Philosophy 110Introduction to Logic and Reasoning
Nicolas FillionAssistant Professor
Simon Fraser University
October 28th, 2013
Transition 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. . .
Transition 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. . .
Transition 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. . .
Transition 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. . .
Transition 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. . .
Transition 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. . .
Transition Systematizing things a bit
Return on the exam and upcoming assignments
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!
Transition Systematizing things a bit
Return on the exam and upcoming assignments
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!
Transition Systematizing things a bit
Return on the exam and upcoming assignments
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!
Transition Systematizing things a bit
Return on the exam and upcoming assignments
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!
Transition Systematizing things a bit
Return on the exam and upcoming assignments
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).
Transition Systematizing things a bit
Return on the exam and upcoming assignments
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).
Transition Systematizing things a bit
Return on the exam and upcoming assignments
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).
Transition Systematizing things a bit
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!
Transition Systematizing things a bit
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!
Transition Systematizing things a bit
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!
Transition Systematizing things a bit
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!
Transition Systematizing things a bit
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!
Transition Systematizing things a bit
Where we’re at, and where we’re going
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!
Transition Systematizing things a bit
Where we’re at, and where we’re going
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!
Transition Systematizing things a bit
Where we’re at, and where we’re going
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!
Transition Systematizing things a bit
Where we’re at, and where we’re going
Predicate logic is harder.
So, in preparation, we will tidy up what we have to make it easierto add to it.
Transition Systematizing things a bit
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]]]]∨]]] .
Transition Systematizing things a bit
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]]]]∨]]] .
Transition Systematizing things a bit
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]]]]∨]]] .
Transition Systematizing things a bit
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]]]]∨]]] .
Transition Systematizing things a bit
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]]]]∨]]] .
Transition Systematizing things a bit
Well-formed formulas
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).
Transition Systematizing things a bit
Well-formed formulas
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).
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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 step above, 4 + 4 is a natural number.Abbreviate this as 8.
So, by (2) and the steps above, 8 + 1 is a natural number.Abbreviate this as 9.
Transition Systematizing things a bit
Well-formed formulas
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 step above, 4 + 4 is a natural number.Abbreviate this as 8.
So, by (2) and the steps above, 8 + 1 is a natural number.Abbreviate this as 9.
Transition Systematizing things a bit
Well-formed formulas
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 step above, 4 + 4 is a natural number.Abbreviate this as 8.
So, by (2) and the steps above, 8 + 1 is a natural number.Abbreviate this as 9.
Transition Systematizing things a bit
Well-formed formulas
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 step above, 4 + 4 is a natural number.Abbreviate this as 8.
So, by (2) and the steps above, 8 + 1 is a natural number.Abbreviate this as 9.
Transition Systematizing things a bit
Well-formed formulas
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 step above, 4 + 4 is a natural number.Abbreviate this as 8.
So, by (2) and the steps above, 8 + 1 is a natural number.Abbreviate this as 9.
Transition Systematizing things a bit
Well-formed formulas
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 step above, 4 + 4 is a natural number.Abbreviate this as 8.
So, by (2) and the steps above, 8 + 1 is a natural number.Abbreviate this as 9.
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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.
Transition Systematizing things a bit
Well-formed formulas
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.
By (3), (W → (R ∨W )) is a wff.
By (3), (P ∨ (W → (R ∨W ))) is a wff.
By (3*), P ∨ (W → (R ∨W )) is a wff.
Governing (or main) operator
From this, we say that the governing (or main) operator is theone last introduced in the recursion.
Transition Systematizing things a bit
Well-formed formulas
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.
By (3), (W → (R ∨W )) is a wff.
By (3), (P ∨ (W → (R ∨W ))) is a wff.
By (3*), P ∨ (W → (R ∨W )) is a wff.
Governing (or main) operator
From this, we say that the governing (or main) operator is theone last introduced in the recursion.
Transition Systematizing things a bit
Well-formed formulas
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.
By (3), (W → (R ∨W )) is a wff.
By (3), (P ∨ (W → (R ∨W ))) is a wff.
By (3*), P ∨ (W → (R ∨W )) is a wff.
Governing (or main) operator
From this, we say that the governing (or main) operator is theone last introduced in the recursion.
Transition Systematizing things a bit
Well-formed formulas
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.
By (3), (W → (R ∨W )) is a wff.
By (3), (P ∨ (W → (R ∨W ))) is a wff.
By (3*), P ∨ (W → (R ∨W )) is a wff.
Governing (or main) operator
From this, we say that the governing (or main) operator is theone last introduced in the recursion.
Transition Systematizing things a bit
Well-formed formulas
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.
By (3), (W → (R ∨W )) is a wff.
By (3), (P ∨ (W → (R ∨W ))) is a wff.
By (3*), P ∨ (W → (R ∨W )) is a wff.
Governing (or main) operator
From this, we say that the governing (or main) operator is theone last introduced in the recursion.
Transition Systematizing things a bit
Well-formed formulas
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.
By (3), (W → (R ∨W )) is a wff.
By (3), (P ∨ (W → (R ∨W ))) is a wff.
By (3*), P ∨ (W → (R ∨W )) is a wff.
Governing (or main) operator
From this, we say that the governing (or main) operator is theone last introduced in the recursion.
Transition Systematizing things a bit
Well-formed formulas
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.
By (3), (W → (R ∨W )) is a wff.
By (3), (P ∨ (W → (R ∨W ))) is a wff.
By (3*), P ∨ (W → (R ∨W )) is a wff.
Governing (or main) operator
From this, we say that the governing (or main) operator is theone last introduced in the recursion.
Transition Systematizing things a bit
Sequents
We often have been writing abstract arguments as follows:
P → (Q → R), P & R ∴ Q
What does the symbol ∴ mean?
It means that we have made an inference from the premises tothe conclusion. It means “therefore.”
However, it does not tells us whether the premises followsfrom the conclusion (in this case, it doesn’t).
How could we say that? In words, we would just be adding “isformally valid”. But in symbols?
Transition Systematizing things a bit
Sequents
We often have been writing abstract arguments as follows:
P → (Q → R), P & R ∴ Q
What does the symbol ∴ mean?
It means that we have made an inference from the premises tothe conclusion. It means “therefore.”
However, it does not tells us whether the premises followsfrom the conclusion (in this case, it doesn’t).
How could we say that? In words, we would just be adding “isformally valid”. But in symbols?
Transition Systematizing things a bit
Sequents
We often have been writing abstract arguments as follows:
P → (Q → R), P & R ∴ Q
What does the symbol ∴ mean?
It means that we have made an inference from the premises tothe conclusion. It means “therefore.”
However, it does not tells us whether the premises followsfrom the conclusion (in this case, it doesn’t).
How could we say that? In words, we would just be adding “isformally valid”. But in symbols?
Transition Systematizing things a bit
Sequents
We often have been writing abstract arguments as follows:
P → (Q → R), P & R ∴ Q
What does the symbol ∴ mean?
It means that we have made an inference from the premises tothe conclusion. It means “therefore.”
However, it does not tells us whether the premises followsfrom the conclusion (in this case, it doesn’t).
How could we say that? In words, we would just be adding “isformally valid”. But in symbols?
Transition Systematizing things a bit
Sequents
We often have been writing abstract arguments as follows:
P → (Q → R), P & R ∴ Q
What does the symbol ∴ mean?
It means that we have made an inference from the premises tothe conclusion. It means “therefore.”
However, it does not tells us whether the premises followsfrom the conclusion (in this case, it doesn’t).
How could we say that? In words, we would just be adding “isformally valid”. But in symbols?
Transition Systematizing things a bit
Sequents
We would like a symbol that means
“therefore it follows necessarily that”.
This symbol is the turnstyle:
`
So, thanks to the knowledge we acquired, we know we could write
P → Q, ¬Q ` ¬P
Transition Systematizing things a bit
Sequents
We would like a symbol that means
“therefore it follows necessarily that”.
This symbol is the turnstyle:
`
So, thanks to the knowledge we acquired, we know we could write
P → Q, ¬Q ` ¬P
Transition Systematizing things a bit
Sequents
We would like a symbol that means
“therefore it follows necessarily that”.
This symbol is the turnstyle:
`
So, thanks to the knowledge we acquired, we know we could write
P → Q, ¬Q ` ¬P
Transition Systematizing things a bit
Sequents
But it would be wrong to rewrite the example from 2 slides earlierwith a turnstyle:
P → (Q → R), P & R ` Q
We can only use ` for formally valid arguments.
You can even intuitively think about ` as saying that an argumentis formally valid in symbols.
Transition Systematizing things a bit
Sequents
But it would be wrong to rewrite the example from 2 slides earlierwith a turnstyle:
P → (Q → R), P & R ` Q
We can only use ` for formally valid arguments.
You can even intuitively think about ` as saying that an argumentis formally valid in symbols.
Transition Systematizing things a bit
Sequents
Generally, consider an argument of the form
p, q, . . . , s ∴ t
We can writep, q, . . . , s ` t
if and only if there is a deduction with t as its conclusion andp, q, . . . , s as its premises.
Then we say that p, q, . . . , s ` t is a sequent.
Transition Systematizing things a bit
Sequents
Generally, consider an argument of the form
p, q, . . . , s ∴ t
We can writep, q, . . . , s ` t
if and only if there is a deduction with t as its conclusion andp, q, . . . , s as its premises.
Then we say that p, q, . . . , s ` t is a sequent.
Transition Systematizing things a bit
Sequents
Generally, consider an argument of the form
p, q, . . . , s ∴ t
We can writep, q, . . . , s ` t
if and only if there is a deduction with t as its conclusion andp, q, . . . , s as its premises.
Then we say that p, q, . . . , s ` t is a sequent.
Transition Systematizing things a bit
Theorems
In science in general, there is a type of statement that is especiallyimportant: theorems.
Theorems are supposed to hold generally, inany circumstances.
In logic, we can say it as follows: theorems are those statementsthat can be proved based on zero premises.
We will even say this:
Theorem
In logic, a theorem is a sequent with no premises.
If p is a theorem, we write:
` p
Transition Systematizing things a bit
Theorems
In science in general, there is a type of statement that is especiallyimportant: theorems. Theorems are supposed to hold generally, inany circumstances.
In logic, we can say it as follows: theorems are those statementsthat can be proved based on zero premises.
We will even say this:
Theorem
In logic, a theorem is a sequent with no premises.
If p is a theorem, we write:
` p
Transition Systematizing things a bit
Theorems
In science in general, there is a type of statement that is especiallyimportant: theorems. Theorems are supposed to hold generally, inany circumstances.
In logic, we can say it as follows: theorems are those statementsthat can be proved based on zero premises.
We will even say this:
Theorem
In logic, a theorem is a sequent with no premises.
If p is a theorem, we write:
` p
Transition Systematizing things a bit
Theorems
In science in general, there is a type of statement that is especiallyimportant: theorems. Theorems are supposed to hold generally, inany circumstances.
In logic, we can say it as follows: theorems are those statementsthat can be proved based on zero premises.
We will even say this:
Theorem
In logic, a theorem is a sequent with no premises.
If p is a theorem, we write:
` p
Transition Systematizing things a bit
Theorems
We have seen examples just before the midterm; in particular, wededuced
P ∨ ¬P
from no premises!
So, we can write` P ∨ ¬P
to say that P ∨ ¬P is a theorem.
Transition Systematizing things a bit
Theorems
We have seen examples just before the midterm; in particular, wededuced
P ∨ ¬P
from no premises!
So, we can write` P ∨ ¬P
to say that P ∨ ¬P is a theorem.