• Inductive reasoning
• Deductive reasoning
• Abductive reasoning

• Inductive reasoning:
  • rules from observations or experiences;
  • generalisation;
  • poor at truth preserving;
  • can lead to undesirable outcomes

• Every time I eat spicy food, I get a stomach-ache.
  Therefore, spicy food causes stomach aches (for me).


• Every sheep I have seen is white.
  Therefore, all sheep are white.

• Deductive reasoning:
  • developing rules based on already known rules;
  • conclusions narrower than underlying rules;
  • easier to correct;
  • starting point — where do we get assumptions?

• All 🐕 dogs have ears, golden retrievers are dogs,
  therefore they have ears.
• All mammals have backbones. Humans are mammals. Therefore, humans have backbones.

• All 🐈 cats can fly. Whiskers is a cat.
  Therefore, Whiskers can fly. 🤔

• Abductive reasoning:
  • is reasoning to the best explanation;
  • It starts from observations or evidence and proposes a plausible hypothesis that would explain them;
  • The conclusion is not guaranteed to be true — it is tentative and revisable.
  • Abductive reasoning contrasts with:
    • Deductive reasoning: guarantees truth if premises are true.
    • Inductive reasoning: generalises from repeated cases.

• While I was reading in bed last night, I started hearing scratching sounds coming from the ceiling. Then, this morning, I noticed there were tiny little droppings on the kitchen floor. I'm pretty sure we have a mice infestation at our house.


• You walk outside in the morning and see the ground is wet. The sky is clear and there is no rain forecast. You conclude that it probably rained overnight.

• Fallacy is a false belief
• Formal/logical fallacy
• Informal fallacy

Example:

    If it is raining, the ground will be wet.

    The ground is wet.

    Therefore, it must be raining.

• A implies B: A ⇒ B
• If A is true, then B is true as well.
• Example:

  If it is raining, then the ground is wet.

We can chain implications:

• If A ⇒ B and B ⇒ C, then A ⇒ C.
• Example:

  If it is raining, then the ground is wet.

  If the ground is wet, then Monty does not want to go out.

  Hence, if it is raining, then Monty does not want to go out.

Consider the following statements:

• To be President of the US, you must have been born in the US.
• Anyone born in the US is an American citizen.

Are the following statements true?

• Any American citizen can become President of the US?
• Any President of the US is an American citizen?

• A ⇒ B means A is a sufficient condition for B.
• If A is true, B must be true.
• But B true does not imply A true.
• Example:

  If I live in Gold Coast, then I live in Australia.

• B ⇒ A means A is a necessary condition for B.
• B cannot be true unless A is also true.
• Example:

  If you pass the exam, then you had sat the exam.

• If A is both necessary and sufficient for B: A ⇔ B
• A ⇔ B can be also written as A iff B (iff short for if and only if)
• Example:

  A quadrilateral is a square ⇔ it has four equal sides and four right angles.

  A quadrilateral has four equal sides and four right angles ⇔ it is a square.

Relationships between statements A and B:

• A necessary but not sufficient for B.
• A sufficient but not necessary for B.
• A equivalent to B (A is both necessary and sufficient for B).
• A neither necessary nor sufficient for B.

• A ⇒ B is equivalent to not B ⇒ not A (\(\lnot B \Ra \lnot A\)).

Example:

Consider the statement: When it is raining, the ground is wet.

Which of the following is logically consistent with this statement?

• When the ground is wet, it is raining
• When it isn't raining, the ground isn't wet
• When the ground isn't wet, it isn't raining.

If it is true, then it must be false;

if it is false, then it must be true.

Therefore, it cannot be classified as either true or false.

If it is true, then it must be false;

if it is false, then it must be true.

Therefore, it cannot be classified as either true or false.

The sentence:

This statement is FALSE

is a self-referential statement.

It talks about its own truth value.

Classical logic assumes:

• Every statement is either TRUE or FALSE.
• A statement cannot be both.

But this sentence breaks that rule.

It creates a contradiction.

The paradox dates back to ancient Greece.

It is often attributed to Epimenides of Crete (~6th-century BC).

He reportedly said:

“All Cretans are liars.”

If he was Cretan, was he telling the truth?

The Liar Paradox influenced modern logic.

It relates to:

• Self-reference
• Formal systems
• Limits of mathematics

A 🐊 crocodile steals a child from a mother on the banks of the Nile. When the mother begs for the child's return, the crocodile makes a deal:

"If you correctly predict what I will do with the child, I will return it. If you predict incorrectly, I will eat it."

The mother replies:

"You will eat my child."

• If you use software, say what you used (e. g. Excel, WolframAlpha, Python, etc. ).
• If you use built-in functions from a specific software, state which ones and on what data.

Example: "I have used STDEV.S function on the number of possums data to find the standard deviation."

• If you use built-in functions from a specific software, state which ones and on what data.

If you use a programming language to make some computations, you can comment it in your document and add the full code as an Appendix.

x = (0:0.25:1);
for j = 1:100
   t = j/25;
   z = sin(pi * x) * cos(2 * pi * t);
   plot(x, z); axis([0,1,-1,1]);
   M(j) = getframe;
end