propositional

10/12/26

Skills to be acquired in this tutorial:

To learn how compound sentences are true or false depending on the truth or falsity of their component sentences.

There is the idea of setting up a code or convention or dictionary
between atomic sentences and capital letters.

There are compound sentences, each of which has a main connective
which connects its components.

There are five sentential logical connectives:

'∼' which translates back to 'it is not the case that...'

'.' which translates back to '... and ...'

'∨' which translates back to '... or ...'

'⊃' which translates back to 'if... then ...'

10/12/26

Skills to be acquired in this tutorial:

Symbolizing compound sentences. Learning about logical connectives, and the notion of the main connective. Recognizing different constructions in English which have the same underlying logical form. Paraphrasing the English into a standard form.

Why this is useful:

It is the next step in learning how to symbolize. Main connectives are very important-- they are central to symbolization, they are central to the semantics, and they are central to derivations.

10/12/26

Skills to be acquired in this tutorial:

To become familiar with the notions of argument, valid, invalid, premise, and conclusion. To learn how to symbolize atomic sentences.

Reading

A. Hausman, H.Kahane, P.Tidman, [2007] Logic and Philosophy Chapters 1, 2.1

Tutorial:

The main role of logic is to assess arguments-- to say whether an individual argument is valid or whether it is invalid. In logic, arguments are taken to consist of two components--premises, and a conclusion.

For example,

This program provides an introduction to logic to the level of Sentential and Predicate Calculus.

The focus of the program is on arguments and the question of whether they are valid. Arguments have the form <list of premises> ∴<conclusion>. An argument is valid if and only if it is not possible for all its premises to be true and its conclusion false at one and the same time; an argument which is not valid is invalid.

Tutorials 1 to 10: sentential logic, using the Hausman syntax.

10/12/26

You now have to tools to appraise propositional arguments.

Let us run through how these might be used with two examples.

Example 1.

Consider the argument

If no human action is free, then no one is responsible for what they do.
If no one is responsible for what they do, no one should be punished.
Therefore
If no human action is free, no one should be punished.

First it should be symbolized

10/12/26

Skills to be acquired

Becoming familiar with common inference patterns and being able to use them via replacement rules. This helps with assessing ordinary everyday reasoning such as that found in the law, in newspapers, in advertisements, etc.

10/12/26

Skills to be acquired:

Learning conditional proof.

The Tutorial:

There is an additional propositional rule of inference-- Conditional Proof-- which is slightly more difficult than the ones that we have met before. It is slightly more difficult in that it requires you to make new assumptions, and the correct new assumptions at that.

10/12/26

1. Modus Ponens (MP)

p⊃q,
p
∴ 
q

2. Modus Tollens (MT)

p⊃q,
~q
∴ 
~p

3. Hypothetical Syllogism (HS)

p⊃q,
q⊃r
∴ 
p⊃r

4. Disjunctive Syllogism(DS)

p∨q,
~p
∴ 
q

5. Constructive Dilemma (CD)

(p⊃q).(r⊃s),
p∨r
∴ 
q∨s