# When is a connective truth functional?

I got this question from Logic, laws of truth, by Nicholas J.J Smith.

He says (page 24) :

"A connective is truth functional if it has the property that the truth or falsity of a compound proposition formed from the connective and some other propositions is completely determined by the truth or falsity of those component propositions."

I don't really seem to be able to appreciate the usefulness of truth-functional connectives.

Perhaps, I don't understand what he is saying in that paragraph, so I would appreciate any explanation of what he is trying to say and why truth-functional connectives are useful.

Also (if you want to) can you guys explain what Nicholas means when he says "...this proposition has no internal structure..."?

• When its truth value only depends on truth values of its components, and not their meaning. For example, natural disjunction is not truth functional: "it will rain tomorrow or it will not rain tomorrow" holds today even though neither "it will rain tomorrow" nor "it will not rain tomorrow" have definitive truth values today. Commented Apr 3, 2019 at 19:46

When is a connective truth functional?

Short answer : when it is defined by a truth table.

Classical propositional logic is a truth-functional logic in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a truth function. On the contrary, modal logic is non-truth-functional.

See an example in Truth Functionality and non-Truth Functional Connectives, comparing :

Agnes will attend law school and so will Bob,

where the truth-value of the compound sentence depends only on the truth-value of the two atomic sentences, with :

Agnes will attend law school and then she will make millions,

where the "and then" connective express a time-dependency between the two atomic sentences.

For different examples, see 6.3.1 Indicative and Counterfactual Conditionals (page 110) of Smith's book.

An example (motivated by your previous question) dealing with the concept of "internal structure" of a statement will be the following.

The statement

"Jim is a bachelor and Jim (the same Jim) is married"

is not a contradiction in propositional logic, because the sentence has the logical form B ∧ M, and this formula is not a contradiction.

In order to discover the contradicition, we need a deeper level of analysis that consider also the semantics of the expressions "is a bachelor" and "is married", in addition to the logical connective "and".

This level of analysis will be available with predicate logic where we can analyze the atomic sentences with a subject-predicate logical form :

Bachelor(Jim) and Married(Jim).

In this case, using the axiom :

Bachelor(x) iff not Married(x),

we may derive the contradiction not expressible in propositional logic.

Nicholas Smith defines the internal structure of arguments as propositions (page 23-4). He then breaks propositions, the internal structure of arguments, into two kinds.

1. Basic propositions which have no parts that are themselves propositions.
2. Compound propositions which are composed of other propositions and connectives between them.

Propositional logic studies the internal structure of compound propositions, but it does not concern itself with the internal structure of basic propositions, that is, it is not interested in the internal structure of basic propositions.

Predicate logic looks at the internal structure of basic propositions.

Here are the questions:

I don't really seem to be able to appreciate the usefulness of truth-functional connectives.

Truth-functional connectives allow one to study compound propositions in propositional logic. These connectives are part of the internal structure that breaks the compound proposition into component propositions and connectives. This is why they are useful.

Perhaps, I don't understand what he is saying in that paragraph, so I would appreciate any explanation of what he is trying to say and why truth-functional connectives are useful.

The truth or falsity of the compound proposition can be determined by examining the truth or falsity of its component propositions and by studying how they are related by the connectives joining those component propositions.

Instead of trying to determine the truth or falsity of a compound proposition, which might be complicated, there is a way to break that compound proposition into simpler component propositions by looking at how the connectives join them together into the compound proposition. That is what makes truth-functional connectives useful. They simplify the problem of determining the truth value of compound propositions.

Also (if you want to) can you guys explain what Nicholas means when he says "...this proposition has no internal structure..."?

Smith discussed three levels of internal structure.

1. An argument has an internal structure made up of propositions.
2. A compound proposition has an internal structure made up of other propositions and connectives studied in propositional logic.
3. A basic proposition has an internal structure as well which is studied in predicate logic.

From the perspective of propositional logic the basic propositions can be viewed as having no internal structure that propositional logic studies.

Smith, N. J. (2012). Logic: The laws of truth. Princeton University Press.

I think both of the other answers are correct, but they answer from the side of people who already understand the system of sentential logic. In the process, they might be skipping over some things that are non-obvious until the system clicks.

It might help to think of the system as a game with the following rules:

1. Everything must be expressed in propositions which are TRUE or FALSE
2. Propositions relate to each other in truth-connective ways that also evaluate to TRUE and FALSE
3. These connectors that are truth-evaluators are truth-functional connectives: if we use "if , then", it takes exactly two terms. If we use "&" (and), it takes exactly two terms; if we use ~(not), it takes exactly one term. And so on.

What makes Smith's account hard to understand and many of the details above hard to grasp is that: This game as a whole is a replacement for natural language. Put another way, don't think about the English words "and", "or", "if", or "not." Don't think about deep meanings for TRUE and FALSE. Instead, think about a system where if we are very precise with our language we can very easily check whether things line up and give the outcome we claim.

In natural language, "Do you have chips and a drink for me?" could mean:

• do you have chips for me and a drink for me
• do you have chips (for anyone) and a drink for me
• are you willing to give me the chips and drink you have?

The constructed language of logic does not have any of these ambiguities. Any time you encounter the word "and" in it, it has a meaning precisely specified by a truth table ( A AND B is TRUE if and only if A is TRUE and B is TRUE; otherwise, it is FALSE).

"truth-functional connectives" is the name for the words in this language that connect propositions in 100% accurate ways vis-a-vis their truth.

real world connectives in spoken English (and other languages) do nothing of that sort. So the value of the term is that it points out the parts that work in our logic game and their difference with connectives in natural language.