any truth-functional operators?No. So we have to symbolize it as

H,P ∴ A

which is invalid. But we shouldn’t conclude from that thatthis argument is formally invalid!

H,P ∴ A

Are there any truth-functional operators?

No. So we have to symbolize it as

H,P ∴ A

which is invalid.

But we shouldn’t conclude from that thatthis argument is formally invalid!

H,P ∴ A

The “problem” is that we only dealtwith operators connectingstatements.We didn’t look at the logic internalto statements.

So, we will need to zoom closer, tocapture logical terms such as all,some, is, etc., and capture relationsbetween predicates!

In this way, many arguments thatlook invalid from the restricted pointto view of statement logic will beshown to be valid!

Predicate logic is harder.

So, in preparation, we will tidy up what we have to make it easierto add to it.

Well-formed formulas

First, we recapitulate what our symbolic alphabet contains:

capital letters such as A,B,C, . . . ,X, Y, Z,A1, A2 . . .

logical operators ¬,∨,&,→,↔“groupers” (, [, {, ), ], }inference markers ∴ , `

From this alphabet, we can form all sorts of strings of symbolssuch as

P (→ &Q]]]]∨]]] .

logical operators ¬,∨,&,→,↔

“groupers” (, [, {, ), ], }inference markers ∴ , `

P (→ &Q]]]]∨]]] .

logical operators ¬,∨,&,→,↔“groupers” (, [, {, ), ], }

inference markers ∴ , `From this alphabet, we can form all sorts of strings of symbolssuch as

P (→ &Q]]]]∨]]] .

We have an analogue of the diagram we’ve seen for our formallanguage:

strings of symbols

well-formed formulas

The key here is that we have explicit rules telling us what is“grammatically correct” (i.e., what is a well-formed formula).

We have an analogue of the diagram we’ve seen for our formallanguage:

strings of symbols

well-formed formulas

The key here is that we have explicit rules telling us what is“grammatically correct” (i.e., what is a well-formed formula).

Before looking at the definition of wff, I want to tell you what kindof definition it is: it’s a recursive definition.

A recursive definition contains three things:

1 A base case: accepting axiomatically that some things satisfythe definition.

2 Recursive rules: rules that you can apply to generate morethings satisfying the definition.

3 A clause that nothing else satisfies the definition.

Example

Recursive definition of natural numbers

1 1 is a natural number.

2 If n and m are natural numbers, then n+m is a naturalnumber.

3 Nothing that can’t be obtained by (1) and (2) is a naturalnumber.

Example

Use the definition to show that 9 is a natural number.

By (1), 1 is a natural number.

So, by (2), 1 + 1 is a natural number. Abbreviate this as 2.

So, by (2) and the step above, 2 + 2 is a natural number.Abbreviate this as 4.

So, by (2) and the steps above, 8 + 1 is a natural number.Abbreviate this as 9.

Example

Well-formed formulas (wff)

1 Any capital letter A,B,C, . . . ,X, Y, Z,A1, A2 . . . is a wff.

2 If p is a wff, then ¬p is a wff.

3 If p and q are wf’s, then (p & q), (p ∨ q), (p → q), and(p ↔ q) are wf’s.

3* If (p) is a wff, then p is a wff (you can drop the outergroupers).

4 Nothing else is a wff.

Example

Using the recursive definition of wff, show thatP ∨ (W → (R ∨W )) is a wff.

By (1), P,R,W are wf’s.

By (3), (R ∨W ) is a wff.