Deductive Arguments
Phil 105, Ben Birkenstock
What is a philosophical argument?
A set of statements given to rationally show a certain conclusion to be true, or
likely to be true; A set of declarative statements, or propositions, which are claims
about how the world is, isn’t or might be, that can be true or false, to show another
statement to be (likely to be) true or false.
– A statement is a (true or false) description of how the world is
– The premises are the statements in an argument that present reasons to
believe the conclusion
– The conclusion is the statement that the argument is trying to support
What is a philosophical argument?
E.g.:
Premise 1: All humans are mortal. (All A’s = B’s)
Premise 2: Kongzi is human; (C = A)
Conclusion: Therefore, Kongzi is mortal. (C = B)
Arguments – Basics
– Can be either sound or unsound, that is they can either provide
good reasons to believe the conclusion is true, or not.
– Two factors to an argument: are premises true, and if true, do
they support the conclusion?
– In other words, in order to be sound, an argument needs:
– 1. A good structure
– 2. True premises.
Arguments – Basics
A successful argument is NOT true, or valid: it is SOUND
– A premise or conclusion (a statement) is true or false
– An argument structure is valid or invalid
– Evidence is strong or weak
– An argument is sound or unsound
Arguments – Basics
Every time we encounter someone giving reasons to believe something , they are
making an argument
– The ability to analyze the assumptions we find in and around ourselves,
– & the ability to discern how it is that assumptions are presented to us as true,
and for us to be able to arrive at true conclusions
– We are doing philosophy all the time!
Watchwords: ‘because, either, if, since, all, none, always, never, often’
When you hear these, someone is probably making an argument!
Arguments – Structure
The structure of an argument is the organization of its premises so
that they relate to each other in a way that supports (or fails to
support) the conclusion.
– If an argument is not structured well, it may have true premises
that do not provide any evidence for the conclusion:
P1. All humans are mortal;
P2. Kongzi is mortal,
C. Therefore, Kongzi is human. (Sound or unsound?)
Arguments – Structure
Two broad types of argument structures: Inductive and deductive
– Deductive arguments aim at validity: they try to be valid in their structure;
try to convince us that their conclusion is certain, is true.
– does not provide new info beyond what is described in the premises
– Inductive arguments aims at coherence: they try to convince us that their
conclusion is probable.
– gives us new information, the premises provide reasons to believe something about the
world that they don’t already tell us. I.e., the conclusion goes beyond the premises
Deductive Arguments – Validity
Deductive arguments have a valid structure:
– If their premises are true, then their conclusion has to be true
In other words, a valid argument is an argument in which there is no way for the
conclusion to be false if the premises are true.
These Are Valid Arguments
(…But are they sound?)
Argument #1
P1. All humans are mortal
P2. Kongzi is human
C. Therefore, Kongzi is mortal.
#2
P1. All humans can fly
P2. Kongzi is human
C. Therefore, Kongzi can fly.
#3
P1. I am on TWU’s campus
P2. TWU’s campus is in Canada
C. Therefore, I am in Canada
#4
P1. I am the Monkey King
P2. The Monkey King grew up in China
C. Therefore, I grew up in China
Testing for Validity vs Soundness
We can tell that an argument can have a valid structure without being sound.
Analyze these arguments. Are they valid? Sound?
Argument #1
P1. All apples are fruits
P2. All fruits come from plants
C. Therefore, all apples come from plants.
#2
P1. All apples are fruits
P2. Some fruits grow on trees
C. Therefore, some apples grow on trees.
#3
P1. If the sky is blue, then I am human
P2. The sky is blue
C. Therefore, I am human
Deductive Arguments – Validity
One way to write out arguments is to use letters to represent the content:
P1. All H are M
P2. K is H
C. Therefore, K is M.
Letters are just a way of standing in for the same statement/content in an
argument’s structure.
Deductive Arguments – Validity
It is very important when making an argument that you keep the same content
exactly the same in your arguments!
P1. All humans are mortal
P2. Kongzi is a wise teacher
C. Therefore, Kongzi is mortal (valid?)
Try writing this argument symbolically to test its validity…
(with letters representing the content)
Deductive Arguments – Validity
P1. All H are M
P2. K is WT
C. Therefore, K is M
P1. All humans are mortal
P2. Kongzi is a wise teacher
C. Therefore, Kongzi is mortal
Are all wise teachers human?
Deductive Arguments – Validity
It is very important when making an argument that you keep the same content
exactly the same in your arguments!
P1. All humans are mortal
P2. Kongzi is a human
C. Therefore, Kongzi’s teaching is mortal.
Let’s test the validity by writing out the argument symbolically…
Deductive Arguments – Validity
P1. All humans are mortal
P2. Kongzi is a human
C. Therefore, Kongzi’s teaching is
mortal (no longer helpful).
P1. All H are M
P2. K is H
C. Therefore, KT is M.
Does Kongzi’s teaching have all the same qualities as Kongzi?
Deductive Arguments – Modus Ponens
As we saw, some premises contain two concepts (“all H are M”).
A “conditional premises” is a premise that unites two statements in a
specific–“conditional”– relationship, where:
– If one statement (the antecedent) is true, then the other sentence (the
consequent) is also true.
– Therefore, a conditional is also known as an “if-then” statement:
– If Kongzi is human, then he is mortal
– If I am at TWU, then I am in Canada
– “if p, then q”
Deductive Arguments – Modus Ponens
The structure of a modus ponens argument, then, is as follows:
P1. If p, then q
P2. P;
C. Therefore, q
(Remember, the letters must always stand for the same content)
P1. If Kongzi is human, then he is mortal.
P2. Kongzi is human;
C. Therefore, Kongzi is mortal.
Testing for Validity and Soundness in Modus Ponens
Modus Ponens is a valid argument structure. If we follow the modus ponens
structure exactly when we construct an argument, then we know that we have a
valid deductive argument. (Remember, modus ponens is just one type of deductive
argument.)
We know a modus ponens argument is sound if we know for sure that BOTH
premises are true.
If we aren’t sure that EITHER premise is true, then we don’t know if the modus
ponens argument is sound.
Testing for Validity and Soundness in Modus Ponens
As mentioned, an argument may be valid without being sound, that is, a structure may be valid
without the content of the reasons being true.
Analyze the following valid modus ponens arguments. Are they sound?
P1. If Covid is not lethal, then it has not killed anyone.
P2. Covid is not lethal;
C. Therefore, Covid has not killed anyone.
P1. If covid is a pandemic, then it kills most of the population.
P2. Covid is a pandemic.
C. Therefore, Covid kills most of the population..
Testing for Validity and Soundness in Modus Ponens
Analyze the following deductive arguments. Are they valid?
P1. If you are at TWU Richmond, then you are in Canada
P2. You are in Canada,
C. Therefore, You are at TWU Richmond.
P. 1 If you are at TWU Richmond, then you are in Canada,
P. 2 You are not at TWU Richmond,
C. Therefore, you are not in Canada.
Try writing them out their
structures using letters to
help you analyze their
validity
Testing for Validity and Soundness in Modus Ponens
Symbols help us to see where the same content is being used to convince us of a conclusion.
If we fill in an argument with letters as placeholders, we will be able to tell whether it accurately
follows a valid structure or not (in this case modus ponens)
Another way of testing validity: is it possible for premises to be true and conclusion false? If so,
this argument is not valid.
Or (If we think that it probably is valid) showing that if the the conclusion were false, at least one
premise would have to be false; this means argument is valid
Modus Tollens
The last fallacy uses something called a negation, the opposite of a statement,
symbolized as not p or not q.
This last invalid structure raises the question of whether inserting a negation into
modus ponens can lead to a valid argument structure. It can, and in fact has a
name of its own: Modus Tollens
P1. If p, then q
P2. Not q,
C. Therefore, not p.
P1. If you are at TWU, then you are in Canada,
P2. You are not in Canada,
C. Therefore, you are not at TWU.
Testing for Validity and Soundness in Modus Tollens
Argument #1
P1. If Kongzi is mortal, then Kongzi is human.
P2. Kongzi is not human;
C. Therefore, Kongzi is not mortal.
#2
P1. If angels are human, then angels are mortal
P2. Angels are not mortal;
C. Therefore, angels are not human
#3
P1. If you are at TWU Rmd, then
you are in Canada
P2. You are not at TWU Rmd;
C. Therefore, you are not in
Canada.
Can an argument be sound without being valid?
Only if it is NOT a deductive argument!
Remember, deductive arguments are arguments that try to have a valid structure,
whose premises guarantee the truth of the conclusion
– Even if a deductive argument argument has a true conclusion statement, if it’s
structure is not valid, it’s premises do not support to its conclusion logically
(see previous slide, first example)
– Therefore, a deductive argument needs both a valid structure and true
premises to be sound
Can an argument be sound without being valid?
A non-deductive (inductive) argument tries to have a structure that supports its
conclusion, whose premises make the truth of the conclusion more probable
– For example:
P1. 30% of 2 million randomly surveyed Canadians are Social Democrats
C. Therefore, 30% of all Canadians are Social Democrats
– In order to be sound, an inductive argument needs true, or probably true
premises, that support its conclusion.
– So, while it needs a logical structure, it does not need a valid structure
Conclusion: Deductive Arguments in daily life
Modus ponens arguments are often hidden in everyday language. While we hear
conditionals all the time – “if it rains tomorrow, I won’t play tennis;” “If motion
exists, there must be an unmoved mover” – these often do not add up to a full
modus ponens argument. Instead, we often have to listen for “causal” words such
as “because,” “since,” “so,” and “therefore,” which sometimes indicate a modus
pollens/tollens argument.
Often, the antecedent and consequent are in reverse order from how you will set
them up when you put them in modus
ponens form
– the word “because,” for example, indicates an antecedent statement, so when
it is in the middle of a sentence, that sentence often has an
antecedent following a consequent…
Conclusion: Deductive Arguments in daily life
It’s difficult to play tennis outdoors in the rain, because the wet ground makes tennis balls bounce
irregularly.
P1. If wet ground makes tennis balls bounce irregularly, then it is difficult to play tennis outdoors
in the rain
P2. The wet ground does make tennis balls bounce irregularly;
C. Therefore, it is difficult to play tennis outdoors in the rain.
Things cannot move themselves; so there must be an unmoved mover.
P1. If things cannot move themselves, there must be an unmoved mover.
P2. Things cannot move themselves;
C. Therefore, there must be an unmoved mover.
Conclusion: Deductive Arguments in daily life
Try to label the following statements as p (for the antecedent) and q (for the
consequent) and then structure them in Modus ponens or tollens form:
Since I cannot focus on zoom, I would prefer to come to class in person
I would rather study from home; so, I do not prefer to study in person.
Because the Russian people do not all agree with Putin, the Russian military
situation is unstable.
I think war is usually cruel, meaningless, and avoidable because my thinking is
influenced by Laozi.
Conclusion: Deductive Arguments in daily life
How are reasons presented for conclusions all around us? How do people,
intentionally or not, present their evidence as supporting their position?
Deduction, with its high standard of absolute truth-retention, is rarely successful in
defending common assumptions and conclusions in popular debates. But it is a
useful tool both in showing what would be needed to defend an argument, as well
as a structure for creating careful arguments.
Finally, the complexity of validity – and philosophical argumentation in general – is
a strong reminder that we are often likely to be unaware of
our own ignorance and bias precisely when we are most sure
about our opinion
