Sentence Semantics I

Propositional Logic

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Sometimes just using a single word to communicate can be a good idea. However, that’s not how we usually do it. How does Semantics deal with this rather obvious fact about meaning?

It is important to think about the kind of meanings and concepts connected with words. We have seen that there is an incredible amount of complex information connected with words.

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Even the simplest imaginable sentence connects with an incredible amount of complex information. 

However, we do not communicate just by using single words. We speak and write in sentences. That is even more complicated.

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If you wanted to, you could go on forever mapping the conceptual connections for a single word! Don’t try it!

Just thinking about all the complicated information flying around in one sentence could drive you crazy! How does Semantics deal with that? Well, first of all, by simplifying everything.

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Propositional Logic treats sentences (or propositions) as atoms and plays around with them. We give examples to help make sense of it, but Propositions are basically simple atoms (much simpler than real atoms!) with labels like p, q, or r.

In Propositional Logic, we treat sentences as propositions.

atom

In Propositional Logic we treat propositions as atoms. Actually, we treat them as simpler than atoms because we don’t think about their internal structure at all. 

In Propositional Logic, truth values are assigned to propositions (sentences). Truth values are either True(T) or False (F). Look at the two following sentences:

(1) Ken hit Jim.

(2) Jim was hit by Ken.

Clearly, these are two different sentences. However, Propositional Logic claims that these different sentences may be converted into identical propositions.

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Active and Passive sentences are treated as equivalent and, therefore, identical propositions. “Rie punched Jun” and “Jun was punched by Rie” amounts to the same thing. It is treated as the same atomic proposition p. It is either true or false. Are you happy with that?

Propositional logic labels propositions, usually starting with the small letter p. If you have two propositions, they are labelled p and q, and so on.

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Say we have a proposition such as “Ben is studying.” We call it p (or q or r etc) and basically forget about its structure and content.

Propositions have truth values. So we can ask, under what conditions is a proposition p true (T) or false (F)? Say we have a proposition p such as the following:

p = Ben is studying.

Then we can say that p = T when Ben is studying and p = F when he is not.

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Try not to think too deeply about meaning for the moment. Obviously, there are problems with saying things are either true or false. Just try to think of propositions as atoms that are either T  (for true) or F (for false).

Of course, there are always problems related to truth values. Can you really be sure he is studying? What if the person who says he is studying is not reliable? However, we imagine a simple world in which we can say straightforwardly whether or not propositions like this are true or false.

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We are trying to imagine very simple worlds where everything is either just true or false. “The Prince is standing” is True, right? 

So, say we add another proposition q to p:

p = Ben is studying.

q = Ben is sleeping.

We can say that p = F if q = T. Do you agree? If Ben is sleeping, he is not studying, right?

Student-sleeping

Try to think of everything as being either true or false. You’re not studying if you’re sleeping, right?

Say we add another proposition r to p and q:

p = Ben is studying.

q = Ben is sleeping.

r = Ben is using his mobile phone.

What happens to p and q if r = T? We can say that p is only true if q is false. Can we say that p is only true if r is false? Can we say that q is only true if p and r are both false? Think about it, but try not to get a headache!

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How about studying and using a mobile phone at the same time. No problem?

Now simple propositions like this can be turned into complex propositions by using Logical Constants or connectives.

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The logical constants or connectives. There is variation in the way they are written, as you will see.

There are a number of ways to write these, but you can get the basic idea from the following:

Conjunction   ∧ (or &) means “and”

Disjunction   ∨ means “or”

Implication   → means ” if … then”

Negation   ¬ means “not”

Equivalence   ≡ means “if and only if (iff)”

Say we have a proposition such as (3) below:

(3) p ∧ q

When is (3) true? Propositional Logic makes use of so-called truth tables to determine whether a proposition is true or not.

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The truth table for the connective ∧. If either of the conjuncts is not true, the whole proposition is not true. Pretty easy?

Here is a truth table for the disjunction:

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We say that p or q is true if either (or both) of the disjuncts is true. This may seem a bit strange because that’s not how we usually use “or” when we speak. Don’t worry about it for the moment.

Here’s a truth table for implication:

truth_table_impl

Say p is “Eri hates studying logic” and q is “Eri has studied logic.” If Eri hates studying logic, we can probably say she has studied logic. So p → q is true if both p and q are also true. It is also treated as true if both p and q are false! And it is true if p is false but q is true. Does that sound OK to you? Now you hate logic, so you don’t care any more, right?

Here’s a truth table for negation which can be written as ~ or  – as well as ¬ so try not to get confused!:

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When is ¬ p true? Well, when p is false. You knew that, right?

Here’s a truth table for equivalence, which can be written as ≡ or ⇔.

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Say you have two propositions, p = “Jun is a man” and q = “Jun is an adult male human.” These are equivalent propositions. Whenever one is true, so is the other. Whenever one is false, so is the other. If one has a different truth value from the other, they are not equivalent statements.

 

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