Constructive Dilemma - is it inclusive or exlusive disjunction?

[MathHelp]

Hi team,

Really struggling with this - even though it seems like it should be obvious.

Constructive Dilemma:

Quote

 

(if p then q) and (if r then s)

p or r

Therefore:

 q or s

 

I can't tell if the "p or q" should be exclusive (so either p or q) or if it is actually inclusive.

I suspect it is an inclusive disjunction:

Quote

 

(If I get money for christmas then I will go on holiday) and (if i get time off work on New Years then I will go to a party) 

I get money for christmas and/or I get time off work on New Year (both are technically possible)

Therefore:

I will go on holiday or I will go to a party (both are technically possible)

 

The main point of uncertainty is while I have just shown an example with constructive dilemma where I used inclusive disjunctions, I am not sure if whether the conclusion might create problems in the larger scheme of things. Logical arguments can obviously be a string of different rules of inference and as I learn more I am wondering I will learn a rule of inference that connects to the constructive dilemma rule but does not work with my example

[Lorentz Jr]

22 minutes ago, MathHelp said:

I can't tell if the "p or q" should be exclusive (so either p or q) or if it is actually inclusive.

I suspect it is an inclusive disjunction:

It is. Incompatibility between p and q doesn't imply anything about r or s.

If silence is golden, then 1 + 1 = 2.

If silence is not golden, then 2 + 2 = 4.

In formal logic, disjunctions are always inclusive. Exclusive disjunctions may be written as "xor". The only ambiguity is in informal conversation.

Edited December 16, 2022 by Lorentz Jr

[studiot]

Can I recommend EJ Lemmon's book

Beginnibng Logic.

It is good because Lemmon goes into both the formal logic and maths way of presenting statements and compare them.

His style is very clear and easy to follow.

[MathHelp]

I am still working through my current textbook that actually explains deductive very well. There has been only one part of the book where I think they could have written things better - which was the explanation of the addition rule of implication.

However, the book does not provide as much information on inductive logic as I would like. Do you know any good books that cover inductive logic well?

Specifically wanting to know more about arguments based on signs and hypothetical reasoning. As far as I can tell the book simply mentions that there is a type of induction based on signs. With hypothetical reasoning it makes the distinction between empirical hypothesises and theoretical hypothesises and gives criteria for tentative acceptance of a hypothesis as adequacy, internal consistency, external consistency, and fruitfulness. Does really help to improve these two methods of reasoning much.

[studiot]

11 hours ago, MathHelp said:

I am still working through my current textbook that actually explains deductive very well. There has been only one part of the book where I think they could have written things better - which was the explanation of the addition rule of implication.

However, the book does not provide as much information on inductive logic as I would like. Do you know any good books that cover inductive logic well?

Specifically wanting to know more about arguments based on signs and hypothetical reasoning. As far as I can tell the book simply mentions that there is a type of induction based on signs. With hypothetical reasoning it makes the distinction between empirical hypothesises and theoretical hypothesises and gives criteria for tentative acceptance of a hypothesis as adequacy, internal consistency, external consistency, and fruitfulness. Does really help to improve these two methods of reasoning much.

I already gave you a 'road map' to explore induction v deduction in your other thread.

However since you are still hazy here is another road map for you to follow.

 

As can bee seen there are several meanings to the terms induce and induction, just within Mathematics and Philisophy/Logic.

At the bottom of the right hand page (291) there is a more general term  - inference which is used even more widely than induction
I would add imply as well.

Which brings me to
Scientific Inference
H Jeffreys
Cambridge University press.

Here are a couple of extracts about induction more widely that are worth reading.

 

The theory of what is called first order induction, as in the first attachment above, will be found in Lemon.

The full theory, including the so called induction axioms, of what is called second order or probabilistic induction will be found in

Computability and Logic

Boolos and Jeffrey  (no s at the end this time)

Cambridge University press

A further type of induction  -  transfinite induction, will be found in any good number theory or foundations of mathematics book eg Stewart and Tall