Pathos, with a dash of Logos and Ethos

Pathos

It might not seem like Coolidge is aiming for his audience's emotional nerve centers in this speech. After all, he's not using a lot of flowery language, metaphors, or talking about dreams or The Fault in Our Stars. That's just not our boy Cal. But he does want to stir up certain feelings in his audience—primarily, their patriotism and their desire for economic prosperity.

For example, right at the beginning of the speech he says: "Realizing that we cannot live unto ourselves alone, we have contributed of our resources and our counsel to the relief of the suffering and the settlement of the disputes among the European nations. Because of what America is and what America has done, a firmer courage, a higher hope, inspires the heart of all humanity" (1.6-7).

What he's doing is justifying American economic and military assistance, but how does he do it? He doesn't just reference globalization or the world economy, he says "we cannot live unto ourselves alone." That's a much more emotionally powerful statement. Plus he frames the assistance as "relief of the suffering," which makes America a paragon of humanitarian virtue. If Americans had to go against their isolationist impulses to get involved in World War I, they needed to feel emotionally invested in the outcome.

He continues on by giving a sort of mini history lesson, as another way of drumming up the audience's emotional connection to the idea of America. He begins by saying that, "We cannot continue these brilliant successes in the future, unless we continue to learn from the past" (2.3). As he goes through specific events in American history, he always presents them as examples of America's greatness.

For instance, he describes westward expansion (which some might see as an aggressive, imperialist, violent process) as: "We made freedom a birthright" (3.7). He says America entered World War I, which wasn't necessarily a popular decision in 1917, "in the defense of our own ideals and in the general cause of liberty" (3.8). And of course, "Throughout all these experiences we have enlarged our freedom, we have strengthened our independence" (4.1). It's all about pride in being American and the ideals that have always been associated with that identity.

On the economy, Coolidge uses recent voting trends as evidence that people were looking for a certain kind of economic policy. Specifically, his kind of economic policy: "No matter what others may want, these people want a drastic economy. They are opposed to waste. They know that extravagance lengthens the hours and diminishes the rewards of their labor. I favor the policy of economy, not because I wish to save money, but because I wish to save people" (15.7-10).

Notice that he's not talking about himself, he's putting the emotions and desires in the people instead. He's reminding them of how they feel. Then he turns it around and shows how strongly he agrees with them.

Logos

Although Coolidge does frequently rely on the emotional pull of his rhetoric to appeal to the audience, he doesn't just leave logic and reason by the wayside. He does at least broadly back up his claims most of the time.

For example, before doling out the emotional punch at the start of the speech, he reminds the audience that "Already we have sufficiently rearranged our domestic affairs so that confidence has returned, business has revived, and we appear to be entering an era of prosperity which is gradually reaching into every part of the nation" (1.5). Well, that's something tangible and provable, which his audience would see as evidence of the success of government policy.

He defends the U.S.'s economic assistance to Europe by saying: "One of the greatest dangers to peace lies in the economic pressure to which people find themselves subjected. One of the most practical things to be done in the world is to seek arrangements under which such pressure may be removed, so that opportunity may be renewed and hope may be revived" (11.5-6). That economic system is helping keep the peace by removing a serious source of tension.

Similarly, he justifies his domestic economy policy with some logical arguments as well: "We do not any longer need war-time revenues […] The property of the country belongs to the people of the country. […] They do not support any privileged class; they do not need to maintain great military forces; they ought not to be burdened with a great array of public employees" (16.4, 8, 10).

Ethos

Most inaugural addresses don't have to waste time convincing the audience of the credibility of the speaker; after all, it's the president. He's got the presidential seal, he's up on the presidential podium, and his mama loves him, she loves him, she…

Sorry, got carried away.

Anyway, Coolidge was too modest a guy to tout his own character or credibility, but he does make lots of appeals to what he thinks is the ultimate governmental authority: the "matchless wisdom" of the Constitution. It's the source document of all our policies and governance structures, he explains, and everyone needs to get with the program and do what it says. He'd just taken an oath to protect and defend it, and he wants us all to do the same.