Hausman[2007] Logic and Philosophy Chapter 8
The sentential rules of derivation carry over unchanged into Predicate Logic
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Hausman[2007] Logic and Philosophy Chapter 8
A few concepts are needed give a simple portrayal of the truth and falsity of predicate logic formulas.
There is the notion of an Interpretation which consists of a Universe together with an account of how the various symbols in the predicate logic formulas apply in this Universe.
There should be a Universe, which is the collection of the objects that the formulas is about. We write, for example,
Universe = {a,b,c}
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To learn how to interpret simple predicate logic formulas as being true or false.
This helps in proving invalidity by the technique of displaying a counter-example.
Hausman[2007] Logic and Philosophy Chapter 8
In sentential logic, we just took it that each of the atomic sentences either is true or is false-- we did not look into the structure of the sentences.
Truth can be discussed in more detail in predicate logic.
A start can be made in predicate logic by taking apart 'atomic' propositions and by re-phrasing what they have to say in a 'entity-has-property' way.
The constant terms a,b,c...h are used to denote entities, the predicates A,B,C...Z are used to denote properties that these entities have, and these are put together by writing the predicate first followed by the term, for example Gb.
Hausman[2007] Logic and Philosophy Chapter 7
In predicate logic, many different styles of expression in English get cast into the same 'property-is-had-by-entity' form. For example,
Beryl is wise. (ie adjective)
Beryl runs. (ie verb)
Beryl is a philosopher. (ie noun)
might be symbolized Wb, Rb, and Pb, respectively.
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To start learning how to symbolize sentences using predicate logic.
Hausman[2007] Logic and Philosophy Chapter 7
There are many valid arguments which cannot be shown to be valid using sentential logic alone. For example,
Beryl is a philosopher.
All philosophers are wise.
Therefore
Beryl is wise.
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You may have derivations of your own that you wish to try. Just type, paste, or drag and drop, them into the panel, select your derivation, and click 'Start from selection'.
You will need to use the correct logical symbols. Here they are
F ∴ F . G ∼ ∨ ⊃ ≡ ∃ ∴
And the right syntax (the premises separated by commas and then a 'therefore' followed by the conclusion).
So, a selection needs to have a form similar to this
F⊃G,G⊃H∴F⊃H
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You now have to tools to appraise sentential arguments.
Let us run through how these might be used with two examples.
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
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Learning conditional proof, and indirect proof.
There are two additional sentential rule of inference-- Conditional Proof and Indirect Proof (Reductio ad absurdum)-- which are slightly more difficult than the ones that we have met before. They are slightly more difficult in that they require you to make new assumptions, and the correct new assumptions at that.
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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.
Hausman[2007] Logic and Philosophy Chapter 4