Deductive reasoning and hypothesis

[musicgold]

TL;DR Summary

I know how deductive reasoing works (syllogism). I know how hypothesis are made and tested.
But I can't seem to understand how these two are linked.

Most references use a syllogism like the one below to explain deductive reasoning. They also say that deductive reasoning uses hypotheses to start with a general statement and examines the possibilities to reach a specific conclusion.

all dogs are animals; all animals have four legs; therefore all dogs have four legs.

1. I haven't found an example of deductive reasoning using a hypotheis. Could you give me an example?
2. How can I use an hypothesis in the case of a syllogism?

Thanks

 

[Simon Bridge]

Hypothesis testing is one way to verify the truth of the presuppositions in a syllogism.
Remember the conclusion is only true if the presups are true and the logic is valid.

So all deductive reasoning use hypotheses, and produces one.
It's how you choose what to test next.

ie. Say we have defined animals to have four legs, we want to know if all dogs are animals... we deduce:

1. all dogs are animals
2. all animals have four legs
3. therefore, all dogs have four legs

Then test 3
ie. sample dogs to see how many legs they have ... if we find a 3-legged dog, then not all dogs are animals.
note: if we don't find any dogs with other than 4 legs, the evidence is inconclusive (see: the problem of induction.)

This is how we go about getting indirect evidence in science.

 

[Stephen Tashi]

Summary:: I know how deductive reasoing works (syllogism).

A syllogism is one method of deductive reasoning, but not the only one.

I know how hypothesis are made and tested.

The word "hypothesis" can have different meanings.

In mathematical logic if a principle of deduction begins with certain "given" infomation, that information is called the "hypothesis".

In the so-called scientific method, a speculation about how a physical phenomena works is called a "hypothesis".

In statistics, an assumption that a certain probability distribution applies is called a "null hypothesis" and procedures for evaluating the assumption are called "hypothesis testing".

There is no definite relation among these three uses of the word "hypothesis". A hypothesis in logic need not be a statement about science and statistical hypothesis testing is not a form of mathematical deduction.

For example, the hypothesis of a mathematical theorem could be "Given right triangle ABC with angle C = 30 degrees". This type of hypothesis is not a hypothesis about a particular natural phenomena.

The "null hypothesis" for a statistical test might be "A randomly chosen male is just as likely to have high blood pressure as a randomly chosen female". This hypothesis does concern a physical phenomenon, but a statistical test of it involves assuming the null hypothesis and computing the probability of observed data given that assumption. Based on the value of that probability a subjective decison is made about whether to reject the null hypothesis. This subjective decision is not an application of a syllogism or other form of mathematical deduction.

 

[Svein]

In mathematical logic the expression $A\Rightarrow B$ ("if A then B") is equivalent to $\sim A\vee B$ ("either A is wrong or B is right").

Even worse - in axiomatic propositional calculus the "implies" is the basis of the axioms which are:

1. $(A\supset (B\supset A))$
2. $(A\supset (B\supset C))\supset ((A\supset B)\supset (A\supset C)))$
3. $((\thicksim B\supset\thicksim A)\supset ((\thicksim B\supset A)\supset B))$

 

[Simon Bridge]

Stephen Tashi wrote:
"The word "hypothesis" can have different meanings."

... to be clear,I am deducing from context that "hypoithesis"in OP refers to "hypothesis testing" in applied math, specifically as taught at the senior secondary or starting tertiary level. Could be wrong.

I guess, still waiting for OP to give feedback.

 

[Mark44]

Summary:: I know how deductive reasoing works (syllogism). I know how hypothesis are made and tested.
But I can't seem to understand how these two are linked.

all dogs are animals; all animals have four legs; therefore all dogs have four legs.

to be clear,I am deducing from context that "hypoithesis"in OP refers to "hypothesis testing" in applied math, specifically as taught at the senior secondary or starting tertiary level. Could be wrong.

I think that "hypothesis testing" is an unintended choice of words. I believe the context here is deductive reasoning, and not how the phrase is normally used in inferential statistics.

In the italicized text, the hypotheses are "all dogs are animals" and "all animals have four legs." The hypotheses are the givens that we assume to be true*. The conclusion is "all dogs have four legs."
Given that we accept that the hypotheses are true, the conclusion necessarily must follow. This can be shown in a variety of ways, such as by the rules of symbolic logic or more simply, by the use of a Venn diagram.

*I should add that one of the hypotheses that we are assuming to be true is clearly not true. Chickens and shrimp are animals, but they don't have four legs.

 

[musicgold]

A syllogism is one method of deductive reasoning, but not the only one.
The word "hypothesis" can have different meanings.

In mathematical logic if a principle of deduction begins with certain "given" infomation, that information is called the "hypothesis".

In the so-called scientific method, a speculation about how a physical phenomena works is called a "hypothesis".

In statistics, an assumption that a certain probability distribution applies is called a "null hypothesis" and procedures for evaluating the assumption are called "hypothesis testing".

There is no definite relation among these three uses of the word "hypothesis". A hypothesis in logic need not be a statement about science and statistical hypothesis testing is not a form of mathematical deduction.

For example, the hypothesis of a mathematical theorem could be "Given right triangle ABC with angle C = 30 degrees". This type of hypothesis is not a hypothesis about a particular natural phenomena.

The "null hypothesis" for a statistical test might be "A randomly chosen male is just as likely to have high blood pressure as a randomly chosen female". This hypothesis does concern a physical phenomenon, but a statistical test of it involves assuming the null hypothesis and computing the probability of observed data given that assumption. Based on the value of that probability a subjective decison is made about whether to reject the null hypothesis. This subjective decision is not an application of a syllogism or other form of mathematical deduction.

This is a great explanation. Thank you.

You are absolutely right; I was confusing myself with the use of word hypothesis. I was trying to see if deductive reasoning uses hyoptheis testing (as used in statisics) to verify the premises or conclusion of a syllogism.

Looks like, in deductive reasoning the word 'hypothsis' means the starting theory.
I found some additional explanation here.
https://www.scribbr.com/methodology/inductive-deductive-reasoning/

Just to complete the loop, is it fair to say that in the case of a syllogism, the premises are the starting hypoteses.

 

[Mark44]

Just to complete the loop, is it fair to say that in the case of a syllogism, the premises are the starting hypoteses.

Yes. As I mentioned already, the hypotheses, or premises, are the given information, and are assumed to be true.